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BEGIN:VEVENT
SUMMARY:Hong Wang (IAS)
DTSTART:20200421T220000Z
DTEND:20200421T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/1
DESCRIPTION:Title: Distinct distances for well-separated sets\nby Hong
Wang (IAS) as part of UCLA analysis and PDE seminar\n\nLecture held in ht
tps://ucla.zoom.us/j/9264073849.\n\nAbstract\nGiven a set E of dimension s
>1\, Falconer conjectured that its distance set \\Delta(E)=\\{|x-y|: x\, y
\\in E\\} should have positive Lebesgue measure. Orponen\, Shmerkin and Ke
leti-Shmerkin proved the conjecture for tightly spaced sets\, for example\
, AD-regular sets.\n\nIn this talk\, we are going to discuss the opposite
type: well-separated sets. This is joint work with Larry Guth and Noam Sol
omon.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ioannis Angelopoulos (Caltech)
DTSTART:20200421T230000Z
DTEND:20200422T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/2
DESCRIPTION:Title: Semi-global constructions of vacuum spacetimes\nby
Ioannis Angelopoulos (Caltech) as part of UCLA analysis and PDE seminar\n\
nLecture held in https://ucla.zoom.us/j/9264073849.\n\nAbstract\nI will de
scribe some techniques for constructing semi-global solutions to the chara
cteristic initial value problem for the vacuum Einstein equations with dif
ferent types of data\, and will also mention some applications as well as
some open problems in the area.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joni Teravainen (Oxford)
DTSTART:20200428T170000Z
DTEND:20200428T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/3
DESCRIPTION:Title: Higher order uniformity of the Möbius function\nby
Joni Teravainen (Oxford) as part of UCLA analysis and PDE seminar\n\nLect
ure held in https://ucla.zoom.us/j/9264073849.\n\nAbstract\nRecently\, Mat
omäki\, Radziwiłł and Tao showed that the Möbius function is discorrel
ated with linear exponential phases on almost all short intervals. I will
discuss joint work where we generalize this result to a much wider class o
f phase functions\, showing that the Möbius function does not correlate w
ith polynomial phases or more generally with nilsequences. I will also dis
cuss applications to superpolynomial word complexity for the Liouville seq
uence and to counting polynomial patterns weighted by the Möbius function
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Beltran (U. Madison Wisconsin)
DTSTART:20200505T220000Z
DTEND:20200505T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/4
DESCRIPTION:Title: Regularity of the centered fractional maximal function<
/a>\nby David Beltran (U. Madison Wisconsin) as part of UCLA analysis and
PDE seminar\n\nLecture held in https://caltech.zoom.us/j/747242458.\n\nAbs
tract\nI will report some recent progress regarding the boundedness of the
map $f \\mapsto |\\nabla M_\\beta f|$ from the endpoint space $W^{1\,1}(\
\mathbb{R}^d)$ to $L^{d/(d-\\beta)}(\\mathbb{R}^d)$\, where $M_\\beta$ den
otes the fractional version of the centered Hardy--Littlewood maximal func
tion. A key step in our analysis is a pointwise relation between the cente
red and non-centered fractional maximal functions at the derivative level\
, which allows to exploit the known techniques in the non-centered case.\n
\nThis is joint work with José Madrid.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luca Spolaor (UCSD)
DTSTART:20200505T230000Z
DTEND:20200506T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/5
DESCRIPTION:Title: Regularity of the free boundary for the two-phase Berno
ulli problem\nby Luca Spolaor (UCSD) as part of UCLA analysis and PDE
seminar\n\nLecture held in https://caltech.zoom.us/j/747242458.\n\nAbstrac
t\nI will describe a recent result obtained in collaboration with G. De Ph
ilippis and B. Velichkov concerning the regularity of the free boundaries
in the two phase Bernoulli problems. The novelty of our work is the analys
is of the free boundary at branch points\, where we show that it is given
by the union of two C1 graphs. This completes the work started by Alt\, Ca
ffarelli\, and Friedman in the 80’s.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Khavinson (U. South Florida)
DTSTART:20200519T230000Z
DTEND:20200520T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/6
DESCRIPTION:Title: Classical Potential Theory from the High Ground of Line
ar Holomorphic PDE\nby Dmitry Khavinson (U. South Florida) as part of
UCLA analysis and PDE seminar\n\nLecture held in https://ucla.zoom.us/j/92
64073849.\n\nAbstract\n"Between two truths of the real domain\, the easies
t and shortest path quite often passes through the complex domain."\n\n
P. Painleve\, 1900.\n\n\nAbstract:
\n\nNewton noticed that the gravitational potential of a spherical mass wi
th constant density equals\, outside the ball\, the potential of the poin
t-mass at the center. Rephrasing\, the gravitational potential of the bal
l with constant mass density continues as a harmonic function inside the b
all except for the center. Fairly recently\, it was noted that the latter
statement holds for any polynomial\, or even for entire densities.\n\nIf a
harmonic in a spherical shell function vanishes on one piece of a line th
rough the center piercing the shell\, then it must vanish on the second pi
ece of that line. Yet\, the similar statement fails for tori.\n\nIf we sol
ve the Dirichlet problem in an ellipse with entire data\, the solution wil
l always be an entire harmonic function. Yet\, if we do that in a domain b
ounded by the curve x^4 + y^4 =1\, with the data as simple as x^2+y^2\, th
e solution will have infinitely many singularities outside the curve. \nWh
ere and why do eigenfunctions of the Laplacian in domains bounded by algeb
raic curves start having singularities?\n\nWe shall discuss these and some
other questions under the unified umbrella of analytic continuation of s
olutions to analytic pde in C^n.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kirsti Biggs (Chalmers U. Technology)
DTSTART:20200526T170000Z
DTEND:20200526T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/7
DESCRIPTION:Title: Ellipsephic efficient congruencing for the moment curve
\nby Kirsti Biggs (Chalmers U. Technology) as part of UCLA analysis an
d PDE seminar\n\nLecture held in https://ucla.zoom.us/j/9264073849.\n\nAbs
tract\nAn ellipsephic set is a subset of the natural numbers whose element
s have digital restrictions in some fixed prime base. Such sets have a fra
ctal structure and can be viewed as p-adic Cantor sets. The particular ell
ipsephic sets that interest us have certain additive properties - for exam
ple\, the set of integers whose digits are squares forms a key motivating
example\, because there are few representations of an integer as the sum o
f two squares.\n\n\nWe obtain discrete restriction estimates for the momen
t curve over ellipsephic sets—in number theoretic terms\, we bound the n
umber of ellipsephic solutions to a Vinogradov system of equations—using
Wooley’s nested efficient congruencing method. These results generalise
previous work of the speaker\, on the quadratic case of this problem\, to
the moment curve of arbitrary degree.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihailis Kolountzakis (U. Crete)
DTSTART:20200602T160000Z
DTEND:20200602T165000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/8
DESCRIPTION:Title: Orthogonal Fourier analysis on domains: methods\, resul
ts and open problems\nby Mihailis Kolountzakis (U. Crete) as part of U
CLA analysis and PDE seminar\n\nLecture held in https://caltech.zoom.us/j/
747242458.\n\nAbstract\nWe all know how to do Fourier Analysis on an inter
val\, on {\\mathbb R}^d\, or other groups. But what if our functions live
on a subset of Euclidean space\, let's say on a regular hexagon in the pla
ne? Can we use our beloved exponentials\, functions of the form e_\\lambda
(x) = \\exp(2\\pi i \\lambda\\cdot x) to analyze the functions defined on
our domain? In other words\, can we select a set of frequencies \\lambda s
uch that the corresponding exponentials form an orthogonal basis for L^2 o
f our domain? It turns out that the existence of such an orthogonal basis
depends heavily on the domain. So the answer is yes\, we can find an ortho
gonal basis of exponentials for the hexagon\, but if we ask the same quest
ion for a disk\, the answer turns out to be no.\n\nFuglede conjectured in
the 1970s that the existence of such an exponential basis is equivalent to
the domain being able to tile space by translations (the hexagon\, that w
e mentioned\, indeed can tile\, while the disk cannot). In this talk we wi
ll track this conjecture and the mathematics created by the attempts to se
ttle it and its variants. We will see some of its rich connections to geom
etry\, number theory and harmonic analysis and some of the spectacular rec
ent successes in our efforts to understand exponential bases. We will emph
asize several problems that are still open.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yakov Shlapentokh-Rothman (Princeton)
DTSTART:20200602T170000Z
DTEND:20200602T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/9
DESCRIPTION:Title: Naked Singularities for the Einstein Vacuum Equations:
The Exterior Solution\nby Yakov Shlapentokh-Rothman (Princeton) as par
t of UCLA analysis and PDE seminar\n\nLecture held in https://caltech.zoom
.us/j/747242458.\nAbstract: TBA\n\nWe will start by recalling the weak cos
mic censorship conjecture. Then we will review Christodoulou's constructio
n of naked singularities for the spherically symmetric Einstein-scalar fie
ld system. Finally\, we will discuss joint work with Igor Rodnianski which
constructs spacetimes corresponding to the exterior region of a naked sin
gularity for the Einstein vacuum equations.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Hughes (U. Bristol)
DTSTART:20200519T220000Z
DTEND:20200519T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/10
DESCRIPTION:Title: Discrete restriction estimates\nby Kevin Hughes (U
. Bristol) as part of UCLA analysis and PDE seminar\n\nLecture held in htt
ps://ucla.zoom.us/j/9264073849.\n\nAbstract\nWe will discuss Wooley's Effi
cient Congruencing approach to discrete restriction estimates for translat
ion-dilation invariant systems of equations. Then we will discuss recent e
stimates for the curve (X\,X^3) which lie just outside of this framework a
s well as that of Decoupling.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (U. Washington)
DTSTART:20201006T220000Z
DTEND:20201006T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/11
DESCRIPTION:Title: Roots of polynomials under repeated differentiation: a
nonlocal evolution equation with infinitely many conservation laws (and s
ome universality phenomena)\nby Stefan Steinerberger (U. Washington) a
s part of UCLA analysis and PDE seminar\n\n\nAbstract\nSuppose you have a
polynomial of degree $p_n$ whose $n$ real roots are roughly distributed li
ke a Gaussian (or some other nice distribution) and you differentiate $t\\
cdot n$ times where $0< t<1$. What's the distribution of the $(1-t)n$ root
s of that $(t\\cdot n)$-th derivative? How does it depend on $t$? We iden
tify a relatively simple nonlocal evolution equation (the nonlocality is g
iven by a Hilbert transform)\; it has two nice closed-form solutions\, a s
hrinking semicircle and a family of Marchenko-Pastur distributions (this s
ounds like random matrix theory and we make some remarks in that direction
). Moreover\, the underlying evolution satisfies an infinite number of con
servation laws that one can write down explicitly. This suggests a lot of
questions: Sean O'Rourke and I proposed an analogous equation for complex-
valued polynomials. Motivated by some numerical simulations\, Jeremy Hosk
ins and I conjectured that $t=1$\, just before the polynomial disappears\,
the shape of the remaining roots is a semicircle and we prove that for a
class of random polynomials. I promise lots of open problems and pretty p
ictures.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjoern Bringmann (UCLA)
DTSTART:20201006T230000Z
DTEND:20201007T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/12
DESCRIPTION:Title: Invariant Gibbs measures for the three-dimensional wav
e equation with a Hartree nonlinearity\nby Bjoern Bringmann (UCLA) as
part of UCLA analysis and PDE seminar\n\n\nAbstract\nIn this talk\, we dis
cuss the construction and invariance of the Gibbs measure for a three-\ndi
mensional wave equation with a Hartree-nonlinearity.\n\nIn the first part
of the talk\, we construct the Gibbs measure and examine its properties. W
e discuss the mutual singularity of the Gibbs measure and the so-called Ga
ussian free field. In contrast\, the Gibbs measure for one or two-dimensio
nal wave equations is absolutely continuous with respect to the Gaussian f
ree field.\n\nIn the second part of the talk\, we discuss the probabilisti
c well-posedness of the corresponding nonlinear wave equation\, which is n
eeded in the proof of invariance. At the moment\, this is the only theorem
proving the invariance of any singular Gibbs measure under a dispersive e
quation.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Khang Huynh (UCLA)
DTSTART:20201020T220000Z
DTEND:20201020T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/13
DESCRIPTION:Title: A geometric trapping approach to global regularity for
2D Navier-Stokes on manifolds\nby Khang Huynh (UCLA) as part of UCLA
analysis and PDE seminar\n\n\nAbstract\nWe use frequency decomposition tec
hniques to give a direct proof of global existence and regularity for the
Navier-Stokes equations on two-dimensional Riemannian manifolds without bo
undary. Our techniques are inspired by an approach of Mattingly and Sinai
which was developed in the context of periodic boundary conditions on a fl
at background\, and which is based on a maximum principle for Fourier coef
ficients. The extension to general manifolds requires several new ideas\,
connected to the less favorable spectral localization properties in our se
tting. Our arguments make use of frequency projection operators\, multilin
ear estimates that originated in the study of the non-linear Schrodinger e
quation\, and ideas from microlocal analysis.\n\nThis is joint work with A
ynur Bulut.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaemin Park (Georgia Tech)
DTSTART:20201013T210000Z
DTEND:20201013T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/14
DESCRIPTION:Title: Radial symmetry in stationary/uniformly-rotating solut
ions to 2D Euler equation\nby Jaemin Park (Georgia Tech) as part of UC
LA analysis and PDE seminar\n\n\nAbstract\nIn this talk\, I will discuss w
hether all stationary/uniformly-rotating solutions of 2D Euler equation mu
st be radially symmetric\, if the vorticity is compactly supported. For a
stationary solution that is either smooth or of patch type\, we prove that
if the vorticity does not change sign\, it must be radially symmetric up
to a translation. It turns out that the fixed-sign condition is necessary
for radial symmetry result: indeed\, we are able to find non-radial sign c
hanging stationary solution with compact support. We have also obtained so
me sharp criteria on symmetry for uniformly-rotating solutions for 2D Eule
r equation and the SQG equation. The symmetry results are mainly obtained
by calculus of variations and elliptic equation techniques\, and the const
ruction of non-radial solution is obtained from bifurcation theory. Part o
f this talk is based on joint work with Javier Gomez-Serrano\, Jia Shi and
Yao Yao\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Bloom (Cambridge)
DTSTART:20201103T180000Z
DTEND:20201103T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/15
DESCRIPTION:Title: Spectral structure and arithmetic progressions\nby
Thomas Bloom (Cambridge) as part of UCLA analysis and PDE seminar\n\n\nAb
stract\nHow much additive structure can we guarantee in sets of integers\,
knowing only their density? The study of which density thresholds are suf
ficient to guarantee the existence of various kinds of additive structures
is an old and fascinating subject with connections to analytic number the
ory\, additive combinatorics\, and harmonic analysis.\n\nIn this talk we w
ill discuss recent progress on perhaps the most well-known of these thresh
olds: how large do we need a set of integers to be to guarantee the existe
nce of a three-term arithmetic progression? In recent joint work with Olof
Sisask we broke through the logarithmic density barrier for this problem\
, establishing in particular that if a set is dense enough such that the s
um of reciprocals diverges\, then it must contain a three-term arithmetic
progression\, establishing the first case of an infamous conjecture of Erd
os.\n\nWe will give an introduction to this problem and sketch some of the
recent ideas that have made this progress possible. We will pay particula
r attention to the ways we exploit 'spectral structure' - understanding co
mbinatorially sets of large Fourier coefficients\, which we hope will have
further applications in number theory and harmonic analysis.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yao Yao (Georgia Tech)
DTSTART:20201118T000000Z
DTEND:20201118T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/16
DESCRIPTION:Title: Two results on the interaction energy\nby Yao Yao
(Georgia Tech) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nFor
any nonnegative density $f$ and radially decreasing interaction potential
$W$\, the celebrated Riesz rearrangement inequality shows the interaction
energy $E[f] = \\int f(x)f(y)W(x-y) dxdy$ satisfies $E[f] \\leq E[f^*]$\,
where $f^*$ is the radially decreasing rearrangement of $f$. It is a natu
ral question to look for a quantitative version of this inequality: if its
two sides almost agree\, how close must $f$ be to a translation of $f^*$?
Previously the stability estimate was only known for characteristic funct
ions. I will discuss a recent work with Xukai Yan\, where we found a simpl
e proof of stability estimates for general densities. \n\nI will also disc
uss another work with Matias Delgadino and Xukai Yan\, where we constructe
d an interpolation curve between any two radially decreasing densities wit
h the same mass\, and show that the interaction energy is convex along thi
s interpolation. As an application\, this leads to uniqueness of steady st
ates in aggregation-diffusion equations with any attractive interaction po
tential for diffusion power $m\\geq 2$\, where the threshold is sharp.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Speck (Vanderbilt)
DTSTART:20201020T230000Z
DTEND:20201021T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/17
DESCRIPTION:Title: Stable big bang formation in general relativity: the c
omplete sub-critical regime\nby Jared Speck (Vanderbilt) as part of UC
LA analysis and PDE seminar\n\n\nAbstract\nThe celebrated theorems of Hawk
ing and Penrose show that under appropriate assumptions on the matter mode
l\, a large\, open set of initial data for Einstein's equations lead to ge
odesically incomplete solutions. However\, these theorems are "soft" in th
at they do not yield any information\nabout the nature of the incompletene
ss\, leaving open the possibilities that \n\ni) it is tied to the blowup o
f some invariant quantity (such as curvature) or \n\nii) it is due to a mo
re sinister phenomenon\, such as\nincompleteness due to lack of informatio
n for how to uniquely continue the solution (this is roughly\nknown as the
formation of a Cauchy horizon). \n\nDespite the "general ambiguity" in th
e mathematical physics literature\, there are heuristic results\, going ba
ck 50 years\, suggesting that whenever a certain "sub-criticality" conditi
on holds\, the Hawking-Penrose incompleteness is caused by the formation o
f a Big Bang singularity\, that is\, curvature blowup along an entire spac
elike hypersurface. In\nrecent joint work with I. Rodnianski and G. Fourno
davlos\, we have given a rigorous proof of the heuristics. More precisely\
, our results apply to Sobolev-class perturbations - without symmetry - of
generalized Kasner solutions whose exponents satisfy the sub-criticality
condition. Our main\ntheorem shows that - like the generalized Kasner solu
tions - the perturbed solutions develop Big Bang singularities. \n\nIn thi
s talk\, I will provide an overview of our result and explain how it is ti
ed to some of the main themes of investigation by the mathematical general
relativity community\, including the remarkable work of Dafermos-Luk on t
he stability of Kerr Cauchy horizons. I will also discuss the new gauge th
at we used in our work\, as well as intriguing connections to other proble
ms concerning stable singularity formation.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleksandr Logunov (Princeton)
DTSTART:20201215T190000Z
DTEND:20201215T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/18
DESCRIPTION:Title: Zero sets of Laplace eigenfunctions\nby Aleksandr
Logunov (Princeton) as part of UCLA analysis and PDE seminar\n\n\nAbstract
\nIn the beginning of 19th century Napoleon set a prize for the best mathe
matical explanation of Chladni’s resonance experiments. Nodal geometry s
tudies the zeroes of solutions to elliptic differential equations such as
the visible curves that appear in these physical experiments. We will disc
uss geometrical and analytic properties of zero sets of harmonic functions
and eigenfunctions of the Laplace operator. For harmonic functions on the
plane there is an interesting relation between local length of the zero s
et and the growth of harmonic functions. The larger the zero set is\, the
faster the growth of harmonic function should be and vice versa. Zero sets
of Laplace eigenfunctions on surfaces are unions of smooth curves with eq
uiangular intersections. Topology of the zero set could be quite complicat
ed\, but Yau conjectured that the total length of the zero set is comparab
le to the square root of the eigenvalue for all eigenfunctions. We will st
art with open questions about spherical harmonics and explain some methods
to study nodal sets.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristian Gonzales-Riquelme (IMPA)
DTSTART:20201117T230000Z
DTEND:20201118T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/19
DESCRIPTION:Title: BV and Sobolev continuity for maximal operators\nb
y Cristian Gonzales-Riquelme (IMPA) as part of UCLA analysis and PDE semin
ar\n\n\nAbstract\nThe regularity of maximal operators has been a topic of\
ninterest in harmonic analysis over the past decades. In this topic we are
interested in what can be said about the variation of a maximal function
Mf given some information about the original function f. In this talk we p
resent\nsome recent results about the continuity of the map $f \\mapsto \\
nabla Mf$ for the uncentered Hardy-Littlewood maximal operator in both the
$BV({\\mathbb R})$ and the $W^{1\,1}_{rad}({\\mathbb R}^d)$ settings.\n\n
This is based on joint works with D. Kosz (BV case) and E. Carneiro and J.
Madrid (radial case).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paata Ivanisvili (NC State)
DTSTART:20201201T230000Z
DTEND:20201202T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/20
DESCRIPTION:Title: Sharpening the triangle inequality in Lp spaces\nb
y Paata Ivanisvili (NC State) as part of UCLA analysis and PDE seminar\n\n
\nAbstract\nThe classical triangle inequality in Lp estimates the norm of
the sum of two functions in terms of the sums of the norms of these functi
ons. Perhaps one drawback of this estimate is that it does not see how "or
thogonal" these functions are. For example\, if f and g are not identicall
y zero and they have disjoint supports then the triangle inequality is pre
tty strict (say for p>1).\n\nMotivated by the L2 case\, where one has a tr
ivial inequality ||f+g||^2 \\leq ||f||^2 + ||g||^2 + 2 |fg|_1\, one can th
ink about the quantity |fg|_1 as measuring the "overlap" between f and g.
What is the correct analog of this estimate in Lp for p different than 2?\
n\nMy talk will be based on a joint work with Carlen\, Frank and Lieb wher
e we obtain one extension of this estimate in Lp\, thereby proving and imp
roving the suggested possible estimates by Carbery\, and another work with
Mooney where we further refine these estimates. The estimates will be pro
vided for all real p's.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuel Carneiro (ICTP)
DTSTART:20201215T180000Z
DTEND:20201215T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/21
DESCRIPTION:Title: Uncertain signs\nby Emanuel Carneiro (ICTP) as par
t of UCLA analysis and PDE seminar\n\n\nAbstract\nWe consider a generalize
d version of the sign uncertainty\nprinciple for the Fourier transform\, f
irst proposed by Bourgain\, Clozel and\nKahane in 2010 and revisited by Co
hn and Gonçalves in 2019\, in connection\nto the sphere packing problem.
In our setup\, the signs of a function and\nits Fourier transform resonate
with a generic given function P outside of\na ball. One essentially wants
to know if and how soon this resonance can\nhappen\, when facing a suitab
le competing weighted integral condition. The\noriginal version of the pro
blem corresponds to the case P=1.\nSurprisingly\, even in such a rough set
up\, we are able to identify sharp\nconstants in some cases. This is a joi
nt work with Oscar Quesada-Herrera\n(IMPA - Rio de Janeiro).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Damanik (Rice)
DTSTART:20201202T000000Z
DTEND:20201202T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/22
DESCRIPTION:Title: Proving Positive Lyapunov Exponents: Beyond Independen
ce\nby David Damanik (Rice) as part of UCLA analysis and PDE seminar\n
\n\nAbstract\nWe discuss the problem of proving the positivity of the Lyap
unov exponent for Schr\\"odinger operators with potentials defined by a hy
perbolic base transformation and a H \\"older continuous sampling function
. Prominent examples of such base transformations are given by the doublin
g map and the Arnold cat map. The talk is based on joint work with Artur A
vila and Zhenghe Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shukun Wu (UIUC)
DTSTART:20201027T210000Z
DTEND:20201027T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/23
DESCRIPTION:Title: On the Bochner-Riesz problem in dimension 3\nby Sh
ukun Wu (UIUC) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe
improve the Bochner-Riesz conjecture in dimension 3 to p>3.25. The main me
thod we used is the iterated polynomial partitioning algorithm. We also ob
serve some relations between wave packets at different scales.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yilin Wang (MIT)
DTSTART:20201208T220000Z
DTEND:20201208T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/24
DESCRIPTION:Title: SLE\, energy duality\, and foliations by Weil-Petersso
n quasicircles\nby Yilin Wang (MIT) as part of UCLA analysis and PDE s
eminar\n\n\nAbstract\nThe Loewner energy for Jordan curves first arises fr
om the small-parameter large deviations of Schramm-Loewner evolution (SLE)
. It is finite if and only if the curve is a Weil-Petersson quasicircle\,
an interesting class of Jordan curves appearing in Teichmuller theory\, ge
ometric function theory\, and string theory with currently more than 20 eq
uivalent definitions. In this talk\, I will show that the large-parameter
large deviations of SLE gives rise to a new Loewner-Kufarev energy\, which
is dual to the Loewner energy via foliations by Weil-Petersson quasicircl
es and exhibits remarkable features and symmetries. Based on joint works w
ith Morris Ang and Minjae Park (MIT) and with Fredrik Viklund (KTH).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Beck (Fordham)
DTSTART:20201103T190000Z
DTEND:20201103T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/25
DESCRIPTION:Title: Two-phase free boundary problems and the Friedland-Hay
man inequality\nby Thomas Beck (Fordham) as part of UCLA analysis and
PDE seminar\n\n\nAbstract\nThe Friedland-Hayman inequality provides a lowe
r bound on the first Dirichlet eigenvalues of complementary subsets of the
sphere. In this talk\, we will describe a variant of this inequality to g
eodesically convex subsets of the sphere with mixed Dirichlet-Neumann boun
dary conditions. Using this inequality\, we prove an almost-monotonicity f
ormula and Lipschitz continuity up to the boundary for the minimizer of a
two-phase free boundary problem. This is joint work with David Jerison and
Sarah Raynor.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Nachman (U. Toronto)
DTSTART:20201110T220000Z
DTEND:20201110T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/26
DESCRIPTION:Title: A Nonlinear Plancherel Theorem with Applications to Gl
obal Well-posedness for the Defocusing Davey-Stewartson Equation and to th
e Inverse Boundary Value Problem of Calderon\nby Adrian Nachman (U. To
ronto) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nThis is joi
nt work with Idan Regev and Daniel Tataru.\n\nThe talk will aim to present
our solutions to 2+\\epsilon open problems.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Forlano (UCLA)
DTSTART:20201124T220000Z
DTEND:20201124T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/27
DESCRIPTION:Title: Normal form approach to the one-dimensional cubic nonl
inear Schr\\"{o}dinger equation in almost critical spaces\nby Justin F
orlano (UCLA) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nIn r
ecent years\, the normal form approach has provided an alternative method
to establishing the well-posedness of solutions to nonlinear dispersive PD
Es\, as compared to using heavy machinery from harmonic analysis. In this
talk\, I will describe how to apply the normal form approach to study the
one-dimensional cubic nonlinear Schr\\"{o}dinger equation (NLS) on the rea
l-line and prove local well-posedness in almost critical Fourier-amalgam s
paces. This involves using an infinite iteration of normal form reductions
(namely\, integration by parts in time) to derive the normal form equatio
n\, which behaves better than NLS for rough functions.\n\nThis is joint wo
rk with Tadahiro Oh (U. Edinburgh).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Dobner (UCLA)
DTSTART:20210106T000000Z
DTEND:20210106T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/28
DESCRIPTION:Title: Extreme values of the argument of the zeta function\nby Alexander Dobner (UCLA) as part of UCLA analysis and PDE seminar\n\n
\nAbstract\nLet $S(t) = \\frac{1}{\\pi}\\Im \\log \\zeta(\\frac{1}{2}+it)$
. The behavior of this function is intimately connected to irregularities
in the locations of the zeros of the zeta function. In particular $S(t)$ m
easures the difference between the "expected" number of zeta zeros up to h
eight $t$ and the actual number of such zeros. I will discuss what is know
n about the distribution of $S(t)$ and prove a new unconditional lower bou
nd on how often $S(t)$ achieves large values.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugenia Malinnikova (Stanford)
DTSTART:20210126T220000Z
DTEND:20210126T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/29
DESCRIPTION:Title: Landis’ conjecture on the decay of solutions to Schr
ödinger equations on the plane.\nby Eugenia Malinnikova (Stanford) as
part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe consider a real-v
alued function on the plane for which the absolute value of the Laplacian
is bounded by the absolute value of the function at each point. In other w
ords\, we look at solutions of the stationary Schrödinger equation with a
bounded potential. The question discussed in the talk is how fast such fu
nction may decay at infinity. We give the answer in dimension two\, in hig
her dimensions the corresponding problem is open.\n\n \n\nThe talk is base
d on the joint work with A. Logunov\, N. Nadirashvili\, and F. Nazarov.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yufei Zhao (MIT)
DTSTART:20210120T000000Z
DTEND:20210120T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/30
DESCRIPTION:Title: Joints of varieties\nby Yufei Zhao (MIT) as part o
f UCLA analysis and PDE seminar\n\n\nAbstract\nWe generalize the Guth-Katz
joints theorem from lines to varieties. A special case of our result says
that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2
})$ joints\, where a joint is a point contained in a triple of these plane
s not all lying in some hyperplane. Our most general result gives upper bo
unds\, tight up to constant factors\, for joints with multiplicities for s
everal sets of varieties of arbitrary dimensions (known as Carbery's conje
cture). Our main innovation is a new way to extend the polynomial method t
o higher dimensional objects.\n\nJoint work with Jonathan Tidor and Hung-H
sun Hans Yu.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgis Moschidis (UC Berkeley)
DTSTART:20210112T180000Z
DTEND:20210112T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/31
DESCRIPTION:Title: The instability of Anti-de Sitter spacetime for the Ei
nstein-scalar field system\nby Georgis Moschidis (UC Berkeley) as part
of UCLA analysis and PDE seminar\n\n\nAbstract\nhe AdS instability conjec
ture provides an example of weak turbulence appearing in the dynamics of t
he Einstein equations in the presence of a negative cosmological constant.
The conjecture claims the existence of arbitrarily small perturbations to
the initial data of Anti-de Sitter spacetime which\, under evolution by t
he vacuum Einstein equations with reflecting boundary conditions at confo
rmal infinity\, lead to the formation of black holes after sufficiently lo
ng time. \n In this talk\, I will present a rigorous proof of the AdS i
nstability conjecture in the setting of the spherically symmetric Einstei
n-scalar field system. The construction of the unstable initial data will
require carefully designing a family of initial configurations of localize
d matter beams and estimating the exchange of energy taking place between
interacting beams over long periods of time\, as well as estimating the de
coherence rate of those beams. I will also discuss possible paths for exte
nding these ideas to the vacuum case.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elena Giorgi (Princeton)
DTSTART:20210116T000000Z
DTEND:20210116T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/32
DESCRIPTION:Title: Electromagnetic-gravitational perturbations of Kerr-Ne
wman spacetime\nby Elena Giorgi (Princeton) as part of UCLA analysis a
nd PDE seminar\n\n\nAbstract\nThe Kerr-Newman spacetime is the most genera
l explicit black hole solution\, and represents a stationary rotating char
ged black hole. Its stability to gravitational and electromagnetic perturb
ations has eluded a proof since the 80s in the black hole perturbation com
munity\, because of "the apparent indissolubility of the coupling between
the spin-1 and spin-2 fields in the perturbed spacetime"\, as put by Chand
rasekhar. We will present a derivation of the Teukolsky and Regge-Wheeler
equations in Kerr-Newman in physical space and use it to obtain a quantita
tive proof of stability.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Betsy Stovall (UW-Madison)
DTSTART:20210302T190000Z
DTEND:20210302T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/33
DESCRIPTION:Title: Existence of extremizers for Fourier restriction opera
tors\nby Betsy Stovall (UW-Madison) as part of UCLA analysis and PDE s
eminar\n\n\nAbstract\nWe learn in first year graduate analysis that an ope
rator from one Banach space to another is continuous if and only if the im
age of the unit ball is a bounded set. In this talk\, we will discuss the
question of whether this image has a point of maximal norm\, in the specif
ic context of certain Fourier restriction operators.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Lewicka (U. Pittsburgh)
DTSTART:20210109T000000Z
DTEND:20210109T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/34
DESCRIPTION:Title: Expansions of averaging operators and applications
\nby Marta Lewicka (U. Pittsburgh) as part of UCLA analysis and PDE semina
r\n\n\nAbstract\nhe following approach of finding solutions to a partial d
ifferential equation Lu=0\, proved to be quite versatile:\n\n(i) develop a
n asymptotic expansion of a suitable family of averaging operators (to be
applied on u)\; the operators are parametrized by the radius \\epsilon of
averaging\, and the coefficient in the expansion that multiplies the appro
priate power of \\epsilon should equal Lu\, the "appropriate power" refers
to the order of L\;\n\n(ii) study the related mean value equation by remo
ving higher order terms in the expansion\;\n\n(iii) interpret the mean val
ue equation as the dynamic programming principle of a two-player game inco
rporating deterministic and stochastic components\;\n\n(iv) pass to the li
mit in the radius of averaging \\epsilon\, in order to recover solutions t
o Lu=0 from the values of the game process.\n\nIn my talk\, I will explain
this approach in the contexts of p-Laplacian and the non-local geometric
p-Laplacian. Other applications include: Robin boundary conditions and wei
ghted Laplace-Beltrami operator on a manifold. In each case\, finding the
appropriate averaging principle is the key starting point in order to deve
lop (i)-(iv).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philip T. Gressman (UPenn)
DTSTART:20210105T230000Z
DTEND:20210106T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/35
DESCRIPTION:Title: Radon-like Transforms\, Geometric Measures\, and Invar
iant Theory\nby Philip T. Gressman (UPenn) as part of UCLA analysis an
d PDE seminar\n\n\nAbstract\nFourier restriction\, Radon-like operators\,
and decoupling theory are three active areas of harmonic analysis which in
volve submanifolds of Euclidean space in a fundamental way. In each case\,
the mapping properties of the objects of study depend in a fundamental wa
y on the "non-flatness" of the submanifold\, but with the exception of cer
tain extreme cases (primarily curves and hypersurfaces)\, it is not clear
exactly how to quantify the geometry in an analytically meaningful way. In
this talk\, I will discuss a series of recent results which shed light on
this situation using tools from an unusually broad range of mathematical
sources.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sigurd Angenent (UW-Madison)
DTSTART:20210202T230000Z
DTEND:20210203T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/36
DESCRIPTION:Title: Nonunique evolution through cones in Mean Curvature Fl
ow and Ricci Flow\nby Sigurd Angenent (UW-Madison) as part of UCLA ana
lysis and PDE seminar\n\n\nAbstract\nFor any integer $k>1$ there exist smo
oth solutions $M_t$ ($t<0$) of MCF that form a one-point singularity at ti
me $t=0$\, after which there exist at least $2k$ forward evolutions $M_t^1
\, \\dots\, M_t^k\, N_t^1\, \\dots\, N_t^k$ ($t>0$) by the flow. The solu
tions $M_t^j$ and $N_t^j$ are topologically distinct. The analogous stat
ement for Ricci Flow also holds\, and I will explain both.\n\nBuilding on
these self similar solutions to MCF\, I will also describe non-self simila
r solutions that have a given cone as their initial data. One conclusion
is that for any $k>1$ there is a smooth self similar solution to MCF that
forms a one point singularity\, and for which the set of possible smooth f
orward evolutions contains a k-dimensional continuum. Another conclusion
is that the set of smooth solutions to MCF whose initial condition is one
of the stationary cones in $\\mathbb{R}^n$ ($n\\in\\{4\, 5\, 6\, 7\\}$) is
infinite dimensional .\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cyrill Muratov (New Jersey Institute of Technology)
DTSTART:20210302T180000Z
DTEND:20210302T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/37
DESCRIPTION:Title: Magnetic skyrmions in the conformal limit\nby Cyri
ll Muratov (New Jersey Institute of Technology) as part of UCLA analysis a
nd PDE seminar\n\n\nAbstract\nWe characterize skyrmions in ultrathin ferro
magnetic films as local minimizers of a reduced micromagnetic energy appro
priate for quasi two-dimensional materials with perpendicular magnetic ani
sotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimizatio
n is carried out in a suitable class of two-dimensional magnetization conf
igurations that prevents the energy from going to negative infinity\, whil
e not imposing any restrictions on the spatial scale of the configuration.
We first demonstrate existence of minimizers for an explicit range of the
model parameters when the energy is dominated by the exchange energy. We
then investigate the conformal limit\, in which only the exchange energy s
urvives and identify the asymptotic profiles of the skyrmions as degree $1
$ harmonic maps from the plane to the sphere\, together with their radii\,
angles and energies. A byproduct of our analysis is a quantitative rigidi
ty result for degree $\\pm 1$ harmonic maps from the two-dimensional spher
e to itself.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (IAS)
DTSTART:20210115T230000Z
DTEND:20210116T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/38
DESCRIPTION:Title: Restriction theory in Fourier analysis\nby Hong Wa
ng (IAS) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nIf a func
tion has Fourier transform supported on a sphere\, what can we say about t
his function?\n\nGiven a collection of long thin tubes pointing in differe
nt directions\, how much do they overlap?\n\nThese two questions are close
ly related. In this talk\, we will discuss how understanding the second qu
estion leads to progress on the first one. More precisely\, we will discus
s Stein's restriction conjecture and Sogge's local smoothing conjecture fo
r the wave equation.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tsviqa Lakrec (U. Jerusalem)
DTSTART:20210216T180000Z
DTEND:20210216T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/39
DESCRIPTION:Title: Equidistribution of affine random walks on some nilman
ifolds\nby Tsviqa Lakrec (U. Jerusalem) as part of UCLA analysis and P
DE seminar\n\n\nAbstract\nWe consider the action of the group of affine tr
ansformations on a nilmanifold. \nGiven a probability measure on this grou
p and a starting point $x$\, a random walk on the nilmanifold is defined.
\nWe study quantitative equidistribution in law of such affine random walk
s on nilmanifolds. \nUnder certain assumptions\, we show that a failure to
have fast equidistribution on a nilmanifold is due to a failure on some f
actor nilmanifold. \nCombined with equidistribution results on the torus\,
this leads to an equidistribution statement on some nilmanifolds\, such a
s Heisenberg nilmanifolds.\n\nThis talk is based on joint works with Weiku
n He and Elon Lindenstrauss.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Seeger (College de France)
DTSTART:20210111T230000Z
DTEND:20210112T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/40
DESCRIPTION:Title: Interpolation results for pathwise Hamilton-Jacobi equ
ations\nby Benjamin Seeger (College de France) as part of UCLA analysi
s and PDE seminar\n\n\nAbstract\nI will show how interpolation methods can
be used to make sense of pathwise Hamilton-Jacobi equations for a wide ra
nge of Hamiltonians and driving paths. The various function spaces describ
e regularity (including Sobolev\, Besov\, Holder\, and variation) as well
as structure. I will also discuss some criteria for a function to be repre
sentable as a difference of convex functions\, a class which plays an impo
rtant role in the theory of pathwise Hamilton-Jacobi equations.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreia Chapouto (U. Edinburgh)
DTSTART:20210112T170000Z
DTEND:20210112T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/41
DESCRIPTION:Title: Invariance of the Gibbs measures for the periodic gene
ralized KdV equations\nby Andreia Chapouto (U. Edinburgh) as part of U
CLA analysis and PDE seminar\n\n\nAbstract\nIn this talk\, we consider the
periodic generalized Korteweg-de Vries equations (gKdV). In particular\,
we study gKdV with the Gibbs measure initial data. The main difficulty lie
s in constructing local-in-time dynamics in the support of the measure. Si
nce gKdV is analytically ill-posed in the $L^2$-based Sobolev support\, we
instead prove deterministic local well-posedness in some Fourier-Lebesgue
spaces containing the support of the Gibbs measures. New key ingredients
are bilinear and trilinear Strichartz estimates adapted to the Fourier-Leb
esgue setting. Once we construct local-in-time dynamics\, we apply Bourgai
n's invariant measure argument to prove almost sure global well-posedness
of the defocusing gKdV with respect to the Gibbs measure and invariance of
the Gibbs measure under the gauged gKdV dynamics.\n\nThis talk is based o
n joint work with Nobu Kishimoto (RIMS\, University of Kyoto).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kihyun Kim (KAIST)
DTSTART:20210129T230000Z
DTEND:20210130T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/42
DESCRIPTION:Title: Blow-up dynamics for the self-dual Chern-Simons-Schrö
dinger equation\nby Kihyun Kim (KAIST) as part of UCLA analysis and PD
E seminar\n\n\nAbstract\nWe consider the blow-up dynamics of the self-dual
Chern-Simons-Schrödinger equation (CSS) under equivariance symmetry. (CS
S) is $L^2$-critical\, has the pseudoconformal symmetry\, and admits a sol
iton $Q$ for each equivariance index $m \\geq 0$. An application of the ps
eudoconformal transformation to $Q$ yields an explicit finite-time blow-up
solution $S(t)$ which contracts at the pseudoconformal rate $|t|$. In the
high equivariance case $m \\geq 1$\, the pseudoconformal blow-up for smoo
th finite energy solutions in fact occurs in a codimension one sense\; it
is stable under a codimension one perturbation\, but also exhibits an inst
ability mechanism. In the radial case $m=0$\, however\, $S(t)$ is no longe
r a finite energy blow-up solution. Interestingly enough\, there are smoot
h finite energy blow-up solutions whose blow-up rates differ from the pseu
doconformal rate by a power of logarithm. We will explore these interestin
g blow-up dynamics (with more focus on the latter) via modulation analysis
. This talk is based on my joint works with Soonsik Kwon and Sung-Jin Oh.\
n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adi Glucksam (U. Toronto)
DTSTART:20210119T230000Z
DTEND:20210120T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/43
DESCRIPTION:Title: Stationary random entire functions and related questio
ns\nby Adi Glucksam (U. Toronto) as part of UCLA analysis and PDE semi
nar\n\n\nAbstract\nThe complex plane acts on the space of entire function
by translations\, taking f(z) to f(z+w). B.Weiss showed in `97 that there
exists a probability measure defined on the space of entire functions\, wh
ich is invariant under this action. In this talk I will present optimal bo
unds on the minimal possible growth of functions in the support of such me
asures and discuss other growth-related problems inspired by this work. In
particular\, I will focus on the question of minimal possible growth-rate
of frequently oscillating subharmonic functions.\nThe talk is partly base
d on a joint work with L. Buhovsky\, A. Logunov\, and M. Sodin.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Polona Durcik (Chapman University)
DTSTART:20210203T000000Z
DTEND:20210203T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/44
DESCRIPTION:Title: Multilinear singular and oscillatory integrals and app
lications\nby Polona Durcik (Chapman University) as part of UCLA analy
sis and PDE seminar\n\n\nAbstract\nWe give an overview of some recent resu
lts in the area of multilinear singular and oscillatory integrals. We disc
uss their connection with certain questions about point configurations in
subsets of the Euclidean space and convergence of some ergodic averages. B
ased on joint works with Michael Christ\, Vjekoslav Kovac\, and Joris Roos
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Goncalves (Bonn)
DTSTART:20210216T190000Z
DTEND:20210216T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/45
DESCRIPTION:Title: Sign Uncertainty\nby Felipe Goncalves (Bonn) as pa
rt of UCLA analysis and PDE seminar\n\n\nAbstract\nI will talk about some
of the recent developments of the sign\nuncertainty principle and its rela
tion with sphere packings and modular\nforms\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lillian Pierce (Duke)
DTSTART:20210319T220000Z
DTEND:20210319T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/46
DESCRIPTION:Title: Counterexamples for high-degree analogues of the Schr
ödinger maximal operator\nby Lillian Pierce (Duke) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nIn 1980 Carleson posed a question on
the minimal regularity of an initial data function that implies pointwise
convergence for the solution of the linear Schrodinger equation. After pr
ogress by many authors\, this was recently resolved (up to the endpoint) b
y Bourgain\, whose counterexample construction for the Schrodinger maximal
operator proved a necessary condition on the regularity\, and Du and Zhan
g\, who proved a sufficient condition. In this talk we describe how Bourga
in's counterexamples can be constructed from first principles. Then we des
cribe a new flexible number-theoretic method for constructing counterexamp
les\, which proves a necessary condition for high-degree analogues of the
Schrodinger maximal operator to be bounded from H^s to\nlocal L^1.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Darren King (U. Texas)
DTSTART:20210316T210000Z
DTEND:20210316T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/47
DESCRIPTION:Title: A capillarity model for soap films\nby Darren King
(U. Texas) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe stu
dy a variational model for soap films based on capillarity theory and its
relation to minimal surfaces. Here\, soap films are modelled\, not as surf
aces\, but as regions of small volume satisfying a homotopic spanning cond
ition.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kasso Okoudjou (Tufts)
DTSTART:20210309T220000Z
DTEND:20210309T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/48
DESCRIPTION:Title: The HRT Conjecture\nby Kasso Okoudjou (Tufts) as p
art of UCLA analysis and PDE seminar\n\n\nAbstract\nIn 1996\, C.~Heil\, J.
~Ramanatha\, and P.~Topiwala conjectured that the (finite) set $\\mathcal{
G}(g\, \\Lambda)=\\{e^{2\\pi i b_k \\cdot}g(\\cdot - a_k)\\}_{k=1}^N$ is l
inearly independent for any non-zero square integrable function $g$ and
subset $\\Lambda=\\{(a_k\, b_k)\\}_{k=1}^N \\subset \\mathbb{R}^2.$ This p
roblem is now known as the HRT Conjecture\, and is still largely unresolv
ed. \n \n\nIn the first part of the talk\, I will give an overview of the
state of the conjecture. I will then introduce an inductive approach to in
vestigate the conjecture\, by attempting to answer the following question.
Suppose the HRT conjecture is true for a function $g$ and a fixed set of
$N$ points $\\Lambda=\\{(a_k\, b_k)\\}_{k=1}^N \\subset \\mathbb{R}^2.$ Fo
r what other point $(a\, b)\\in \\mathbb{R}^2\\setminus \\Lambda$ will the
HRT remain true for the same function $g$ and the new set of $N+1$ points
$\\Lambda'=\\Lambda \\cup \\{(a\, b)\\}$? I will illustrate this inductiv
e argument on special classes of sets $\\Lambda$ when $N\\leq 4$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuomas Hytonen (U. Helsinki)
DTSTART:20210209T180000Z
DTEND:20210209T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/49
DESCRIPTION:Title: Extrapolation of compactness on weighted spaces\nb
y Tuomas Hytonen (U. Helsinki) as part of UCLA analysis and PDE seminar\n\
n\nAbstract\nThe extrapolation theorem of Rubio de Francia is one of the m
ost powerful tools in the theory of weighted norm inequalities: it allows
one to deduce an inequality (often but not necessarily: the bounded of an
operator) on all weighted L^p spaces with a range of p\, by checking it ju
st for one exponent p (but all relevant weights). My topic is an analogous
method for extrapolation of compactness. In a relatively soft way\, it re
covers several recent results about compactness of operators on weighted s
paces and also gives some new ones. I expect there to be many more applica
tions to discover.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kyeongsik Nam (UCLA)
DTSTART:20210223T220000Z
DTEND:20210223T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/50
DESCRIPTION:Title: Spectral large deviations for sparse random matrices\nby Kyeongsik Nam (UCLA) as part of UCLA analysis and PDE seminar\n\n\n
Abstract\nThe large deviation problem for the spectrum of random matrices
has attracted immense interest. It was first studied for GUE and GOE\, whi
ch are exactly solvable\, and subsequently studied for Wigner matrices wit
h general distributions. Once the sparsity is induced (i.e. each entry is
multiplied by the independent Bernoulli distribution\, Ber(p))\, eigenvalu
es can exhibit a drastically different behavior. For a large class of Wign
er matrices\, including Gaussian ensembles and the adjacency matrix of Erd
os-Renyi graphs\, dense behavior ceases to hold near the constant average
degree of sparsity\, p~1/n (up to a poly-logarithmic factor). In this talk
\, I will talk about the spectral large deviation for Gaussian ensembles w
ith a sparsity p=1/n. Joint work with Shirshendu Ganguly.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathan Keller (Bar Ilan University)
DTSTART:20210420T170000Z
DTEND:20210420T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/51
DESCRIPTION:Title: The mysteries of low-degree Boolean functions\nby
Nathan Keller (Bar Ilan University) as part of UCLA analysis and PDE semin
ar\n\n\nAbstract\nAnalysis of Boolean functions studies functions on the d
iscrete cube {-1\,1}^n\, aiming at understanding what the structure of the
(discrete) Fourier transform tells us about the function. In this talk we
focus on the structure of low-degree functions on the discrete cube\, nam
ely\, on functions whose Fourier coefficients are concentrated on low degr
ees. While such functions look very simple\, we are surprisingly far from
understanding them well\, even in the most basic first-degree case. \nWe s
hall present several results on first-degree Boolean functions\, including
the recent proof of Tomaszewski's conjecture (1986) which asserts that an
y first-degree function (viewed as a random variable) lies within one stan
dard deviation from its expectation with probability at least 1/2. Then we
shall discuss several core open questions\, which boil down to understand
ing\, what does the knowledge that a low-degree function is bounded\, or i
s two-valued\, tell us about its structure.\n\nBased on joint work with Oh
ad Klein\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramon van Handel (Princeton)
DTSTART:20210406T220000Z
DTEND:20210406T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/52
DESCRIPTION:Title: The extremals of the Alexandrov-Fenchel inequality
\nby Ramon van Handel (Princeton) as part of UCLA analysis and PDE seminar
\n\n\nAbstract\nIt is a basic fact of convexity that the volume of convex
bodies is a polynomial\, whose coefficients (mixed volumes) define a large
family of natural geometric parameters. A fundamental result of convex ge
ometry\, the Alexandrov-Fenchel inequality\, states that these coefficient
s are log-concave. This result proves to have striking connections with ot
her areas of mathematics\, such as combinatorics and algebraic geometry.\n
\nThere is a long-standing problem surrounding the Alexandrov-Fenchel ineq
uality that has remained open since the original works of Minkowski (1903)
and Alexandrov (1937): in what cases is equality attained? This question
corresponds to the solution of certain unusual isoperimetric problems\, wh
ose extremal bodies turn out to be numerous and strikingly bizarre. With Y
. Shenfeld\, we recently succeeded to settle this problem completely in th
e setting of convex polytopes\, as well as to develop new tools for the st
udy of general convex bodies. In this talk\, I aim to sketch what the extr
emals look like and to indicate some combinatorial\, analytic\, and geomet
ric issues that arise in their characterization.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleksiy Klurman (Bristol)
DTSTART:20210420T180000Z
DTEND:20210420T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/53
DESCRIPTION:Title: On the zeros of Fekete polynomials\nby Oleksiy Klu
rman (Bristol) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nSin
ce its discovery by Dirichlet in the nineteenth century\, Fekete polynomia
ls (with coefficients being Legendre symbols) and their zeros attracted co
nsiderable attention\, in particular\, due to their intimate connection wi
th putative Siegel zero and small class number problem.\n\nThe goal of thi
s talk is to discuss what we knew\, know and would like to know about zero
s of such (and\, time permitting\, related) polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Zorin-Kranich (Bonn)
DTSTART:20210427T170000Z
DTEND:20210427T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/54
DESCRIPTION:Title: Decoupling for quadratic forms\nby Pavel Zorin-Kra
nich (Bonn) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nI will
talk about how decoupling inequalities benefit from\nscale-dependent Bras
camp-Lieb inequalities. The main result describes\nthe sharp decoupling ex
ponents for all manifolds that can be represented\nas graphs of tuples of
quadratic forms. Joint work with Shaoming Guo\,\nChangkeun Oh\, and Ruixia
ng Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dorin Bucur (Université de Savoie)
DTSTART:20210504T170000Z
DTEND:20210504T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/55
DESCRIPTION:Title: Rigidity results for measurable sets\nby Dorin Buc
ur (Université de Savoie) as part of UCLA analysis and PDE seminar\n\n\nA
bstract\nLet $\\Omega \\subset \\R^d$ be a set with finite Lebesgue measur
e such that\, for a fixed radius $r>0$\, the Lebesgue measure of $\\Omega
\\cap B _ r (x)$ is equal to a positive constant when $x$ varies in the
essential boundary of $\\Omega$. We prove that $\\Omega$ is a ball (or a
finite union of equal balls) provided it satisfies a nondegeneracy condi
tion\, which holds in particular for any set of diameter larger than $r$ w
hich is either open and connected\, or of finite perimeter and indecomposa
ble. This is a joint work with Ilaria Fragala.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefanie Petermichl (University of Toulouse)
DTSTART:20210504T180000Z
DTEND:20210504T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/56
DESCRIPTION:Title: The matrix-weighted Hardy-Littlewood maximal function
is unbounded\nby Stefanie Petermichl (University of Toulouse) as part
of UCLA analysis and PDE seminar\n\n\nAbstract\nIn a joint work with Nazar
ov\, Skreb and Treil\, we highlight a marked\ndifference in the presence o
f a matrix weight between the Doob type\nmaximal operator in the dyadic se
tting (with absolute values outside)\nand the dyadic Hardy-Littlewood type
maximal operator (with absolute\nvalues inside). The former is $L^2$ boun
ded while the latter is not.\nFirst\, it will be discussed how to interpre
t these operators in a\nspace with matrix weight. For this\, we will use c
onvex bodies to\nreplace absolute values. (equivalent to the more familiar
\nChrist-Goldberg type definition). We will also discuss the Carleson\nEmb
edding Theorems that are the natural partners of these maximal\noperators
and observe a different behaviour as well.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Izabella Laba (University of British Columbia)
DTSTART:20210518T220000Z
DTEND:20210518T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/57
DESCRIPTION:Title: Tiling the integers with translates of one tile: the C
oven-Meyerowitz tiling conditions for three prime factors\nby Izabella
Laba (University of British Columbia) as part of UCLA analysis and PDE se
minar\n\n\nAbstract\nIt is well known that if a finite set of integers A t
iles the integers by translations\, then the translation set must be perio
dic\, so that the tiling is equivalent to a factorization A+B=Z_M of a fin
ite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling
period M has at most two distinct prime factors\, each of the sets A and B
can be replaced by a highly ordered "standard" tiling complement. It is n
ot known whether this behaviour persists for all tilings with no restricti
ons on the number of prime factors of M.\n\nIn joint work with Itay Londne
r\, we proved that this is true when M=(pqr)^2 is odd. (We are currently f
inalizing the even case.) In my talk I will discuss this problem and intro
duce the main ingredients in the proof.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soeren Fournais (Aarhus University)
DTSTART:20210518T230000Z
DTEND:20210519T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/58
DESCRIPTION:Title: Energy of the Dilute Bose Gas in 3D\nby Soeren Fou
rnais (Aarhus University) as part of UCLA analysis and PDE seminar\n\n\nAb
stract\nIn this talk\, we will review recent progress on the energy of the
3 dimensional dilute Bose gas. It has recently become possible to verify
the old prediction by Bogoliubov and Lee-Huang-Yang of the first correctio
n term to the ground state energy of the interacting gas in the thermodyna
mic limit. \nIf time permits\, I will also discuss the relation of these e
nergy results to proofs of "Bose-Einstein condensation” on density depen
dent length scales.\n\nThis is joint work with Jan Philip Solovej.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pablo Shmerkin (University of British Columbia)
DTSTART:20210406T230000Z
DTEND:20210407T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/59
DESCRIPTION:Title: Explicit and nonlinear variants of Bourgain's projecti
on theorem\nby Pablo Shmerkin (University of British Columbia) as part
of UCLA analysis and PDE seminar\n\n\nAbstract\nBourgain's projection the
orem is a significant extension of his celebrated discretized sum-product
theorem. After reviewing the original formulation of the projection theore
m\, I will present an explicit version\, an extension to parametrized fami
lies of smooth maps\, and applications to the Falconer distance set proble
m. Partly based on joint work in progress with Hong Wang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guofang Wei (UCSB)
DTSTART:20210608T170000Z
DTEND:20210608T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/61
DESCRIPTION:Title: Fundamental Gap Estimate for Convex Domains\nby Gu
ofang Wei (UCSB) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nI
n their celebrated work\, B. Andrews and J. Clutterbuck proved the fundame
ntal gap conjecture that difference of first two eigenvalues of the Lapla
cian with Dirichlet boundary condition on convex domain with diameter D in
the Euclidean space is greater than or equal to $3\\pi^2/D^2$. In sever
al joint works with X. Dai\, Z. He\, S. Seto\, L. Wang (in various subse
ts) the estimate is generalized\, showing the same lower bound holds for
convex domains in the unit sphere. In sharp contrast\, in recent joint w
ork with T. Bourni\, J. Clutterbuck\, X. Nguyen\, A. Stancu and V. Wheele
r\, we prove that the product of the fundamental gap with the square of th
e diameter can be arbitrarily small for convex domains of any diameter in
hyperbolic space. Very recently\, jointed with X. Nguyen\, A. Stancu\,
we show that even for horoconvex domains in the hyperbolic space\, the p
roduct of their fundamental gap with the square of their diameter has no p
ositive lower bound.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tamar Ziegler (HUJI)
DTSTART:20210608T180000Z
DTEND:20210608T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/62
DESCRIPTION:Title: Some applications of analysis over finite fields\n
by Tamar Ziegler (HUJI) as part of UCLA analysis and PDE seminar\n\nLectur
e held in https://caltech.zoom.us/j/99420414248.\n\nAbstract\nWe describe
how one can use equidistribution properties of families of polynomials def
ined over finite fields to derive some interesting effective results in al
gebra. For example : given an ideal J generated by m complex homogeneous p
olynomials of degree < d\, we show that J is contained in an ideal J’ ge
nerated by C(m) homogeneous polynomials of degree < d that form a regular
sequence\, where C(m) is polynomial in m. All terms will be defined and
explained in the talk.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Weigt (Aalto University)
DTSTART:20210312T180000Z
DTEND:20210312T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/63
DESCRIPTION:Title: Endpoint regularity of the dyadic and the fractional m
aximal function\nby Julian Weigt (Aalto University) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nThe well-known open $W^{1\,1}$-probl
em for maximal operators asks if the\nbound\n$\n\\|\\nabla Mf\\|_{L^1(\\ma
thbb R^d)}\n\\leq C_d\n\\|\\nabla f\\|_{L^1(\\mathbb R^d)}\n$\nholds for t
he uncentered and the centered Hardy-Littlewood maximal\noperator.\nWe pro
ve the variants\n$\n\\mathop{\\mathrm{var}}(M^{\\mathrm d}f)\n\\leq C_d\n\
\mathop{\\mathrm{var}}(f)\n$\nfor the dyadic maximal operator $M^{\\mathrm
d}$ and\n$\n\\|\\nabla M_\\alpha f\\|_{L^{d/(d-\\alpha)}(\\mathbb R^d)}\n
\\leq C_{d\,\\alpha}\n\\|\\nabla f\\|_{L^1(\\mathbb R^d)}\n$\nfor the unce
ntered and the centered fractional Hardy-Littlewood maximal\noperator $M_\
\alpha$ if $0<\\alpha \\lt d$.\n\nThe latter bound has thus far been known
to hold only for\n$1\\leq\\alpha \\lt d$.\n\nThe techniques are rather el
ementary.\nThe proof for the the fractional Hardy-Littlewood maximal opera
tor uses\n$\\alpha>0$ to organize the optimal balls in a dyadic manner\nan
d then reduce to the setting of dyadic cubes and apply the proof from\n$M^
{\\mathrm d}$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Galkowski (University College London)
DTSTART:20210511T170000Z
DTEND:20210511T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/64
DESCRIPTION:Title: Geodesic beams and Weyl remainders\nby Jeffrey Gal
kowski (University College London) as part of UCLA analysis and PDE semina
r\n\n\nAbstract\nIn this talk we discuss quantitative improvements for Wey
l remainders\nunder dynamical assumptions on the geodesic flow. We conside
r a variety\nof Weyl type remainders including asymptotics for the eigenva
lue\ncounting function as well as for the on and off diagonal spectral\npr
ojector. These improvements are obtained by combining the geodesic\nbeam a
pproach to understanding eigenfunction concentration together\nwith an app
ropriate decomposition of the spectral projector into\nquasimodes for the
Laplacian. One striking consequence of these\nestimates is a quantitativel
y improved Weyl remainder on all product\nmanifolds. This is joint work wi
th Y.Canzani\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Harrop-Griffiths (UCLA)
DTSTART:20210601T220000Z
DTEND:20210601T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/65
DESCRIPTION:Title: Some recent progress on integrable PDEs\nby Benjam
in Harrop-Griffiths (UCLA) as part of UCLA analysis and PDE seminar\n\nLec
ture held in https://ucla.zoom.us/j/9264073849.\n\nAbstract\nIn this talk
we present some recent progress on integrable PDEs. We first consider the
well-posedness of the cubic NLS and mKdV on the line. We then discuss resu
lts for some related ODE and PDE models. This is joint work with Rowan Kil
lip and Monica Visan.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Terence Tao (UCLA)
DTSTART:20210409T230000Z
DTEND:20210410T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/66
DESCRIPTION:Title: Sendov's conjecture for sufficiently high degree polyn
omials\nby Terence Tao (UCLA) as part of UCLA analysis and PDE seminar
\n\n\nAbstract\nIn 1958\, Blagovest Sendov made the following conjecture:
if a polynomial $f$ of degree $n \\geq 2$ has all of its zeroes in the uni
t disk\, and $a$ is one of these zeroes\, then at least one of the critica
l points of $f$ lies within a unit distance of $a$. Despite a large amoun
t of effort by many mathematicians and several partial results (such as th
e verification of the conjecture for degrees $n \\leq 8$)\, the full conje
cture remains unresolved. In this talk we present a new result that estab
lishes the conjecture whenever the degree $n$ is larger than some sufficie
ntly large absolute constant $n_0$. A result of this form was previously
established in 2014 by Degot assuming that the distinguished zero $a$ stay
ed away from the origin and the unit circle. To handle these latter cases
we study the asymptotic limit as $n \\to \\infty$ using techniques from p
otential theory (and in particular the theory of balayage)\, which has con
nections to probability theory (and Brownian motion in particular). Apply
ing unique continuation theorems in the asymptotic limit\, one can control
the asymptotic behavior of both the zeroes and the critical points\, whic
h allows us to resolve the case when $a$ is near the origin via the argume
nt principle\, and when $a$ is near the unit circle by careful use of Tayl
or expansions to gain fine asymptotic control on the polynomial $f$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiri Artstein (Tel-Aviv University)
DTSTART:20210525T170000Z
DTEND:20210525T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/67
DESCRIPTION:Title: Transportation of measure with respect to non-traditio
nal cost functions\nby Shiri Artstein (Tel-Aviv University) as part of
UCLA analysis and PDE seminar\n\nLecture held in https://ucla.zoom.us/j/9
264073849.\n\nAbstract\nWe will discuss some old and new transportation of
measure results\, concentrating on the differences between the classical
(quadratic\, and more generally – finite-valued) cost functions and the
case of so-called "non-traditional" costs\, when the cost considered is al
lowed to assume infinite values (that is\, some moves are prohibited).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shiri Artstein (Tel-Aviv University)
DTSTART:20210520T180000Z
DTEND:20210520T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/68
DESCRIPTION:Title: Polarity\, non-traditional measure transport\, and a n
ew Rockafellar-type theorem\nby Shiri Artstein (Tel-Aviv University) a
s part of UCLA analysis and PDE seminar\n\nLecture held in Meeting ID: 973
5874 4971\, Passcode: 015836.\n\nAbstract\nTransportation of measure is a
classical technique for proving many geometric and analytic results. The
case where the cost considered is allowed to assume infinite values (that
is\, some moves are prohibited) is less well studied. However\, the so-cal
led “polar-cost”\, which induces the polarity transform on geometric c
onvex function (a less-known-cousin of the Legendre transform) is such a c
ost. In this talk we will discuss function classes and transforms induced
by costs\, their associated cost-subgradients and optimal transportation.
We will discuss a new result\, characterizing plans which admit a “poten
tial”\, applicable to such “non-traditional” cost functions. If time
permits\, we will also discuss an analogue of the Brenier/McCann theorem\
, which holds whenever two measures are strongly-compatible. All definitio
ns and notions will be explained throughout the talk\, as well as examples
and intuition\, and no prior specialized knowledge in the theory of measu
re transport is assumed.\n\nUCLA Distinguished Women in Math Lecture Serie
s\n\nMeeting ID: 973 5874 4971\, Passcode: 015836\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Garnett (UCLA)
DTSTART:20211116T220000Z
DTEND:20211116T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/69
DESCRIPTION:Title: Carleson measure estimates for bounded harmonic functi
ons\, without Ahlfors regularity assumptions.\nby John Garnett (UCLA
) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nLet $\\Omega$ be
a domain in $R^{d+1}$ where $d \\geq 1$. It is known that (using definit
ions given at the start of the talk) if $\\Omega$ satisfies a corkscrew
condition and $\\partial \\Omega$ is $d$-Ahlfors\, then the following are
equivalent:\n\n(a) a square function Carleson measure estimate holds fo
r all bounded harmonic functions on $\\Omega\;$\n\n(b) an $\\varepsilon$-a
pproximation property holds for all such functions and all $0 < \\varepsil
on < 1\;$\n\n(c) $\\partial \\Omega$ is uniformly rectifiable.\n\n Here we
explore (a) and (b) when $\\partial \\Omega$ is not required to be Ahlfo
rs regular.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Feldman (University of Utah)
DTSTART:20211012T210000Z
DTEND:20211012T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/70
DESCRIPTION:Title: Limit shapes of Bernoulli-type free boundaries in peri
odic media\nby William Feldman (University of Utah) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nI will discuss some simplified model
s for the shape of liquid droplets on rough solid surfaces\, especially Be
rnoulli-type free boundary problems. In these models small scale roughnes
s leads to large scale non-uniqueness\, hysteresis\, and anisotropies. In
technical terms we need to understand laminating/foliating families of pl
ane-like solutions\, this is related to ideas of Aubry-Mather theory\, but
\, unlike most results in that area\, we need to consider local (but not g
lobal) energy minimizers.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayla Gafni (University of Mississippi)
DTSTART:20211019T210000Z
DTEND:20211019T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/71
DESCRIPTION:Title: Uniform Distribution and Incidence Theory\nby Ayla
Gafni (University of Mississippi) as part of UCLA analysis and PDE semina
r\n\n\nAbstract\nThe Szemeredi-Trotter Incidence Theorem\, a central resul
t in geometric combinatorics\, bounds the number of incidences between n p
oints and m lines in the Euclidean plane. Replacing lines with circles le
ads to the unit distance problem\, which asks how many pairs of points in
a planar set of n points can be at a unit distance. The unit distance pro
blem breaks down in dimensions 4 and higher due to degenerate configuratio
ns that attain the trivial bound. However\, nontrivial results are possib
le under certain structural assumptions about the point set. In this talk
\, we will introduce a quantitative version of uniform distribution and us
e that property to obtain nontrivial bounds on unit distances and point-hy
perplane incidences in higher-dimensional Euclidean space. This is based
on joint work with Alex Iosevich and Emmett Wyman.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monica Visan (UCLA)
DTSTART:20211026T210000Z
DTEND:20211026T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/72
DESCRIPTION:Title: Orbital stability of KdV multisolitons in $H^{-1}$
\nby Monica Visan (UCLA) as part of UCLA analysis and PDE seminar\n\n\nAbs
tract\nWe introduce a variational characterization of multisoliton\nsoluti
ons to the Korteweg-de Vries equation that is meaningful in\n$H^{-1}$\, wh
ich is the space of optimal well-posedness for this\nequation. As a conse
quence\, we obtain orbital stability of\nmultisoliton solutions in $H^{-1}
$. This is based on joint work with\nRowan Killip.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jarkko Kari (University of Turku)
DTSTART:20211005T180000Z
DTEND:20211005T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/73
DESCRIPTION:Title: Low complexity tilings of the plane\nby Jarkko Kar
i (University of Turku) as part of UCLA analysis and PDE seminar\n\n\nAbst
ract\nA two-dimensional configuration is a coloring of the infinite grid Z
^2 using a finite number of colors. For a finite subset D of Z^2\, the D-p
atterns of a configuration are the patterns of shape D that appear in the
configuration. The number of distinct D-patterns of a configuration is a n
atural measure of its complexity. We consider low-complexity configuration
s where the number of distinct D-patterns is at most |D|\, the size of the
shape. We use algebraic tools to study periodicity of such configurations
[1]. In the case D is a rectangle - or in fact any convex shape - we esta
blish that a uniformly recurrent configuration that has low-complexity wit
h respect to shape D must be periodic [2]. This implies an algorithm to de
termine if a given collection of mn rectangular patterns of size mxn admit
a configuration containing only these patterns. Without the complexity bo
und the question is the well-known undecidable domino problem. We also sho
w\, for an arbitrary shape D\, that a low-complexity configuration must be
periodic if it comes from the well-known Ledrappier subshift\, or from a
wide family of other similar algebraic subshifts [3].\n\nReferences\n[1] J
. Kari\, M. Szabados. An Algebraic Geometric Approach to Nivat’s Conject
ure. Information and Computation 271\, pp. 104481 (2020).\n[2] J. Kari\, E
. Moutot. Decidability and Periodicity of Low Complexity Tilings. Theory o
f Computing Systems (in Press).\n[3] J. Kari\, E. Moutot. Nivat’s conjec
ture and pattern complexity in algebraic subshifts. Theoretical Computer S
cience 777\, pp. 379–386 (2019).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jared Speck (Vanderbilt Univeristy)
DTSTART:20211130T180000Z
DTEND:20211130T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/74
DESCRIPTION:Title: Advances in the mathematical theory of shock waves
\nby Jared Speck (Vanderbilt Univeristy) as part of UCLA analysis and PDE
seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuming Paul Zhang (UCSD)
DTSTART:20211102T220000Z
DTEND:20211102T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/75
DESCRIPTION:Title: Optimal Estimates on the Propagation of Reactions with
Fractional Diffusion\nby Yuming Paul Zhang (UCSD) as part of UCLA ana
lysis and PDE seminar\n\n\nAbstract\nWe study the reaction-fractional-diff
usion equation $u_t+(-\\Delta)^s u=f(u)$ with ignition and monostable reac
tions $f$\, and $s\\in (0\,1)$. We obtain the first optimal bounds on the
propagation of front-like solutions in the cases where no traveling fronts
exist. Our results cover most of these cases\, and also apply to propagat
ion from localized initial data. This is a joint work with A. Zlatos.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Changkeun Oh (University of Wisconsin-Madison)
DTSTART:20211123T220000Z
DTEND:20211123T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/76
DESCRIPTION:Title: Decoupling inequalities for quadratic forms and beyond
\nby Changkeun Oh (University of Wisconsin-Madison) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nIn this talk\, I will present some r
ecent progress on decoupling inequalities for some translation- and dilati
on-invariant systems (TDI systems in short). In particular\, I will emphas
ize decoupling inequalities for quadratic forms. If time permits\, I will
also discuss some interesting phenomenon related to Brascamp-Lieb inequali
ties that appears in the study of a cubic TDI system. Joint work with Shao
ming Guo\, Pavel Zorin-Kranich\, and Ruixiang Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruoci Sun (Karlsruhe Institute of Technology)
DTSTART:20211005T170000Z
DTEND:20211005T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/77
DESCRIPTION:Title: Complete integrability of the Benjamin–Ono equation
on the multi-soliton manifolds\nby Ruoci Sun (Karlsruhe Institute of T
echnology) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (University of Chicago)
DTSTART:20211102T210000Z
DTEND:20211102T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/78
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation withou
t cutoff\nby Luis Silvestre (University of Chicago) as part of UCLA an
alysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Or Shalom (HUJI)
DTSTART:20211130T190000Z
DTEND:20211130T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/79
DESCRIPTION:Title: A structure theorem for Gowers-Host-Kra seminorms for
non-finitely generated countable abelian groups of unbounded torsion\n
by Or Shalom (HUJI) as part of UCLA analysis and PDE seminar\n\n\nAbstract
\nFurstenberg's famous proof of Szemeredi's theorem leads to a natural que
stion about the convergence and limit of some multiple ergodic averages. I
n the case of $\\mathbb{Z}$-actions these averages were studied by Host-Kr
a and Ziegler. They show that the limiting behavior of such multiple ergod
ic average is determined on a certain factor that can be given the structu
re of an inverse limit of nilsystems (i.e. rotations on a nilmanifold). Th
is structure result can be generalized to $\\mathbb{Z}^d$ actions (where t
he average is taken over a Folner sequence)\, but the non-finitely generat
ed case is still open. The only progress prior to our work is due to Berge
lson Tao and Ziegler\, who studied actions of the infinite direct sum $\\m
athbb{Z}/p\\mathbb{Z}$. In our work we generalize this further to the case
where the sum is taken over different primes (the most interesting case i
s when the multiset of primes is unbounded). We will explain how this case
is significantly different from the work of Bergelson Tao and Ziegler by
describing a new phenomenon that only happens in these settings. Moreover\
, we will discuss a generalized version of nilsystems that plays a role in
our work and some corollaries. If time allows we will also discuss the gr
oup actions of other abelian groups.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tongou Yang (UBC)
DTSTART:20211207T220000Z
DTEND:20211207T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/80
DESCRIPTION:Title: Decoupling for smooth surfaces in $\\mathbb R^3$\n
by Tongou Yang (UBC) as part of UCLA analysis and PDE seminar\n\n\nAbstrac
t\nFor each $d\\geq 0$\, we prove decoupling inequalities in $\\mathbb R\n
^3$ for the graphs of all bivariate polynomials of degree at most $d$ with
\nbounded coefficients\, with the decoupling constant depending uniformly
in d\nbut not the coefficients of each individual polynomial. As a consequ
ence\,\nwe prove a decoupling inequality for (a compact piece of) every sm
ooth\nsurface in $\\mathbb R^3$\, which in particular solves a conjecture
of\nBourgain\, Demeter and Kemp.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Shkoller (UC Davis)
DTSTART:20211109T230000Z
DTEND:20211110T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/81
DESCRIPTION:Title: Simultaneous development of shocks and cusps for 2D co
mpressible Euler from smooth initial data\nby Steve Shkoller (UC Davis
) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hyunju Kwon (IAS)
DTSTART:20211019T220000Z
DTEND:20211019T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/82
DESCRIPTION:Title: Euler flows with local energy dissipation\nby Hyun
ju Kwon (IAS) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Chang (Princeton)
DTSTART:20211022T220000Z
DTEND:20211022T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/83
DESCRIPTION:Title: The Kakeya needle problem for rectifiable sets\nby
Alan Chang (Princeton) as part of UCLA analysis and PDE seminar\n\n\nAbst
ract\nWe show that the classical results about rotating a line segment in
arbitrarily small area\, and the existence of a Besicovitch and a Nikodym
set hold if we replace the line segment by an arbitrary rectifiable set. T
his is joint work with Marianna Csörnyei.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura Cladek (UCLA)
DTSTART:20211109T220000Z
DTEND:20211109T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/84
DESCRIPTION:Title: Additive energy of regular measures in one and higher
dimensions\, and the fractal uncertainty principle\nby Laura Cladek (U
CLA) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe obtain new
bounds on the additive energy of (Ahlfors-David type) regular measures in
both one and higher dimensions\, which implies expansion results for sums
and products of the associated regular sets\, as well as more general non
linear functions of these sets. As a corollary of the higher-dimensional r
esults we obtain some new cases of the fractal uncertainty principle in od
d dimensions. This is joint work with Terence Tao.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annina Iseli
DTSTART:20220104T220000Z
DTEND:20220104T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/85
DESCRIPTION:Title: Projection theorems for linear-fractional families of
projections\nby Annina Iseli as part of UCLA analysis and PDE seminar\
n\n\nAbstract\nMarstrand’s theorem (1954) states that given a Borel set
A in the\nEuclidean plane\, the Hausdorff dimension of the image of A unde
r the\northogonal projection onto a line L equals the smaller of 1 and dim
A\,\nfor almost every line L that contains the origin. This theorem has si
nce\nbeen generalized to higher dimensions as well as to various different
\nspaces that carry natural families of projection mappings.\nIn the first
part of this talk\, I will recall some of these\ngeneralizations and the
different methods used to proving them. In the\nsecond part\, I am going t
o present some recent (joint with A.\nLukyanenko) about projection theorem
s for families of projections that\nare induced by either Möbius transfor
mations or real linear fractional\ntransformations.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarah Peluse (Princeton)
DTSTART:20220111T230000Z
DTEND:20220112T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/86
DESCRIPTION:Title: Polynomial and multidimensional configurations in dens
e sets\nby Sarah Peluse (Princeton) as part of UCLA analysis and PDE s
eminar\n\n\nAbstract\nSeveral of the most important problems in combinator
ial number theory ask for the size of the largest subset of an abelian gro
up or interval of integers lacking points in some 'arithmetic' configurati
on. One example of such a question is\, "What is the largest subset of {1\
,...\,N} with no nontrivial k-term arithmetic progressions x\,x+y\,...\,x+
(k-1)y?". Gowers initiated the study of higher order Fourier analysis whil
e seeking to answer this question and used it to give the first reasonable
quantitative bounds. In this talk\, I'll discuss what higher order Fourie
r analysis is and why it is relevant to the study of arithmetic progressio
ns and other configurations\, including 'polynomial' and 'multidimensional
' configurations\, and survey recent progress on problems related to the p
olynomial and multidimensional generalizations of Szemer\\'edi's theorem.\
n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marina Iliopoulou (University of Kent)
DTSTART:20220111T220000Z
DTEND:20220111T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/87
DESCRIPTION:Title: Sharp L^p estimates for oscillatory integral operators
of arbitrary signature\nby Marina Iliopoulou (University of Kent) as
part of UCLA analysis and PDE seminar\n\n\nAbstract\nThe restriction probl
em in harmonic analysis asks for L^p bounds on the Fourier transform of fu
nctions defined on curved surfaces. In this talk\, we will present restric
tion estimates for hyperbolic paraboloids\, that depend on the signature o
f the paraboloids. These estimates still hold\, and are sharp\, in the var
iable coefficient regime. This is joint work with Jonathan Hickman.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Malabika Pramanik (UBC)
DTSTART:20220208T230000Z
DTEND:20220209T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/88
DESCRIPTION:Title: On projections and circles\nby Malabika Pramanik (
UBC) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nThis will be
a survey of two classes of problems in analysis:\nmeasuring the size of pr
ojections of sets\, and counting incidences of\ncircles in the plane. I wi
ll mention a few landmark results in each area\nand discuss recently disco
vered connections between the two.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Zahl (UBC)
DTSTART:20220301T220000Z
DTEND:20220301T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/89
DESCRIPTION:Title: A Kaufman-type restricted projection theorem in R^3\nby Joshua Zahl (UBC) as part of UCLA analysis and PDE seminar\n\n\nAbst
ract\nIn this talk\, I will discuss the proof of a conjecture in projectio
n theory posed by Fässler and Orponen. If K is a set in R^3 of Hausdorff
dimension at most one and if \\gamma is a space curve that obeys a natural
non-degeneracy condition\, then Fässler and Orponen conjectured that for
a typical v \\in \\gamma\, the dimension of the projection K.v must be di
m(K). We resolve this conjecture by proving a Kaufman-type bound on the di
mension of the set of exceptional projections.\n\nWhile Fässler and Orpon
en's conjecture is a question in geometric measure theory\, the solution u
ses ideas from harmonic analysis. In particular\, we resolve the conjectur
e by proving L^p bounds on the Wolff circular maximal function for familie
s of rough curves. This is joint work with Orit Raz\, Malabika Pramanik\,
and Tongou Yang\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stan Palasek (UCLA)
DTSTART:20220222T220000Z
DTEND:20220222T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/90
DESCRIPTION:Title: Quantitative regularity theory for the Navier-Stokes e
quations in critical spaces\nby Stan Palasek (UCLA) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nAn important question in the theory
of the incompressible Navier-Stokes equations is whether boundedness of th
e velocity in various norms implies regularity of the solution. Critical n
orms are conjectured to be (roughly) the threshold between positive and ne
gative answers to this question. Of particular interest are 3D solutions i
n the critical endpoint space $L_t^\\infty L_x^3$ for which Escauriaza-Ser
egin-Sverak famously proved global regularity. Recently Tao improved upon
this result by proving quantitative bounds on the solution and conditions
on a hypothetical blowup. In this talk we discuss the quantitative approac
h to regularity including some sharper results in the axisymmetric case\,
as well as extensions to other critical spaces and to higher dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristina Pereyra (UNM)
DTSTART:20220308T220000Z
DTEND:20220308T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/91
DESCRIPTION:Title: Haar Multipliers Revisited\nby Cristina Pereyra (U
NM) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nHaar multiplie
rs are akin to pseudo-differential operators where the trigonometric funct
ions are replaced by Haar functions. We are interested in their boundednes
s properties. We will focus on some particular examples\, the t-Haar multi
pliers\, for which the theory is well understood on Lebesgue spaces and wi
ll discuss recent progress regarding weighted inequalities. This is work i
n progress joint with Daewon Chung\, Claire Huang\, Jean Moraes and Brett
Wick.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Friedrich Klaus (KIT)
DTSTART:20220125T180000Z
DTEND:20220125T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/92
DESCRIPTION:Title: Well-posedness for the KdV hierarchy\nby Friedrich
Klaus (KIT) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe sh
ow well-posedness for the KdV hierarchy at H^{-1} regularity and for the G
ardner hierarchy at L^2 regularity\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Young-heon Kim (UBC)
DTSTART:20220315T210000Z
DTEND:20220315T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/93
DESCRIPTION:Title: The Stefan problem and optimal transport along the Bro
wnian motion\nby Young-heon Kim (UBC) as part of UCLA analysis and PDE
seminar\n\n\nAbstract\nWe discuss an optimal Brownian stopping problem fr
om a given initial distribution where the target distribution is free and
is conditioned to satisfy a given density height constraint. This is a var
iant of optimal transport problem where transport is constrained to occur
following the Brownian motion\, and the transport plan is given by when ea
ch particle is prescribed to stop. The solutions to this optimization prob
lem then generate solutions to the Stefan problem\, a free boundary proble
m of the heat equation that describes supercooled fluid freezing (St1) or
ice melting (St2)\, depending on the type of cost for optimality. The free
zing (St1) case has not been well understood in the literature beyond one
dimension\, while our result gives a well-posedness of weak solution in ge
neral dimensions\, with naturally chosen initial data. We also give a new
connection between the freezing and melting Stefan problems. This is joint
work with Inwon Kim (UCLA).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Hochman (HUJI)
DTSTART:20220125T170000Z
DTEND:20220125T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/94
DESCRIPTION:Title: Host-type equidistribution results\nby Michael Hoc
hman (HUJI) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nGive t
wo somewhat hyperbolic maps f\,g of a manifold\, fix an invariant probabil
ity measure mu for f\, and act on mu\, or on mu-typical points\, by g. As
suming the maps f\,g are not too closely related\, one expects the orbit t
o equidistribute for some natural measure. Examples of this kind begin wit
h Cassel's and Schmidts theorems on normality of numbers in the ternary Ca
ntor set\, and more recently in Host's theorem about measures on tori inva
riant under endomoirphisms. In the talk\, I will discuss some new results
of this type which extend Host's theorem to its natural generality. The ma
in focus will be on the method of proof\, which relies on soft ideas from
equidistribution theory\, fractal geometry and harmonic analysis\, and som
e basic linear algebra.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dallas Albritton (IAS)
DTSTART:20220222T230000Z
DTEND:20220223T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/95
DESCRIPTION:Title: Non-uniqueness of Leray solutions of the forced Navier
-Stokes equations\nby Dallas Albritton (IAS) as part of UCLA analysis
and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Colombo (EPFL)
DTSTART:20220118T190000Z
DTEND:20220118T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/96
DESCRIPTION:Title: Nonuniqueness results from 2D Euler equations to 3D Na
vier-Stokes equations\nby Maria Colombo (EPFL) as part of UCLA analysi
s and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yu Deng (USC)
DTSTART:20220118T180000Z
DTEND:20220118T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/97
DESCRIPTION:Title: Mathematical wave turbulence and propagation of chaos<
/a>\nby Yu Deng (USC) as part of UCLA analysis and PDE seminar\n\nAbstract
: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihaela Ifrim (University of Wisconsin)
DTSTART:20220208T220000Z
DTEND:20220208T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/98
DESCRIPTION:Title: The time-like minimal surface equation in Minkowski sp
ace: low regularity solutions\nby Mihaela Ifrim (University of Wiscons
in) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frederick Manners (UCSD)
DTSTART:20220201T233000Z
DTEND:20220202T003000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/99
DESCRIPTION:Title: Iterated Cauchy--Schwarz arguments and true complexity
\nby Frederick Manners (UCSD) as part of UCLA analysis and PDE seminar
\n\n\nAbstract\nThis talk is about useful facts that can be proved by repe
ated application of the Cauchy--Schwarz inequality. For example\, it is s
tandard that expressions $\\sum_{x\,y} f(x\,y) a(x) b(y)$ are controlled b
y the matrix norm $\\sum_{x\,y\,x'\,y'} f(x\,y) f(x\,y') f(x'\,y) f(x'\,y'
)$\, and an elementary proof is by applying Cauchy--Schwarz twice. Simila
rly in additive combinatorics\, counting three-term arithmetic progression
s (x\,x+y\,x+2y) (i.e.\, averages $\\sum_{x\,y} f_1(x) f_2(x+y) f_3(x+2y)$
) is controlled by the Gowers $U^2$-norm $\\sum_{x\,y\,x'\,y'} f(x+y) f(x+
y') f(x'+y) f(x'+y')$: generalizations of this are the starting point of G
owers' proof of Szemeredi's theorem.\n\nHowever\, seemingly simple general
izations of this statement quickly become subtle. For example\, linear co
nfigurations $(x\, x+z\, x+y\, x+y+z\, x+2y+3z\, 2x+3y+6z)$ are controlled
by the $U^2$-norm (and so by Fourier analysis) but it is not at all strai
ghtforward to prove this just with Cauchy--Schwarz\; whereas controlling $
(x\, x+z\, x+y\, x+y+z\, x+2y+3z\, 13x+12y+9z)$ requires the $U^3$-norm (i
.e.\, quadratic Fourier analysis) and this can be proved just with Cauchy-
-Schwarz. A conjecture of Gowers and Wolf (resolved by the joint efforts
of various authors) gives a condition to determine when a configuration is
controlled by the $U^k$-norm\, but the proofs require deep structure theo
rems and (unlike Cauchy--Schwarz arguments) give very weak bounds.\n\nIn t
his talk\, I will describe how it is (sometimes) possible to find the miss
ing Cauchy--Schwarz arguments by "mining proofs". The equality cases of t
hese Cauchy--Schwarz inequalities correspond (it turns out) to facts about
functional equations. For example\, the 3-term progression case states t
he following: if $f_1\,f_2\,f_3$ are functions such that $f_1(x)+f_2(x+h)+
f_3(x+2h) = 0$ for all $x\,h$\, then each $f_i$ must be affine-linear. Th
is statement is not completely obvious but has a short elementary proof.\n
\nGiven such an elementary proof\, sometimes we can reverse the process to
find an iterated Cauchy--Schwarz proof of the corresponding inequality --
albeit a very long and complicated one that would be hard to discover by
hand\, and requiring a proof of a very specific type. This answers the Go
wers--Wolf question with polynomial bounds\, and hopefully other questions
where the availability of complicated Cauchy--Schwarz arguments is a limi
ting factor.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zane Li (IU)
DTSTART:20220215T220000Z
DTEND:20220215T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/100
DESCRIPTION:Title: A decoupling interpretation of an old argument for Vi
nogradov's Mean Value Theorem\nby Zane Li (IU) as part of UCLA analysi
s and PDE seminar\n\n\nAbstract\nThere are two proofs of Vinogradov's Mean
Value Theorem (VMVT)\, the harmonic analysis decoupling proof by Bourgain
\, Demeter\, and Guth from 2015 and the number theoretic efficient congrue
ncing proof by Wooley from 2017. While there has been recent work illustra
ting the relation between these two methods\, VMVT has been open since 193
5. It is then natural to ask: What does old partial progress on VMVT look
like in harmonic analysis language? How similar or different does it look
from current decoupling proofs? We talk about an old argument that shows V
MVT "asymptotically" due to Karatsuba and interpret this in decoupling lan
guage. This is ongoing work in progress with Brian Cook\, Kevin Hughes\, O
livier Robert\, Akshat Mudgal\, and Po-Lam Yung.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sung-Jin Oh (UC Berkeley)
DTSTART:20220301T230000Z
DTEND:20220302T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/101
DESCRIPTION:Title: A tale of two tails\nby Sung-Jin Oh (UC Berkeley)
as part of UCLA analysis and PDE seminar\n\n\nAbstract\nIn this talk\, I
will introduce a general method for understanding the late-time tail for s
olutions to wave equations on asymptotically flat spacetimes with odd spat
ial dimensions. A particular consequence of the method is a re-proof of Pr
ice’s law-type results\, which concern the sharp decay rate of the late-
time tails on stationary spacetimes. Moreover\, the method also applies to
dynamical spacetimes. In this case\, I will explain how the late-time tai
ls are in general different(!) from the stationary case in the presence of
dynamical and/or nonlinear perturbations of the problem. This is joint wo
rk with Jonathan Luk (Stanford).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sohrab Shahshahani (UMass\, Amherst)
DTSTART:20220201T223000Z
DTEND:20220201T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/102
DESCRIPTION:Title: Tidal energy in Newtonian two-body motion\nby Soh
rab Shahshahani (UMass\, Amherst) as part of UCLA analysis and PDE seminar
\n\n\nAbstract\nIn this talk we discuss the tidal energy for the motion of
two\n gravitating incompressible fluid balls with free boundaries\, obeyi
ng the\n Euler-Poisson equations. When the fluids are replaced by point\n
masses\, the conic curve describing the trajectories of the bodies are\n k
nown according to the classical analysis of Newton. We will consider the\n
effect of replacing point masses by fluid balls in this analysis. This is
\n joint work with Shuang Miao from Wuhan University.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Krause (King's College London)
DTSTART:20220315T220000Z
DTEND:20220315T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/103
DESCRIPTION:by Ben Krause (King's College London) as part of UCLA analysis
and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART:20220412T210000Z
DTEND:20220412T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/104
DESCRIPTION:Title: A general viewpoint on finite point configurations\nby Alex Iosevich (University of Rochester) as part of UCLA analysis and
PDE seminar\n\n\nAbstract\nWe are to study the existence of finite point
configurations inside compact sets of a given Hausdorff dimension. These p
roblems can be viewed as generalizations of the Falconer distance problem\
, and also thin-set versions of point configuration problems studied\, by
Bourgain\, Furstenberg\, Katznelson\, Weiss\, Ziegler\, and others. We are
going to describe a rather general combinatorial paradigm that allows one
to reduce the existence of a variety of point configurations to certain F
ourier Integral Operator estimates.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dominique Maldague (MIT)
DTSTART:20220524T210000Z
DTEND:20220524T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/105
DESCRIPTION:Title: Small cap decoupling for the moment curve in R^3\
nby Dominique Maldague (MIT) as part of UCLA analysis and PDE seminar\n\n\
nAbstract\nI will present the full solution to a small cap decoupling prob
lem for the moment curve in R^3 motivated by a question about exponential
sums. In particular\, we prove Conjecture 2.5 in dimension 3 from the orig
inal small cap decoupling paper (https://arxiv.org/pdf/1908.09166.pdf) of
Demeter\, Guth\, and Wang. Decoupling for the moment curve involves the fo
llowing set-up. Begin with a function $f$ with Fourier transform supported
on a small neighborhood of a curve. Break the curve up into pieces which
are approximately linear blocks. Then we estimate the size of $f$ in terms
of an expression with the Fourier projections onto each of these blocks.
This is possible since the Fourier projections of $f$ onto different block
s cannot both be large for a long time\, which we exploit using a high-low
frequency argument. This is based on in-progress work in collaboration wi
th Larry Guth.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noam Lifshitz (HUJI)
DTSTART:20220405T170000Z
DTEND:20220405T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/106
DESCRIPTION:Title: Product free sets in A_n\nby Noam Lifshitz (HUJI)
as part of UCLA analysis and PDE seminar\n\n\nAbstract\nA subset of a gro
up is said to be product free if it does not contain the product of two el
ements in it. We consider how large can a product free subset of $A_n$ be?
This problem was considered by Gowers and improved by Eberhard. It appear
s as number 4 in Green's list of his 100 favorite open problems. In the ta
lk we will completely solve the problem by determining the largest product
free subset of $A_n$. \n\nOur proof combines a representation theoretic a
rgument due to Gowers\, with an analytic tool called hypercontractivity fo
r global functions. We also make use of a dichotomy between structure and
pseudorandomness of functions over the symmetric group.\nBased on a joint
work with Peter Keevash and Dor Minzer\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sean Eberhard (Cambridge)
DTSTART:20220607T180000Z
DTEND:20220607T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/107
DESCRIPTION:Title: Random polynomials and random matrices\nby Sean E
berhard (Cambridge) as part of UCLA analysis and PDE seminar\n\n\nAbstract
\nI will talk about some recent results about random polynomials (irreduci
bility and Galois groups) and random discrete matrices. I will outline a p
roof\, conditional on the extended Riemann hypothesis\, that random matric
es have irreducible characteristic polynomial with high probability and Ga
lois group >= A_n. The method uses (a) the prime ideal theorem to reduce t
he global problem about the matrix over Z to a local problem about matrice
s mod p\, and (b) recent results about random matrices over finite fields
to conclude.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ram Band (Technion)
DTSTART:20220426T180000Z
DTEND:20220426T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/108
DESCRIPTION:Title: Neumann domains\nby Ram Band (Technion) as part o
f UCLA analysis and PDE seminar\n\n\nAbstract\nThe nodal set of a Laplacia
n eigenfunction forms a partition of the underlying manifold.\nAnother nat
ural partition is based on the gradient vector field of the eigenfunction.
\nExplicitly\, we take all the gradient flow lines which are connected to
saddle points of the eigenfunction.\nThese lines partition the manifold to
submanifolds which are called Neumann domains (you may try to guess the r
eason for this name\, or wait for the talk \;)\nWe present some results ob
tained so far for Neumann domains - their count\, geometric properties and
spectral position.\nWe also compare the Neumann domain results to the ana
logous ones within the nodal domain study.\n\nThe talk is based on joint w
orks with Philippe Charron\, Graham Cox\, Sebastian Egger\, David Fajman a
nd Alexander Taylor.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajula Srivastava (UWM)
DTSTART:20220517T210000Z
DTEND:20220517T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/109
DESCRIPTION:Title: Two Analogues of the Euclidean Spherical Maximal Func
tion on Heisenberg Groups\nby Rajula Srivastava (UWM) as part of UCLA
analysis and PDE seminar\n\n\nAbstract\nWe shall discuss sharp (up to end
points) $L^p\\to L^q$ estimates for local maximal operators associated wit
h dilates of two different surfaces on Heisenberg groups. The first is the
``horizontal sphere" of codimension two. The second is the Kor\\'anyi sp
here: a surface of codimension one compatible with the non-isotropic dilat
ion structure on the group but with points of vanishing curvature. We shal
l examine the geometry of these surfaces in light of two different notions
of curvature and compare their effect on the estimates for the correspond
ing maximal operators. The Heisenberg group structure will play a crucial
role in our arguments. However\, the theory of Oscillatory Integral Operat
ors will be central despite the non-Euclidean setting. We shall also discu
ss two new counterexamples which imply the sharpness of our results (up to
endpoints). Partly based on joint work with Joris Roos and Andreas Seeger
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jongchon Kim (CityU)
DTSTART:20220503T223000Z
DTEND:20220503T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/110
DESCRIPTION:Title: Nikodym sets for spheres and related maximal function
s\nby Jongchon Kim (CityU) as part of UCLA analysis and PDE seminar\n\
n\nAbstract\nAny set containing a sphere centered at every point cannot ha
ve 0 Lebesgue measure. This is a consequence of the L^p boundedness of the
spherical maximal function. On the other hand\, there exist sets of 0 Leb
esgue measure which contain a large family of spheres\, which may be consi
dered as Kakeya/Nikodym sets for spheres. This talk will be a survey of su
ch sets and their Hausdorff dimension\, and related maximal functions. It
will be based on an ongoing joint work with Alan Chang and Georgios Dosidi
s.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yotam Smilansky (Rutgers)
DTSTART:20220517T220000Z
DTEND:20220517T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/111
DESCRIPTION:Title: Order and disorder in multiscale substitution tilings
\nby Yotam Smilansky (Rutgers) as part of UCLA analysis and PDE semina
r\n\n\nAbstract\nThe study of aperiodic order and mathematical models of q
uasicrystals is concerned with ways in which disordered structures can nev
ertheless manifest aspects of order. In the talk I will describe examples
such as the aperiodic Penrose and pinwheel tilings\, together with several
geometric\, dynamical\, functional and spectral properties that enable us
to measure how far such constructions are from demonstrating lattice-like
behavior. A particular focus will be given to new results on multiscale s
ubstitution tilings\, a class of tilings that was recently introduced join
tly with Yaar Solomon.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiwu Lin (Georgia Tech)
DTSTART:20220510T210000Z
DTEND:20220510T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/112
DESCRIPTION:Title: The existence of Prandtl-Batchelor flows on disk and
annulus\nby Zhiwu Lin (Georgia Tech) as part of UCLA analysis and PDE
seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ibrahim Ekren (Florida State University)
DTSTART:20220510T220000Z
DTEND:20220510T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/113
DESCRIPTION:Title: Prediction problems and second order equations\nb
y Ibrahim Ekren (Florida State University) as part of UCLA analysis and PD
E seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Schrecker (UCL)
DTSTART:20220329T170000Z
DTEND:20220329T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/114
DESCRIPTION:Title: Self-similar gravitational collapse for the Euler-Poi
sson equations\nby Matthew Schrecker (UCL) as part of UCLA analysis an
d PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryan Hynd (UPenn)
DTSTART:20220329T180000Z
DTEND:20220329T190000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/115
DESCRIPTION:Title: Asymptotic flatness of Morrey extremals\nby Ryan
Hynd (UPenn) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Decio (NTNU)
DTSTART:20220503T213000Z
DTEND:20220503T223000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/116
DESCRIPTION:Title: Zeros of Steklov eigenfunctions\nby Stefano Decio
(NTNU) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nA Steklov
eigenfunction in a bounded domain is a harmonic function whose normal deri
vative at the boundary is proportional to the function itself\, or in othe
r words it is the harmonic extension of an eigenfunction of the Dirichlet-
to-Neumann operator. The focus of the talk will be the study of the zero s
ets of such objects. I will show that there are many zeros near the bounda
ry and I will discuss upper and lower bounds on the Hausdorff measure of t
he zero set.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiumin Du (Northwestern)
DTSTART:20220412T220000Z
DTEND:20220412T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/117
DESCRIPTION:Title: Falconer's distance set problem\nby Xiumin Du (No
rthwestern) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nA clas
sical question in geometric measure theory\, introduced by Falconer in the
80s is\, how large does the Hausdorff dimension of a compact subset in Eu
clidean space need to be to ensure that the Lebesgue measure of its set of
pairwise Euclidean distances is positive. In this talk\, I'll report some
recent progress on this problem\, which combines several ingredients incl
uding Orponen's radial projection theorem\, Liu's L^2 identity obtained us
ing a group action argument\, and the refined decoupling theory. This is b
ased on joint work with Alex Iosevich\, Yumeng Ou\, Hong Wang\, and Ruixia
ng Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Anderson (Stanford)
DTSTART:20220531T220000Z
DTEND:20220531T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/118
DESCRIPTION:Title: Nonlinear interactions of waves from distant sources<
/a>\nby John Anderson (Stanford) as part of UCLA analysis and PDE seminar\
n\n\nAbstract\nPhysical systems are often idealized as being isolated beca
use very distant events ought not have a significant influence. Mathematic
ally\, this often translates to solving problems with localized data. In t
his talk\, I will discuss results which make this intuitive idealization r
igorous. Indeed\, we study the effects that distant perturbations have on
solutions to nonlinear wave equations. We prove a stability statement\, wh
ich requires analyzing the spacetime geometry of the interaction of waves
originating from distant sources. I also hope to describe some of the addi
tional difficulties involved in extending these results to the physically
interesting case of the Einstein vacuum equations of general relativity. T
his is joint work with Federico Pasqualotto (Duke University).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Shapiro (Princeton)
DTSTART:20220531T210000Z
DTEND:20220531T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/119
DESCRIPTION:Title: Monotonicity theorems for integer-valued fields and d
elocalization in two-dimensions\nby Jacob Shapiro (Princeton) as part
of UCLA analysis and PDE seminar\n\n\nAbstract\nInteger-valued fields are
restricted to take values in Z and usually their Gibbs factor depends only
on the gradient of the field. When the Gibbs factor is such that the typi
cal value of the gradients is much larger than 1 (the spacing of points in
Z)\, the integer constraint becomes less relevant so the field behaves as
if it were real-valued and “delocalizes”. In 2D\, this delocalization
is associated with the Berezinskii–Kosterlitz–Thouless phase of the d
ual O(2) spin model. I will explain these notions for various models and p
resent recent monotonicity theorems for fluctuations which are important t
o establish the delocalized phase.\n\nJoint with: Michael Aizenman\, Matan
Harel and Ron Peled.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hongki Jung (Indiana)
DTSTART:20220419T213000Z
DTEND:20220419T223000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/120
DESCRIPTION:Title: A small cap decoupling for the twisted cubic\nby
Hongki Jung (Indiana) as part of UCLA analysis and PDE seminar\n\n\nAbstra
ct\nSmall cap decouplings deal with decoupling estimates for caps that are
smaller than the canonical size. In 2019\, Demeter\, Guth and Wang studie
d small cap decoupling for exponential sums with frequency points supporte
d on the cubic moment curve. In this talk\, I will discuss the proof of $L
^{10}$ small cap decoupling for general functions\, which involves inciden
ce estimates for tubes and planks in $\\mathbb{R}^3$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiajie Chen (Caltech)
DTSTART:20220419T223000Z
DTEND:20220419T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/121
DESCRIPTION:Title: On the competition between advection and vortex stret
ching\nby Jiajie Chen (Caltech) as part of UCLA analysis and PDE semin
ar\n\n\nAbstract\nWhether the 3D incompressible Euler equations can develo
p a finite-time singularity from smooth initial data is an outstanding ope
n problem. The presence of vortex stretching is the primary source of a po
tential finite-time singularity. However\, to construct a singularity\, th
e effect of the advection is one of the obstacles. In this talk\, we will
first show some examples in incompressible fluids about the competition be
tween advection and vortex stretching. Then we will discuss the De Gregori
o (DG) model\, which adds an advection term to the Constantin-Lax-Majda mo
del to model this competition. In an effort to establish singularity forma
tion in incompressible fluids\, we develop a novel approach based on dynam
ic rescaling formulation. Using this approach\, we construct finite time s
ingularities of the DG model on the real line from smooth initial data and
on a circle from C^{\\alpha} initial data with any $0<\\alpha < 1$. On th
e other hand\, for $C^1$ initial data with the same sign and symmetry prop
erties as those of the blowup solution\, we prove that the solution of the
DG model on a circle exists globally.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tamara Grava (Bristol)
DTSTART:20220607T170000Z
DTEND:20220607T180000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/122
DESCRIPTION:Title: Gibbs ensemble for Integrable Systems\, a case study
: the Ablowitz Laddik lattice\nby Tamara Grava (Bristol) as part of UC
LA analysis and PDE seminar\n\n\nAbstract\nWe consider discrete integrab
le systems with random initial data and connect them with the theory of ra
ndom matrices.\nIn particular we consider the defocusing nonlinear Schr
odinger equation in its integrable version\, that is called Ablowitz Ladik
lattice. In the random initial data setting the Lax matrix of the Ab
lowitz Ladik lattice turns into a random matrix that is related to the c
ircular beta-ensemble at high temperature. We obtain the density of st
ates of the random Lax matrix\, when the size of the matrix goes to infini
ty\, by establishing a mapping to the one-dimensional log-gas. The dens
ity of states is obtained via a particular solution of the double-conflu
ent Heun equation.\nJoint work with Guido Mazzuca https://arxiv.org/pdf/21
07.02303.pdf\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Krause (King's College London)
DTSTART:20220524T220000Z
DTEND:20220524T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/123
DESCRIPTION:Title: Discrete Analogues in Harmonic Analysis: Equidistribu
tion of Exponential Sums and a Theorem of Stein-Wainger\nby Ben Krause
(King's College London) as part of UCLA analysis and PDE seminar\n\n\nAbs
tract\nIn this talk I will review the theory of maximally modulated oscill
atory singular integrals after Stein-Wainger\, and then will use equidistr
ibution-type results for exponential sums to adapt the Stein-Wainger theor
y to the discrete setting.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yifeng Yu (UC Irvine)
DTSTART:20221018T210000Z
DTEND:20221018T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/124
DESCRIPTION:Title: Existence of effective burning velocity in cellular f
low for curvature G-equation\nby Yifeng Yu (UC Irvine) as part of UCLA
analysis and PDE seminar\n\n\nAbstract\nG-equation is a popular level set
model in turbulent combustion\, and\nbecomes an advective mean curvature
type evolution equation when the curvature effect is considered:\n$$\nG_t
+ \\left(1-d\\\, \\Div{\\frac{DG}{|DG|}}\\right)_+|DG|+V(x)\\cdot DG=0.\n$
$\nIn this talk\, I will show the existence of effective burning velocity
under the above curvature G-equation model when $V$ is a two dimensional
cellular flow. Our proof combines PDE methods with a dynamical analysis o
f the Kohn-Serfaty deterministic game characterization of the curvature G-
equation based on the special structure of the cellular flow. This is a jo
int with Hongwei Gao\, Ziang Long and Jack Xin.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Minh-Binh Tran (Texas A&M)
DTSTART:20221025T210000Z
DTEND:20221025T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/125
DESCRIPTION:Title: Some Recent Results On Wave Turbulence\nby Minh-B
inh Tran (Texas A&M) as part of UCLA analysis and PDE seminar\n\n\nAbstrac
t\nWave turbulence describes the dynamics of both classical and non-classi
cal nonlinear waves out of thermal equilibrium. Recent mathematical inte
rests on wave turbulence theory have the roots from the works of Bourgain\
, Staffilani and Colliander-Keel-Staffilani-Takaoka-Tao. In this talk\, I
will present some of our recent results on wave turbulence theory. The tal
k is based on my joint work with Bensoussan (UTD)\, Staffilani (MIT)\, Sof
fer (Rutgers)\, Pomeau (ENS Paris).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Agnieszka Zelerowicz (UC Riverside)
DTSTART:20221129T233000Z
DTEND:20221130T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/126
DESCRIPTION:Title: Lorentz gases on quasicrystals\nby Agnieszka Zele
rowicz (UC Riverside) as part of UCLA analysis and PDE seminar\n\nLecture
held in Caltech Linde 310.\n\nAbstract\nThe Lorentz gas was originally int
roduced as a model for the movement of electrons in metals.\n\n It consist
s of a massless point particle (electron) moving through Euclidean space b
ouncing off a given set of scatterers $\\mathcal{S}$ (atoms of the metal)
with elastic collisions at the boundaries $\\partial \\mathcal{S}$. If the
set of scatterers is periodic in space\, then the quotient system\, which
is compact\, is known as the Sinai billiard. There is a great body of wor
k devoted to Sinai billiards and in many ways their dynamics is well under
stood.\n\n In contrast\, very little is known about the behavior of the Lo
rentz gases with aperiodic configurations of scatterers which model quasic
rystals and other low-complexity aperiodic sets. This case is the focus of
our joint work with Rodrigo Trevi\\~no. \n\nWe establish some dynamical p
roperties which are common for the periodic and quasiperiodic billiards. W
e also point out some significant differences between the two. The novelty
of our approach is the use of tiling spaces to obtain a compact model of
the aperiodic Lorentz gas on the plane.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Wu (UCLA)
DTSTART:20220927T210000Z
DTEND:20220927T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/127
DESCRIPTION:Title: The gradient flow structure of the Landau equation\nby Jeremy Wu (UCLA) as part of UCLA analysis and PDE seminar\n\nLecture
held in UCLA Math 6221.\n\nAbstract\nThe Landau equation is one of the co
rnerstones of kinetic theory. It describes the evolution of a gas of plasm
a particles. Complementing its physical relevance\, the mathematical theor
y of the Landau equation is very deep\, yet incomplete owing to the compet
ing effects of quasilinear diffusion and quadratic growth. Global regulari
ty has eluded researchers because of this competition and a related open q
uestion is global uniqueness of weak solutions. This talk introduces the g
radient flow structure of the Landau equation to set the foundation for an
approach to answering this problem. The construction of the metric which
induces the gradient flow structure builds upon the dynamic formulation of
classical Wasserstein metrics. This is based on joint work with José A.
Carrillo\, Matias G. Delgadino\, and Laurent Desvillettes.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Lawrie (MIT)
DTSTART:20221108T220000Z
DTEND:20221108T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/128
DESCRIPTION:Title: The soliton resolution conjecture for equivariant wav
e maps\nby Andrew Lawrie (MIT) as part of UCLA analysis and PDE semina
r\n\n\nAbstract\nI will present a joint work with Jacek Jendrej (CRNS\, So
rbonne Paris Nord) on equivariant wave maps with values in the two-sphere.
We prove that every finite energy solution resolves\, as time passes\, in
to a superposition of harmonic maps (solitons) and radiation\, settling th
e soliton resolution problem for this equation. It was proved in works of
Côte\, and Jia-Kenig\, that such a decomposition holds along a sequence
of times. We show the resolution holds continuously-in-time via a “no-re
turn” lemma based on the virial identity. The proof combines a modulatio
n analysis of solutions near a multi-soliton configuration with concentrat
ion compactness techniques. As a byproduct of our analysis we prove that t
here are no pure multi-solitons in equivariance class k=1 and no elastic c
ollisions between pure multi-solitons in the higher equivariance classes.\
n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juhi Jang (USC)
DTSTART:20220927T220000Z
DTEND:20220927T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/129
DESCRIPTION:Title: On slowly rotating star solutions\nby Juhi Jang (
USC) as part of UCLA analysis and PDE seminar\n\nLecture held in UCLA Math
6221.\n\nAbstract\nIn this talk we will review recent progress on the loc
al and global dynamics of Newtonian stars governed by the compressible Eul
er-Poisson system and discuss mathematical constructions of slowly rotatin
g star solutions bifurcating from the non-rotating ones. In the case of no
n-isentropic stars\, we introduce a new ad hoc perturbative strategy to ov
ercome the loss of regularity and variational structure caused by the vari
able entropy. If time permits\, we will also discuss recent uniqueness and
orbital stability of McCann’s uniformly rotating binary star solutions
and its application to binary galaxies. The talk is based on joint works w
ith T. Makino\, W. Strauss\, Y. Wu and J. Seok.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nets Katz (Caltech)
DTSTART:20221011T210000Z
DTEND:20221011T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/130
DESCRIPTION:Title: A proto-inverse Szemer\\'edi Trotter theorem\nby
Nets Katz (Caltech) as part of UCLA analysis and PDE seminar\n\nLecture he
ld in Caltech Linde 310.\n\nAbstract\nThe symmetric case of the Szemer\\'e
di-Trotter theorem says that any configuration of N lines and N points in
the plane has at most O(N^{4/3}) incidences. We describe a recipe involvin
g just O(N^{1/3}) parameters which sometimes (that is\, for some choices o
f the parameters) produces a configuration of N point and N lines. (Otherw
ise\, we say the recipe fails.) We show that any near-extremal example for
Szemer\\'edi Trotter is densely related to a successful instance of the r
ecipe. We discuss the relation of this statement to the inverse Szemer\\'e
di Trotter problem. (joint work in progress with Olivine Silier.)\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Perlmutter (UCLA)
DTSTART:20221011T220000Z
DTEND:20221011T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/131
DESCRIPTION:Title: The scattering transform\, a harmonic analysis perspe
ctive on neural networks\nby Michael Perlmutter (UCLA) as part of UCLA
analysis and PDE seminar\n\nLecture held in Caltech Linde 310.\n\nAbstrac
t\nThe scattering transform is a mathematical model of convolutional neura
l networks (CNNs) initially introduced (for Euclidean data) by Mallat in 2
012. This work models the filter convolutions of a CNN as a wavelet transf
orm and uses methods from harmonic analysis to analyze the stability and i
nvariance of CNNs to certain group actions. I will introduce Mallat’s co
nstruction and explain how it has improved our understanding of CNNs. Then
\, in the second half of my talk\, I will discuss recent generalizations o
f the scattering transform to graphs\, manifolds\, and other measure space
s. These generalized scattering transforms utilize wavelets constructed fr
om the spectral decomposition of a suitable Laplacian. I will also discuss
a diffusion maps-based method\, with a provable convergence rate\, for im
plementing the manifold scattering transform from finitely samples of an u
nknown manifold.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shengwen Gan (MIT)
DTSTART:20221129T223000Z
DTEND:20221129T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/132
DESCRIPTION:Title: The restricted projection to planes in R^3.\nby S
hengwen Gan (MIT) as part of UCLA analysis and PDE seminar\n\nLecture held
in Caltech Linde 310.\n\nAbstract\nIn this talk\, I will discuss a conjec
ture made by Fässler and Orponen on the restricted\nprojection to planes
in R^3. I will first talk about the Falconer-type exceptional set estimate
in R^2\, and then I will talk about the proof of the conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Hall (Notre Dame)
DTSTART:20221004T210000Z
DTEND:20221004T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/133
DESCRIPTION:Title: Random matrices and heat flow on polynomials\nby
Brian Hall (Notre Dame) as part of UCLA analysis and PDE seminar\n\n\nAbst
ract\nSeveral recent results have demonstrated a “model deformation phen
omenon” in random matrix theory\, in which the limiting eigenvalue distr
ibutions of two different random matrix models are\, in certain cases\, re
lated by push-forward under an explicit\, canonical map of the plane to it
self. The prototype example is the case of the circular and semicircular l
aws\, which are related by push-forward under the map z —> 2Re(z). There
are by now several broad families of examples extending this simple case.
\n\nI will discuss a conjecture\, developed with Ching Wei Ho\, that prov
ides a finite-N version of the model deformation phenomenon\, at the level
of characteristic polynomials. Specifically\, the conjecture says that ap
plying the heat operator to the characteristic polynomial of one random ma
trix gives a polynomial whose bulk distribution of zeros resembles that of
a different random matrix. As an example\, consider applying the heat ope
rator for time 1/N to the characteristic polynomial of an NxN GUE matrix.
We believe that the zeros of the resulting polynomial will be almost surel
y asymptotically uniform over the unit disk. Thus\, the heat operator can
turn the semicircular law into the circular law. I will explain the conjec
ture and describe some rigorous results in this direction.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wojciech Ozanski (Florida State University)
DTSTART:20221018T220000Z
DTEND:20221018T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/134
DESCRIPTION:Title: Well-posedness of logarithmic spiral vortex sheets\nby Wojciech Ozanski (Florida State University) as part of UCLA analysis
and PDE seminar\n\n\nAbstract\nWe will discuss a family of 2D logarithmic
spiral vortex sheets which include the celebrated spirals introduced by P
randtl (Vortr¨age aus dem Gebiete der Hydro- und Aerodynamik\, 1922) and
by Alexander (Phys. Fluids\, 1971). We will discuss a recent result regard
ing a complete characterization of such spirals in terms of weak solutions
of the 2D incompressible Euler equations. Namely\, we will explain that a
spiral gives rise to such solution if and only if two conditions hold acr
oss every spirals: a velocity matching condition and a pressure matching c
ondition\, which provides the first rigorous mathematical framework for th
e spirals since their introduction by Prandtl in 1922\, despite significan
t progress of the theory of vortex sheets and the Birkhoff-Rott equations.
We will also discuss well-posedness of the symmetric Alexander spiral wit
h two branches\, despite recent evidence for the contrary\, as well as an
existence result of nonsymmetric spirals.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thierry Laurens (UCLA)
DTSTART:20221115T220000Z
DTEND:20221115T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/135
DESCRIPTION:Title: Sharp well-posedness for the Benjamin--Ono equation\nby Thierry Laurens (UCLA) as part of UCLA analysis and PDE seminar\n\n
\nAbstract\nWe will discuss a proof of sharp well-posedness for the Benjam
in--Ono equation in the class of H^s spaces\, on both the line and the cir
cle. This result was previously unknown on the line\, while on the circle
it was obtained recently by Gérard\, Kappeler\, and Topalov. This is jo
int work with Rowan Killip and Monica Visan.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Timothy Buttsworth (U. Penn)
DTSTART:20221122T220000Z
DTEND:20221122T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/136
DESCRIPTION:Title: The prescribed cross curvature equation on the three-
sphere\nby Timothy Buttsworth (U. Penn) as part of UCLA analysis and P
DE seminar\n\n\nAbstract\nFor a given Riemannian manifold\, the cross curv
ature tensor is a symmetric (0\,2)-tensor field which describes how close
the underlying geometry is to being hyperbolic. The cross curvature was in
troduced by Chow and Hamilton in 2004\; they hoped that the corresponding
cross curvature flow could be used to continuously deform an arbitrary Rie
mannian metric of negative sectional curvature into one of constant negati
ve sectional curvature. In this talk\, I will discuss the 'prescribed cros
s curvature equation'\, which is the underlying inhomogeneous steady-state
version of the cross curvature flow. About this problem\, Hamilton conjec
tured that any positive symmetric tensor on the three-sphere was the cross
curvature of exactly one Riemannian metric. I will discuss some recent re
sults which support the existence component of this conjecture\, and refut
e the uniqueness component. Joint work with Artem Pulemotov.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yannick Sire (Johns Hopkins University)
DTSTART:20230110T220000Z
DTEND:20230110T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/137
DESCRIPTION:Title: Nematic Liquid crystal flows with free boundary\n
by Yannick Sire (Johns Hopkins University) as part of UCLA analysis and PD
E seminar\n\n\nAbstract\nI will introduce a new parabolic system for the f
low of nematic liquid crystals\, enjoying a free boundary condition. After
recent works related to the construction of blow-up solutions for several
critical parabolic problems (such as the Fujita equation\, the heat flow
of harmonic maps\, liquid crystals without free boundary\, etc...)\, I wil
l construct a physically relevant weak solution blowing-up in finite time.
We make use of the so-called inner/outer parabolic gluing. Along the way\
, I will present a set of optimal estimates for the Stokes operator with N
avier slip boundary conditions. I will state several open problems related
to the partial regularity of the system under consideration. This is join
t work with F.-H. Lin (NYU)\, Y. Zhou (JHU) and J. Wei (UBC).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandros Eskenazis (Sorbonne)
DTSTART:20221108T230000Z
DTEND:20221109T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/138
DESCRIPTION:Title: Regularity for weighted convex isoperimetric problems
\nby Alexandros Eskenazis (Sorbonne) as part of UCLA analysis and PDE
seminar\n\n\nAbstract\nWe shall discuss results and open questions pertain
ing to the regularity (and irregularity) of solutions of weighted isoperim
etric-type problems over the class of symmetric convex sets. Based on join
t work with G. Moschidis (EPFL).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ewelina Zatorska (ICL)
DTSTART:20221025T193000Z
DTEND:20221025T203000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/139
DESCRIPTION:Title: The dissipative Aw-Rascle system: existence theory an
d hard-congestion limit\nby Ewelina Zatorska (ICL) as part of UCLA ana
lysis and PDE seminar\n\n\nAbstract\nIn this talk I am going to analyze th
e compressible dissipative hydrodynamic model of crowd motion or of granul
ar flow. The model resembles the famous Aw-Rascle model of traffic\, excep
t that the difference between the actual and the desired velocities (the o
ffset function) is a gradient of the density function\, and not a scalar.
This modification gives rise to a dissipation term in the momentum equatio
n that vanishes when the density is equal to zero.\nI will compare the dis
sipative Aw-Rascle system with the compressible Euler and compressible Nav
ier-Stokes equations\, and back it up with two existence and ill-posedness
results. In the last part of my talk I will explain the proof of conjectu
re made by Lefebvre-Lepot and Maury\, that the hard congestion limit of th
is system (with singular offset function) leads to congested compressible/
incompressible Euler equations.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (UCLA)
DTSTART:20221101T210000Z
DTEND:20221101T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/140
DESCRIPTION:Title: Sticky Kakeya sets in R^3\nby Hong Wang (UCLA) as
part of UCLA analysis and PDE seminar\n\n\nAbstract\nA Kakeya set is a se
t of points in R^n which contains a unit line segment in every direction.
The Kakeya conjecture states that the Hausdorff dimension of any Kakeya se
t is n. We study a special collection of the Kakeya sets\, namely the sti
cky Kakeya sets\, where the line segments in nearby directions stay close.
Joint with Josh Zahl\, we show that the sticky Kakeya sets in R^3 has Ha
usdorff dimension 3.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deniz Bilman (University of Cincinnati)
DTSTART:20230131T210000Z
DTEND:20230131T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/141
DESCRIPTION:Title: Wave patterns generated by large-amplitude rogue wave
s and their universal character\nby Deniz Bilman (University of Cincin
nati) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nIt is known
from our recent work that both fundamental rogue wave solutions (with Pete
r Miller and Liming Ling) and multi-pole soliton solutions (with Robert Bu
ckingham) of the nonlinear Schrödinger (NLS) equation exhibit the same un
iversal asymptotic behavior in the limit of large order in a shrinking reg
ion near their peak amplitude point\, despite the quite different boundary
conditions these solutions satisfy at infinity. This behavior is describe
d by a special solution of again the NLS equation that also satisfies ordi
nary differential equations from the Painlev\\’e-III hierarchy. We revie
w these results and show that this profile also arises universally from ar
bitrary background fields. We then show how rogue waves and solitons of ar
bitrary orders can be placed within a common analytical framework in which
the "order" becomes a continuous parameter\, allowing one to tune continu
ously between types of solutions satisfying different boundary conditions.
In this framework\, solitons and rogue waves of increasing integer orders
alternate as the continuous order parameter increases. We show that in a
bounded region of the space-time of size proportional to the order\, these
solutions all appear to be the same when the order is large. However\, i
n the unbounded complementary region one sees qualitatively different asym
ptotic behavior along different sequences. This is joint work with Peter M
iller (U. Michigan).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Buckingham (University of Cincinnati)
DTSTART:20230228T210000Z
DTEND:20230228T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/142
DESCRIPTION:Title: Universality of High-Order Rogue Waves\nby Robert
Buckingham (University of Cincinnati) as part of UCLA analysis and PDE se
minar\n\n\nAbstract\nWe will discuss a series of recent results indicating
that high-order rogue-wave behavior is universally described for a variet
y of different equations and initial conditions by a family of functions c
onnected to the Painleve-III hierarchy and first encountered by Suleimanov
in 2017. This is joint work with Deniz Bilman\, Bob Jenkins\, and Peter
Miller.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katy Craig (UC Santa Barbara)
DTSTART:20230221T220000Z
DTEND:20230221T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/144
DESCRIPTION:Title: Nonlocal particle approximations of the porous medium
equation and applications to sampling and two-layer neural networks\n
by Katy Craig (UC Santa Barbara) as part of UCLA analysis and PDE seminar\
n\n\nAbstract\nGiven a desired target distribution and an initial guess of
its samples\, what is the best way to evolve the locations of the samples
so that they accurately represent the desired distribution? A classical s
olution to this problem is to evolve the samples according to Langevin dyn
amics\, a stochastic particle method for the Fokker-Planck equation. In to
day’s talk\, I will contrast this with a nonlocal\, deterministic partic
le method inspired by the porous medium equation. Using the Wasserstein gr
adient flow structure of the equations and Serfaty’s scheme of Gamma-con
vergence of gradient flows\, I will show that\, as the number of samples i
ncreases and the interaction scale goes to zero\, the interacting particle
system indeed converges to a solution of the porous medium equation. I wi
ll close by discussing practical implications of this result to both sampl
ing and the training dynamics two-layer neural networks. This is based on
joint work with Karthik Elamvazhuthi\, Matt Haberland\, and Olga Turanova.
\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Mohammadi (UC San Diego)
DTSTART:20230221T230000Z
DTEND:20230222T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/145
DESCRIPTION:Title: Dynamics on homogeneous spaces: a quantitative accoun
t\nby Amir Mohammadi (UC San Diego) as part of UCLA analysis and PDE s
eminar\n\n\nAbstract\nRigidity phenomena in homogeneous spaces have been e
xtensively studied over the past few decades with several striking results
and applications. We will give an overview of activities pertaining to th
e quantitative aspect of the analysis in this context with an emphasis on
recent developments and applications. This is based on joint works with El
on Lindenstrauss and Zhiren Wang.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Davit Harutyunyan (UC Santa Barbara)
DTSTART:20230214T220000Z
DTEND:20230214T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/147
DESCRIPTION:Title: On Geometric rigidity of thin domains\nby Davit H
arutyunyan (UC Santa Barbara) as part of UCLA analysis and PDE seminar\n\n
\nAbstract\nA famous theorem of Reshetnyak states that if the gradient of
a Sobolev field belongs to the group of proper rotations SO(n)\, then \nth
e field has to be affine. Friesecke\, James and Mueller proved a quantitat
ive version of this statement in a celebrated work in 2002\,\nwhich is the
so-called Geometric Rigidity Estimate (GRE). A linearization is the Korn
inequality in linear Elasticity. It turned out that \nthe "best" constant
in the estimate is tied with the actual physical rigidity of the domain. I
n this presentation\, we will discuss the GRE and Korn inequality \n(the l
inearization of GRE) for thin domains\, where the question is to find the
asymptotics of the "best" constant in terms of the domain \nthickness.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Song (Caltech)
DTSTART:20230110T230000Z
DTEND:20230111T000000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/148
DESCRIPTION:Title: Entropy\, first eigenvalue and stability of the hyper
bolic plane\nby Antoine Song (Caltech) as part of UCLA analysis and PD
E seminar\n\n\nAbstract\nConsider a closed surface of genus at least 2 end
owed with a Riemannian metric g\, and let (S\,g) be its universal cover. T
here are two important invariants for (S\,g): the first eigenvalue \\lambd
a of the Laplacian and the volume entropy h\, which measures the exponenti
al growth rate of the volume of geodesic balls. We can normalize g so that
h=1. Then a classical inequality states that \\lambda is at most 1/4. Whe
n g is a hyperbolic metric\, equality holds. We will discuss a stability p
roperty for the hyperbolic plane: if \\lambda is close to the upper bound
1/4\, then (S\,g) is close to the hyperbolic plane in a Benjamini-Schramm
topology.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Peszek (University of Warsaw)
DTSTART:20230124T190000Z
DTEND:20230124T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/149
DESCRIPTION:Title: Heterogeneous gradient flows with applications to col
lective dynamics\nby Jan Peszek (University of Warsaw) as part of UCLA
analysis and PDE seminar\n\n\nAbstract\nIn 2001 F. Otto discovered a (now
adays well-known) relationship between the continuity equation and gradien
t flows with respect to the 2-Wasserstein metric. This connection provides
a convenient description of many new and classical models and PDEs includ
ing Keller-Segel and Fokker-Planck as well as models of first-order collec
tive dynamics. \nI am going to present a recent work (joint with David Poy
ato)\, wherein we introduce the so-called fibered 2-Wasserstein metric (wh
ich admits only transportation along fibers controlled by a prescribed pro
babilistic distribution) and explore its applicability in gradient flows.
Based on such a metric\, we develop the notion of heterogeneous gradient f
lows\, and prove that they are equivalent to solutions of parameterized co
ntinuity equations. Lastly\, I will present a collection of applications r
anging from mixtures of fluids\, to multispecies models of collective dyna
mics\, and to (the essential) applications in alignment models.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaoyutao Luo (Duke)
DTSTART:20230124T200000Z
DTEND:20230124T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/150
DESCRIPTION:Title: Illposedness for vortex patches of the Euler and alph
a-SQG equations\nby Xiaoyutao Luo (Duke) as part of UCLA analysis and
PDE seminar\n\n\nAbstract\nI will talk about joint work with A. Kiselev (D
uke) on patch solutions of the Euler and alpha-SQG equations. It is well-k
nown that the vortex patch of the 2D Euler equation is globally well-posed
in non-endpoint Holder spaces. We prove that the Euler vortex patch is il
l-posed at the C^2 endpoint by showing the existence of a patch with C^2 i
nitial data such that the curvature of the patch boundary becomes infinite
instantaneously. The alpha-SQG equations are a family of active scalar in
terpolating the 2D Euler and SQG equations. In contrast to the Euler case\
, we show that the alpha-SQG patch\, in a suitable regime of regularity\,
is ill-posed in all non-L^2 Sobolev spaces and Holder spaces.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fazel Hadadifard (University of California\, Riverside)
DTSTART:20230207T223000Z
DTEND:20230207T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/151
DESCRIPTION:Title: Sharp time asymptotics for the quasi-geostrophic equa
tion and near plane waves of reaction-diffusion models\nby Fazel Hadad
ifard (University of California\, Riverside) as part of UCLA analysis and
PDE seminar\n\nLecture held in Linde Hall 310\, Caltech.\n\nAbstract\nThe
long-term dynamics of the equations arising in fluid mechanics is a ubiqui
tous and well-studied subject\, and several methods have been developed. I
n this talk\, we introduce the scaled variable method of Gallay-Wayne. We
expand the method to cover a wider range of equations/models. \n\nThe met
hod is then applied to the quasi-geostrophic equation and the Boussinesq s
ystem\, both subject to fractional dissipation. We also present the stabil
ity of the plane wave equations in higher dimensions. The method produces
sharp time rates\, the leading order terms as well as sharp asymptotics.\n
\nOur work\, joint with Prof. A. Stefanov\, generalizes the classical wor
ks on the Navier-Stokes system. Since the Green's functions in the fractio
nal dissipation context are not sufficiently decaying at infinity\, the c
enter-stable manifold construction of Gallay-Wayne appears to be out of re
ach. Instead\, we rely on appropriate a priori estimates for the solutions
(both in weighted and unweighted settings) to derive the asymptotic profi
les.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (University of California\, Los Angeles)
DTSTART:20230207T233000Z
DTEND:20230208T003000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/152
DESCRIPTION:Title: Radial projections in the plane\nby Hong Wang (Un
iversity of California\, Los Angeles) as part of UCLA analysis and PDE sem
inar\n\n\nAbstract\nLet $x$ be a point in the plane\, and the radial proje
ction $\\pi_x$ is defined by $\\pi_x(y)= \\frac{x-y}{|x-y|}$ for any $y\\n
eq x\\in \\mathbb{R}^2$. Suppose that $X$ is a Borel set in the plane and
is not contained in any line\, then we show that there exists a point $x\\
in X$ such that $\\pi_x (X)$ has dimension equal to $\\min \\{ \\dim_H X\,
1\\}$. This is joint work with Tuomas Orponen and Pablo Shmerkin.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seung-Yeon Ryoo (Princeton)
DTSTART:20230411T200000Z
DTEND:20230411T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/153
DESCRIPTION:Title: On embedding finitely generated groups of polynomial
growth into Euclidean spaces\nby Seung-Yeon Ryoo (Princeton) as part
of UCLA analysis and PDE seminar\n\n\nAbstract\nIt is well-known that a fi
nitely generated group of \npolynomial growth embeds bilipschitzly into\nE
uclidean space (or Hilbert space) if and only if it is virtually \nabelian
. Thus\, in the not virtually abelian case\,\nthe Euclidean distortion of
the ball of radius $n$ in the group grows \nto infinity as $n\\to\\infty$.
\nWe may therefore ask: what is the precise asymptotics of the Euclidean \
ndistortion of $n$-balls?\nAnd what role does the dimension of the target
Euclidean space play in \nthe distortion?\nWe compute the (infinite-dimens
ional) Euclidean distortion of \n$n$-balls to be a constant multiple of $\
\sqrt{\\log n}$\,\nby establishing for nilpotent Lie groups the classical
Dorronsoro \ntheorem\, which measures the $L^p$ norm of the fractional Lap
lacian\nof a function in terms of a singular integral measuring the local
\ndeviation of the function from suitable polynomials at all points\nand a
t all scales. We then show that\, in the special case of lattices \nof Car
not groups\, the target dimension essentially does not affect\nthe distort
ion\, by constructing embeddings that simultaneously \noptimize the distor
tion and target dimension.\nThis construction involves a combination of th
e Lovász Local Lemma\, \nthe concentration of measure on the Euclidean sp
here\,\nand a version of the Nash-Moser iteration scheme pioneered by Tao
(2018).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonas Luhrmann (Texas A&M)
DTSTART:20230307T223000Z
DTEND:20230307T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/154
DESCRIPTION:Title: On co-dimension one stability of the soliton for the
1D focusing cubic Klein-Gordon equation\nby Jonas Luhrmann (Texas A&M)
as part of UCLA analysis and PDE seminar\n\nLecture held in Linde Hall 18
7\, Caltech.\n\nAbstract\nSolitons are particle-like solutions to dispersi
ve evolution equations\nwhose shapes persist as time goes by. In some situ
ations\, these solitons\nappear due to the balance between nonlinear effec
ts and dispersion\, in\nother situations their existence is related to top
ological properties of\nthe model. Broadly speaking\, they form the buildi
ng blocks for the\nlong-time dynamics of dispersive equations.\n\nIn this
talk I will present joint work with W. Schlag on long-time decay\nestimate
s for co-dimension one type perturbations of the soliton for the\n1D focus
ing cubic Klein-Gordon equation (up to exponential time scales)\,\nand I w
ill discuss our previous work on the asymptotic stability of the\nsine-Gor
don kink under odd perturbations. While these two problems are\nquite simi
lar at first sight\, we will see that they differ by a subtle\ncancellatio
n property\, which has significant consequences for the\nlong-time dynamic
s of the perturbations of the respective solitons.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andreia Chapouto (UCLA)
DTSTART:20230307T233000Z
DTEND:20230308T003000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/155
DESCRIPTION:Title: Disproving the Deift conjecture: the loss of almost p
eriodicity\nby Andreia Chapouto (UCLA) as part of UCLA analysis and PD
E seminar\n\nLecture held in Linde Hall 187\, Caltech.\n\nAbstract\nIn 200
8\, Deift conjectured that almost periodic initial data leads to almost pe
riodic solutions to the Korteweg-de Vries equation (KdV). In this talk\, w
e show that this is not always the case. Namely\, we construct almost peri
odic initial data whose KdV evolution remains bounded but loses almost per
iodicity at a later time\, by building on the new observation that the con
jecture fails for the Airy equation.\nThis is joint work with Rowan Killip
and Monica Visan\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Greenfeld (IAS)
DTSTART:20230509T200000Z
DTEND:20230509T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/156
DESCRIPTION:Title: The structure of translational tilings\nby Rachel
Greenfeld (IAS) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nT
ranslational tiling is a covering of a space (e.g.\, Euclidean space) usin
g translated copies of a building block\, called a "tile''\, without any p
ositive measure overlaps. What are the possible ways that a space can be t
iled? One of the most well known conjectures in this area is the periodic
tiling conjecture. It asserts that any tile of Euclidean space can tile th
e space periodically. In a joint work with Terence Tao\, we disprove the p
eriodic tiling conjecture in high dimensions. \nIn the talk\, I will surve
y the study of the periodic tiling conjecture\, motivate our recent result
and discuss our counterexample as well as new developments.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Yu (University of Florida)
DTSTART:20230404T190000Z
DTEND:20230404T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/157
DESCRIPTION:Title: Infinitely many solutions to the isentropic system of
gas dynamics\nby Cheng Yu (University of Florida) as part of UCLA ana
lysis and PDE seminar\n\n\nAbstract\nIn this talk\, I will discuss the non
-uniqueness of global weak solutions to the isentropic system of gas dynam
ics. In particular\, I will show that for any initial data belonging to a
dense subset of the energy space\, there exists infinitely many global wea
k solutions to the isentropic Euler equations for any 1 < γ ≤ 1 + 2/n.
The proof is based on a generalization of convex integration techniques an
d weak vanishing viscosity limit of the Navier-Stokes equations. This talk
is based on the joint work with M. Chen and A. Vasseur.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petronela Radu (University of Nebraska)
DTSTART:20230404T200000Z
DTEND:20230404T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/158
DESCRIPTION:Title: Analytical\, geometrical\, and applied aspects in non
local frameworks\nby Petronela Radu (University of Nebraska) as part o
f UCLA analysis and PDE seminar\n\n\nAbstract\nThe emergence of nonlocalit
y as a successful framework for capturing a variety of different physical
phenomena has catalyzed research in many directions at the applied\, compu
tational\, as well as at the theoretical levels. While models formulated w
ith the classical continuum mechanics theory have brought huge development
s in technology and science over the last century\, the new frontier requi
res tackling discontinuous\, singular\, or irregular behavior encountered
in many applications such as deformations and damage of solid bodies\, pha
se transitions and image processing. To this end\, the study of systems th
at allow low-regularity (possibly discontinuous) solutions becomes the cri
tical center-piece. In this talk I will present basic nonlocal formulation
s for elasticity\, diffusion\, conservation laws\, as well as some geometr
ic aspects for studying curvature for boundaries that lack (classical) C^2
regularity. For the corresponding nonlocal systems of equations we will d
iscuss recent results (most of them belonging to the nonlinear realm) that
we have obtained with our students and collaborators\, as well as ongoing
problems and future directions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Bedrossian (UCLA)
DTSTART:20230418T223000Z
DTEND:20230418T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/159
DESCRIPTION:Title: Lower bounds on the top Lyapunov exponent of Galerkin
-Navier-Stokes and other stochastic differential equations\nby Jacob B
edrossian (UCLA) as part of UCLA analysis and PDE seminar\n\nLecture held
in Caltech - Linde 255.\n\nAbstract\nWe review our recent joint work with
Alex Blumenthal and Sam Punshon-Smith\, which introduced methods for obtai
ning strictly positive lower bounds on the top Lyapunov exponent of high-d
imensional\, stochastic differential equations such as the weakly damped L
orenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equat
ions. This hallmark of chaos has long been observed in these models\, howe
ver\, no mathematical proof had previously been made for either determinis
tic or stochastic forcing. The method is a combination of a new identity c
onnecting the Lyapunov exponents to a Fisher information of the stationary
measure of the "projective process" with an L1-based uniform hypoelliptic
regularity estimate. We will also discuss some related results\, such as
dichotomies regarding Lyapunov exponents of general non-dissipative SDEs w
ith applications to chaotic charged particle motion (joint with Chi-Hao Wu
) and other applications of uniform hypoelliptic estimates\, such as sharp
estimates on the spectral gap of Markov semigroups (joint with Kyle Liss)
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Kitagawa (Michigan State University)
DTSTART:20230425T200000Z
DTEND:20230425T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/160
DESCRIPTION:Title: Monge solutions of nontwisted optimal transport on no
nstrictly convex boundaries\nby Jun Kitagawa (Michigan State Universit
y) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nIn the optimal
transport (Monge-Kantorovich) problem\, the existence of a single-valued o
ptimal map is guaranteed under certain conditions on the cost function and
measures. However if the cost is ambient Euclidean distance squared restr
icted to the boundary of a convex body\, a result of Gangbo and McCann dem
onstrates there may be nice measures for which there is no singled-valued
optimal map. In this talk I discuss a recent result of ours showing that w
hen the transported measures have sufficiently small optimal transport cos
t\, there exists a single-valued optimal map\, when the body is $C^1$ and
convex (but not necessarily strictly convex). This result is sharp in the
sense that the claim can fail for a non-$C^1$ domain\, even if it is unifo
rmly convex. This talk is based on joint work with Seonghyeon Jeong.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sameer Iyer (UC Davis)
DTSTART:20230530T200000Z
DTEND:20230530T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/161
DESCRIPTION:Title: Reversal in the Stationary Prandtl Equations\nby
Sameer Iyer (UC Davis) as part of UCLA analysis and PDE seminar\n\n\nAbstr
act\nWe investigate reversal and recirculation for the stationary Prandtl
equations. Reversal describes the solution after the Goldstein singularity
\, and is characterized by spatio-temporal regions in which $u > 0$ and $u
< 0$. The classical point of view of regarding the Prandtl equations as a
n evolution $x$ completely breaks down. Instead\, we view the problem as a
quasilinear\, mixed-type\, free-boundary problem. Joint work with Nader M
asmoudi.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enno Lenzmann (U. Basel)
DTSTART:20230523T200000Z
DTEND:20230523T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/162
DESCRIPTION:Title: Turbulence in completely integrable PDEs: The Caloger
o-Moser derivative NLS\nby Enno Lenzmann (U. Basel) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nI will discuss a new type of a deriv
ative nonlinear Schrödinger equation on the line\, which can be seen as a
continuum version of completely integrable Calogero-Moser many-body syste
ms in classical mechanics. The resulting NLS exhibits many intriguing feat
ures such as a Lax pair structure on Hardy spaces\, L^2-criticality\, and
turbulent solutions. In my talk\, I will focus on the dynamics of multi-so
liton solutions. We prove global-in-time existence (which is a large data
result) and\, more strikingly\, we show that these multi-solitons always e
xhibit an unbounded growth of Sobolev norms (turbulence) as time tends to
infinity. This talk is based on joint work with Patrick Gérard (Orsay).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louise Gassot (ENS)
DTSTART:20230523T190000Z
DTEND:20230523T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/163
DESCRIPTION:Title: Zero-dispersion limit for the Benjamin-Ono equation o
n the torus\nby Louise Gassot (ENS) as part of UCLA analysis and PDE s
eminar\n\n\nAbstract\nWe discuss the zero-dispersion limit for the Benjami
n-Ono equation on the torus given a bell-shaped initial data. We prove tha
t the solutions admit a weak limit as the dispersion parameter tends to ze
ro\, which is explicit and constructed from the Burgers' equation. The app
roach relies on the complete integrability for the Benjamin-Ono equation f
rom Gérard\, Kappeler and Topalov\, and also on the spectral study of the
Lax operator associated to the initial data in the zero-dispersion limit.
\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Kehle (ETH)
DTSTART:20230418T213000Z
DTEND:20230418T223000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/164
DESCRIPTION:Title: Turbulence for quasilinear waves on Schwarzschild-AdS
\nby Christoph Kehle (ETH) as part of UCLA analysis and PDE seminar\n\
nLecture held in Caltech - Linde 255.\n\nAbstract\nIn this talk\, I will
present upcoming work proving a "weak turbulent" instability for quasiline
ar wave equations on Schwarzschild-AdS black holes. The instability is gov
erned by a stably trapped 3-mode interaction transferring energy from low-
to high-frequency modes. Our result is motivated by the question of the st
ability of black holes in the presence of a negative cosmological constant
. This is joint work with Georgios Moschidis.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sung-Jin Oh (Berkeley)
DTSTART:20230516T223000Z
DTEND:20230516T233000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/165
DESCRIPTION:Title: Codimension one stability of the catenoid under the h
yperbolic vanishing mean curvature flow\nby Sung-Jin Oh (Berkeley) as
part of UCLA analysis and PDE seminar\n\nLecture held in Caltech.\n\nAbstr
act\nThe catenoid is one of the simplest examples of a minimal hypersurfac
e\, next to the hyperplane. In this talk\, we will view the catenoid as a
stationary solution to the hyperbolic vanishing mean curvature flow\, whic
h is the hyperbolic analog of the (elliptic) minimal hypersurface equation
\, and study its nonlinear stability under no symmetry assumptions. The ma
in result\, which is a recent joint work with Jonas Luhrmann and Sohrab Sh
ahshahani\, is that with respect to a "codimension one" set of initial dat
a perturbations of the n-dimensional catenoid\, the corresponding flow asy
mptotes to an adequate translation and Lorentz boost of the catenoid for n
greater than or equal to 5. Note that the codimension one condition is ne
cessary and sharp in view of the fact that the catenoid is an index 1 mini
mal hypersurface. \n\nAmong the key challenges of the present problem comp
ared to the more classical stability problems for nontrivial stationary so
lutions are: (1) the quasilinearity of the equation\, (2) the slow (polyno
mial) decay of the catenoid at infinity\, and (3) the lack of symmetry ass
umptions. To address these challenges\, we introduce several new ideas\, s
uch as a geometric construction of modulated profiles\, smoothing of modul
ation parameters\, and a robust framework for proving decay for the radiat
ion.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohit Bansil (UCLA)
DTSTART:20230502T200000Z
DTEND:20230502T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/166
DESCRIPTION:Title: The Master Equation in Mean Field Games\nby Mohit
Bansil (UCLA) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nA M
ean Field Game is a differential game (in the sense of game theory) where
instead of a finite number of players we have a continuous distribution of
(infinitely) many players\, however we make the simplifying assumption th
at all players are identical.\n\nIn this talk we consider the existence an
d uniqueness of Nash Equilibrium in Mean Field Games. We show why the stud
y of Nash Equilibrium naturally leads to the study of a Hamilton-Jacobi eq
uation over the space of measures called the master equation\, whose solut
ions give rise to Nash Equilibrium for our game.\n\nFor mean field games t
here isn't a general theory of viscosity solutions analogous to Hamilton-J
acobi equations in finite dimensions. Motivated by this we revisit the cla
s-\nsical solution theory (as opposed to viscosity solutions) of Hamilton
Jacobi equations and identify a symmetry that extends the well-posedness t
heory into new regimes. This\nsymmetry also yields results for the master
equation in mean field games.\n\nWe will see that there are two natural ty
pes of noise that one can impose in a Mean Field Game\, individual noise a
nd common noise\, which correspond to cases where the noise of each player
is independent and identical respectively. Individual noise has a regular
izing effect that is utilized in most well-posedness results for the maste
r equation.\nWe explore well-posedness for the master equation in the case
without individual noise\, under a monotonicity condition.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marianna Russkikh (Caltech)
DTSTART:20230502T210000Z
DTEND:20230502T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/167
DESCRIPTION:Title: Dimers and embeddings\nby Marianna Russkikh (Calt
ech) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe introduce
a concept of ‘t-embeddings’ of weighted bipartite planar graphs. We be
lieve that these t-embeddings always exist and that they are good candidat
es to recover the complex structure of big bipartite planar graphs carryin
g a dimer model. We also developed a relevant theory of discrete holomorph
ic functions on t-embeddings\; this theory unifies Kenyon’s holomorphic
functions on T-graphs and s-holomorphic functions coming from the Ising mo
del. We provide a meta-theorem on convergence of the height fluctuations t
o the Gaussian Free Field.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lei Wu (Lehigh)
DTSTART:20230516T210000Z
DTEND:20230516T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/168
DESCRIPTION:Title: Ghost effect from Boltzmann theory\nby Lei Wu (Le
high) as part of UCLA analysis and PDE seminar\n\nLecture held in Caltech.
\n\nAbstract\nThe hydrodynamic limit aims to derive fluid equations (such\
nas the Euler and Navier-Stokes equations) from kinetic theory (such as th
e Boltzmann and Landau equations) in a rigorous manner. This is a key ingr
edient for addressing the Hilbert Sixth Problem. As the Knudsen number (w
hich measures mean free path) approaches zero\, almost all standard fluid
equations can be derived through proper scaling. Our work presents an unus
ual hydrodynamic limit that shows genuine kinetic effects\, known as the g
host effect. The density and\ntemperature of order are coupled with the v
elocity of order which acts like a "ghost" that can't be observed at the
fluid level. This suggests that standard fluid mechanics is incomplete in
describing many-particle systems even at the continuum regime. This is joi
nt work with Raffaele Esposito\, Yan Guo and Rossana Marra\, and is mainly
based on preprints https://arxiv.org/abs/2301.09427 and https://arxiv.org
/abs/2301.09560 .\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Quentin Berger (Sorbonne)
DTSTART:20230530T210000Z
DTEND:20230530T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/169
DESCRIPTION:Title: The Stochastic Heat Equation with multiplicative Lév
y noise\nby Quentin Berger (Sorbonne) as part of UCLA analysis and PDE
seminar\n\n\nAbstract\nI will introduce the Stochastic Heat Equation with
multiplicative noise and I will discuss its well-posedness and some of it
s properties. This has been well studied when the noise is Gaussian but it
is only recently that the case of non-Gaussian (Lévy) noise has been con
sidered. \nThis is based on joint work with Carsten Chong (Columbia) and H
ubert Lacoin (IMPA).\nDisclaimer: I come from a probability/statistical me
chanics background\, but I plan on introducing all objects that are not ne
cessarily familiar to analysts.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hung Tran (U. Wisc. Madison)
DTSTART:20240206T220000Z
DTEND:20240206T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/170
DESCRIPTION:Title: Periodic homogenization of Hamilton-Jacobi equations:
some recent progress.\nby Hung Tran (U. Wisc. Madison) as part of UCL
A analysis and PDE seminar\n\n\nAbstract\nI first give a quick introductio
n to front propagations\, Hamilton-Jacobi equations\, level-set forced mea
n curvature flows\, and homogenization theory. I will then show the optima
l rates of convergence for homogenization of both first-order and second-o
rder Hamilton-Jacobi equations. Based on joint works with J. Qian\, T. Spr
ekeler\, and Y. Yu.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Changyou Wang (Purdue)
DTSTART:20231010T210000Z
DTEND:20231010T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/171
DESCRIPTION:Title: Analysis on Isotropic-Nematic Phase Transition and Li
quid Crystal Droplet\nby Changyou Wang (Purdue) as part of UCLA analys
is and PDE seminar\n\n\nAbstract\nIn this talk\, I will discuss the phase
transition phenomena between the isotropic and nematic states within the f
ramework of Ericksen theory of liquid crystals with variable degrees of or
ientations. Treating it as the singular perturbation problems within the
Gamma convergence theory\, we will show that the sharp interface formed be
tween isotropic and nematic states is an area minimizing surface. Under su
itable assumptions either on the strong anchoring boundary values on the b
oundary of a bounded domain or the volume constraint of nematic regions in
the entire space\, we also show that the limiting nematic liquid configur
ation in the nematic region is a minimizer of the corresponding Oseen-Fran
k energy with either homeotropic or planar anchoring on the free sharp int
erface pending on the relative sizes of leading Frank elasticity coefficie
nts. This is a joint work with Fanghua Lin.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Leng (UCLA)
DTSTART:20231017T210000Z
DTEND:20231017T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/172
DESCRIPTION:Title: The equidistribution of nilsequences\nby James Le
ng (UCLA) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nConsider
a Nilpotent Lie group $G$ and a discrete subgroup $\\Gamma$ such that the
topological quotient $G/\\Gamma$ is compact. Certain problems in arithmet
ic combinatorics are concerned with an equidistribution theory on $G/\\Gam
ma$. This theory studies the behavior of orbits $g^n\\Gamma$ and classific
ation of their limit sets in $G/\\Gamma$. \n\nIn 2012\, Green and Tao prov
ed a quantitative equidistribution theory on $G/\\Gamma$\, achieving polyn
omial bounds on the rate of equidistribution and with exponent single expo
nential in the dimension of $G$. In this talk\, we go over a recent result
\, which improves the bounds to have exponent polynomial in the dimension
of $G$. We also discuss implications of this result to arithmetic combinat
orics. A key obstruction that the proof of this result overcomes is "induc
tion on dimensions"\, which also seem to appear elsewhere in higher order
Fourier analysis over $\\mathbb{Z}/N\\mathbb{Z}$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rowan Killip (UCLA)
DTSTART:20231003T210000Z
DTEND:20231003T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/173
DESCRIPTION:Title: The Benjamin--Ono equation\nby Rowan Killip (UCLA
) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nThe BO equation
is an effective model for interfacial waves in fluids of infinite depth. L
ike its shallow-water cousin\, the Korteweg--de Vries equation\, BO is com
pletely integrable\; however\, the relevant spectral theory is far removed
from the comfortable familiarity of Sturm--Liouville equations. After des
cribing this model and its integrable structures\, we will then present a
sharp well-posedness theory and a slew of new virial-type identities. This
talk is based on joint work with Thierry Laurens and Monica Visan.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Jacobs (U. Michigan)
DTSTART:20231128T220000Z
DTEND:20231128T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/174
DESCRIPTION:Title: Lagrangian solutions to the Porous Media Equation (an
d friends)\nby Matthew Jacobs (U. Michigan) as part of UCLA analysis a
nd PDE seminar\n\n\nAbstract\nMany works have been devoted to understandin
g and predicting the time evolution of a growing population of cells (bact
erial colonies\, tumors\, etc...). At the macroscopic scale\, cell growth
is typically modeled through Porous Media type equations that describe t
he change in cell density. While these cell growth PDEs have been studied
since the 70s\, our understanding is far from complete\, particularly in t
he case where there are several distinct cell populations.\n\nAn important
open question is whether it is possible for two populations that were sep
arated at initial time to become mixed during the flow. For instance\, can
tumor cells get mixed into healthy cell regions? \n\nIn this talk\, I wi
ll show that it is possible to construct non-mixing solutions to these equ
ations. The key is to construct the Lagrangian flow map along the pressur
e gradient generated by the Porous Media Equation. The main obstruction i
s the fact that the pressure gradient is not sufficiently regular to apply
any generic theory for Lagrangian flows. To overcome this difficulty\, w
e develop a new argument combining features of the Porous Media Equation w
ith the quantitative Lagrangian flow theory of Crippa and De Lellis.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Lawrie (MIT)
DTSTART:20231107T210000Z
DTEND:20231107T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/175
DESCRIPTION:Title: Dynamics of kink clusters for scalar fields in dimens
ion 1+1\nby Andrew Lawrie (MIT) as part of UCLA analysis and PDE semin
ar\n\n\nAbstract\nI will present joint work with Jacek Jendrej. We conside
r classical scalar fields in dimension 1+1 with a symmetric double-well se
lf-interaction potential\, covering\, for example\, the phi-4 model and th
e sine-Gordon equation. Such equations admit non-trivial static solutions
called kinks and antikinks. We define a kink cluster to be a solution appr
oaching\, for large positive times\, a superposition of alternating kinks
and antikinks whose velocities converge to zero and mutual distances grow
to infinity. Our main result is a determination of the leading order asymp
totic behavior of any kink cluster. Our results are partially inspired by
the notion of "parabolic motions" in the Newtonian n-body problem. We expl
ain this analogy and its limitations. We also explain the role of kink clu
sters as universal profiles for the formation/annihilation of multi-kink c
onfigurations.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tongou Yang (UCLA)
DTSTART:20231003T220000Z
DTEND:20231003T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/176
DESCRIPTION:Title: Maximal planar Radon transform via local smoothing\,
and an elliptical maximal operator\nby Tongou Yang (UCLA) as part of U
CLA analysis and PDE seminar\n\n\nAbstract\nWe prove maximal operator boun
ds for a multi-parameter family of nondegnerate planar curves via local sm
oothing. Using a slight twist\, we are also able to obtain a sharp estimat
e on the unrotated elliptical maximal operator. This is joint work with Sh
aoming Guo and Mingfeng Chen.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jake Fillman (Texas State)
DTSTART:20231020T200000Z
DTEND:20231020T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/177
DESCRIPTION:Title: The Spectrum of the Unitary Almost-Mathieu Operator\nby Jake Fillman (Texas State) as part of UCLA analysis and PDE seminar
\n\nLecture held in Math 6943.\n\nAbstract\nWe introduce the unitary almos
t-Mathieu operator\, which is a family of one-dimensional quasi-periodic q
uantum walks obtained from an isotropic two-dimensional quantum walk in a
uniform magnetic field. This operator family exhibits several remarkable f
eatures: its spectrum is a Cantor subset of the unit circle\, and it exper
iences a metal-insulator transition as the strength of the hopping terms i
s varied. We will discuss background information\, the origins of the mode
l\, its interesting spectral features\, and some key ideas needed in proof
s of the main results. [Joint work with Christopher Cedzich\, Darren C. On
g\, and Zhenghe Zhang]\n\nNote: due to technical issues it may not be poss
ible to livestream this talk.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tristan Buckmaster (NYU)
DTSTART:20231114T210000Z
DTEND:20231114T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/178
DESCRIPTION:Title: Smooth Imploding Solutions for 3D Compressible Fluids
\nby Tristan Buckmaster (NYU) as part of UCLA analysis and PDE seminar
\n\n\nAbstract\nIn recent work by Merle-Rodnianski-Szeftel\, the authors c
onstructed smooth self-similar imploding solutions to the isentropic compr
essible Euler equations for almost every adiabatic exponent. The result wa
s also used to construct asymptotically self-similar imploding solutions t
o the compressible Navier-Stokes equations for the case of mildly decaying
density at infinity. The papers left open two natural questions: whether
exact self-similar imploding solutions exist for all adiabatic exponents a
nd whether singularities can form for the compressible Navier-Stokes equat
ions in the case of density constant at infinity. During this talk I will
present joint work with Gonzalo Cao-Labora and Javier Gomez-Serrano that w
ill resolve both of these questions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Garnett (UCLA)
DTSTART:20231107T220000Z
DTEND:20231107T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/179
DESCRIPTION:Title: H^1-BMO duality revisited\nby John Garnett (UCLA
) as part of UCLA analysis and PDE seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Engelstein (U. Minnesota)
DTSTART:20231205T210000Z
DTEND:20231205T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/180
DESCRIPTION:Title: The Robin problem on rough domains\nby Max Engels
tein (U. Minnesota) as part of UCLA analysis and PDE seminar\n\n\nAbstract
\nRobin boundary conditions for elliptic operators model a diffusion conta
ined by a semipermeable membrane (think oxygen being absorbed into the lun
g). Despite huge advances in understanding both the Neumann and Dirichlet
problems in rough domains\, the Robin problem is still mostly not understo
od. \n\nWe construct a ``Robin harmonic measure" for any elliptic operator
in a broad class of domains and prove the surprising fact that this measu
re is mutually absolutely continuous with respect to surface measure\, eve
n when the boundary of the domain is fractal. Along the way we will also a
ddress some older conjectures about partially reflecting Brownian motion.\
n\nThis is joint work with Guy David (Paris Saclay)\, Stefano Decio (IAS)\
, Svitlana Mayboroda (ETH/UMN) and Marco Michetti (Paris Saclay).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/180/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zaher Hani (U. Michigan)
DTSTART:20240227T210000Z
DTEND:20240227T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/181
DESCRIPTION:Title: Hilbert’s sixth problem for nonlinear waves\nby
Zaher Hani (U. Michigan) as part of UCLA analysis and PDE seminar\n\n\nAb
stract\nHilbert’s sixth problem asks for a mathematically rigorous justi
fication of the macroscopic laws of statistical physics from the microscop
ic laws of dynamics. The classical setting of this problem asks for the ju
stification of Boltzmann’s kinetic equation from Newtonian particle dyna
mics. This justification has been proven for short times\, starting with t
he work of Lanford in 1975\, but its long time justification remains one o
f the biggest open problems in kinetic theory.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/181/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linfeng Li (UCLA)
DTSTART:20231031T210000Z
DTEND:20231031T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/182
DESCRIPTION:Title: A regularity result for the free boundary compressibl
e Euler equations of a liquid\nby Linfeng Li (UCLA) as part of UCLA an
alysis and PDE seminar\n\n\nAbstract\nWe derive a priori estimates for the
compressible free boundary Euler equations in the case of a liquid withou
t surface tension. We provide a new weighted functional framework which le
ads to the improved regularity of the flow map by using the Hardy inequali
ty. One of main ideas is to decompose the initial density function. It is
worth mentioning that in our analysis we do not need the higher order wave
equation for the density.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jesse Gell-Redman (U. Melbourne)
DTSTART:20240109T220000Z
DTEND:20240109T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/183
DESCRIPTION:Title: Microlocal methods in scattering for nonlinear evolut
ion equations\nby Jesse Gell-Redman (U. Melbourne) as part of UCLA ana
lysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nI will disc
uss a new methodology for proving small data scattering for the nonlinear
Schrödinger equation\, which avoids the use of Strichartz estimates\, and
uses instead methods from microlocal analysis. This methodology is flexi
ble and can in principle be applied to massive wave propagation as in the
Klein-Gordon or massive Dirac equations. This is joint work with Andrew H
assell and Sean Gomes and with Dean Baskin and Moritz Doll\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/183/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruixiang Zhang (UC Berkeley)
DTSTART:20240305T210000Z
DTEND:20240305T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/184
DESCRIPTION:Title: A new conjecture to unify Fourier restriction and Boc
hner-Riesz\nby Ruixiang Zhang (UC Berkeley) as part of UCLA analysis a
nd PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nThe Fourier restri
ction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space m
apping properties of certain oscillatory integral operators. They both are
central in harmonic analysis\, are open in dimensions $\\geq 3$\, and not
ably have the same conjectured exponents. In the 1970s\, H\\"{o}rmander as
ked if a more general class of operators (known as H\\"{o}rmander type ope
rators) all satisfy the same $L^p$-boundedness as in the above two conject
ures. A positive answer to H\\"{o}rmander's question would resolve the abo
ve two conjectures and have more applications such as in the manifold sett
ing. Unfortunately H\\"{o}rmander's question is known to fail in all dimen
sions $\\geq 3$ by the work of Bourgain and many others. It continues to f
ail in all dimensions $\\geq 3$ even if one adds a ``positive curvature''
assumption which one does have in restriction and Bochner-Riesz settings.
Bourgain showed that in dimension $3$ one always has the failure unless a
derivative condition is satisfied everywhere. Joint with Shaoming Guo and
Hong Wang\, we generalize this condition to arbitrary dimension and call i
t ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz
by conjecturing that any H\\"{o}rmander type operator satisfying Bourgain
's condition should have the same $L^p$-boundedness as in those two conjec
tures. As evidence\, we prove that the failure of Bourgain's condition imm
ediately implies the failure of such an $L^p$-boundedness in every dimensi
on. We also prove that current techniques on the two conjectures apply equ
ally well in our conjecture and make some progress on our conjecture that
consequently improves the two conjectures in higher dimensions. I will tal
k about some history and some interesting components in our proof.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/184/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Royce Pineau (UC Berkeley)
DTSTART:20231128T190000Z
DTEND:20231128T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/185
DESCRIPTION:Title: Sharp Hadamard well-posedness for the incompressible
free boundary Euler equations\nby Benjamin Royce Pineau (UC Berkeley)
as part of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAb
stract\nI will talk about a recent preprint in which we establish an optim
al local well-posedness theory in $H^s$ based Sobolev spaces for the free
boundary incompressible Euler equations on a connected fluid domain. Some
components of this result include: (i) Local well-posedness in the Hadamar
d sense\, i.e.\, local existence\, uniqueness\, and the first proof of con
tinuous dependence on the data\, all in low regularity Sobolev spaces\; (i
i) Enhanced uniqueness: A uniqueness result which holds at the level of th
e Lipschitz norm of the velocity and the $C^{1\,\\frac{1}{2}}$ regularity
of the free surface\; (iii) Stability bounds: We construct a nonlinear fu
nctional which measures\, in a suitable sense\, the distance between two s
olutions (even when defined on different domains) and we show that this di
stance is propagated by the flow\; (iv) Energy estimates: We prove essenti
ally scale invariant energy estimates for solutions\, relying on a newl
y constructed family of refined elliptic estimates\; (v) Continuation crit
erion: We give the first proof of a continuation criterion at the same sca
le as the classical Beale-Kato-Majda criterion for the Euler equation on t
he whole space. Roughly speaking\, we show that solutions can be continued
as long as the velocity is in $L_T^1W^{1\,\\infty}$ and the free surface
is in $L_T^1C^{1\,\\frac{1}{2}}$\; (vi) A novel proof of the construction
of regular solutions. \n \n Our entire approach is in the Eulerian framew
ork and can be adapted to work in relatively general fluid domains. This i
s based on joint work with Mihaela Ifrim\, Daniel Tataru and Mitchell Tayl
or.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Katie Marsden (EPFL)
DTSTART:20240109T210000Z
DTEND:20240109T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/186
DESCRIPTION:Title: Global Solutions for the Half-Wave Maps Equation at C
ritical Regularity\nby Katie Marsden (EPFL) as part of UCLA analysis a
nd PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nIn this talk I wil
l discuss a small data-global wellposedness result for the three-dimension
al Half-Wave Maps equation in the critical Besov space. The Half-Wave Maps
equation is a nonlocal equation into the sphere\, with a close link to th
e better-known Wave Maps equation. The global wellposedness in dimensions
greater than or equal to 4 is already known\, however the 3 dimensional ca
se presents new difficulties due to the loss of a key Strichartz estimate.
To overcome this we use a simplified version of Tao’s gauge transformat
ion for the wave maps equation\, and a new argument involving commuting ve
ctor fields and Sterbenz’s improved Strichartz estimates for functions w
ith angular regularity. Naturally the use of these estimates comes at a co
st\, and we are forced to assume additional angular regularity on the init
ial data.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rena Badreddine (U. Paris-Saclay)
DTSTART:20240116T210000Z
DTEND:20240116T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/187
DESCRIPTION:Title: The Calogero-Sutherland Derivative NLS Equation\n
by Rena Badreddine (U. Paris-Saclay) as part of UCLA analysis and PDE semi
nar\n\nLecture held in MS 6627.\n\nAbstract\nWe consider a type of nonloca
l nonlinear derivative\nSchrödinger equation on the torus\, called the Ca
logero-Sutherland DNLS\nequation. We derive an explicit formula to the sol
ution of this\nnonlinear PDE. Moreover\, using the integrability tools\, w
e establish\nthe global well-posedness of this equation in all the Hardy-S
obolev\nspaces $H^s_+(\\mathbb{T})$\, $s\\geq 0$\, down to the critical re
gularity\nspace\, and under a mass assumption on the initial data for the\
nfocusing equation\, and for arbitrary initial data for the defocusing\neq
uation. Finally\, a sketch of the proof for extending the flow to the\ncri
tical regularity $L^2_+(\\mathbb{T})$ will be presented.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/187/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dominique Maldague (MIT)
DTSTART:20240109T190000Z
DTEND:20240109T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/188
DESCRIPTION:Title: Wave envelope estimates in Fourier restriction theory
\nby Dominique Maldague (MIT) as part of UCLA analysis and PDE seminar
\n\nLecture held in MS 6627.\n\nAbstract\nWave packet decomposition allows
us to express functions with restricted frequency support as a superposit
ion of wave packets (simpler functions which are localized in both space a
nd frequency)\, with one "active" wave packet per direction. I will explai
n the significance of a new type of inequality called a wave envelope esti
mate\, which provides detailed information about the possible overlap patt
erns of wave packets that maximize the L^p norm. Wave envelope estimates w
ere first introduced in the work of Guth-Wang-Zhang (GWZ) proving the shar
p L^4 square function estimate for the cone in R^3. Guth-Maldague subseque
ntly introduced a stopping time algorithm based on the amplitude of the fu
nction compared to its square function which yielded a refined version of
the GWZ wave envelope estimates. Our so-called amplitude-dependent wave en
velope estimate simultaneously implies both sharp decoupling and sharp squ
are function estimates. Applications include sharp small cap decoupling es
timates for the cone\, new estimates for the size of exceptional sets in t
he 3D restricted projections problem\, and a sharp multiplier-type problem
for the moment curve.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/188/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kyeongsik Nam (KAIST)
DTSTART:20240206T210000Z
DTEND:20240206T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/189
DESCRIPTION:Title: Universality of log-correlated fields\nby Kyeongs
ik Nam (KAIST) as part of UCLA analysis and PDE seminar\n\nLecture held in
MS 6627.\n\nAbstract\nLog-correlation naturally appears in diverse object
s such as random matrices and random discrete geometries. In this talk\, I
will give an overview on the theory of log-correlated fields and talk abo
ut recent progress on it. This is based on the joint work with Shirshendu
Ganguly.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/189/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Becker (Bonn)
DTSTART:20240123T210000Z
DTEND:20240123T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/190
DESCRIPTION:Title: A degree one Carleson operator along the paraboloid\nby Lars Becker (Bonn) as part of UCLA analysis and PDE seminar\n\nLect
ure held in MS 6627.\n\nAbstract\nCarleson proved in 1966 that the Fourier
series of any square integrable\nfunction converges pointwise to the func
tion\, by establishing boundedness\nof the maximally modulated Hilbert tra
nsform from L^2 into weak L^2. This\ntalk is about a generalization of his
result\, where the Hilbert transform\nis replaced by a singular integral
operator along a paraboloid.\nI will review the history of extensions of C
arleson's theorem\, and then\ndiscuss the two main ingredients needed to d
educe our result: sparse\nbounds for singular integrals along the parabolo
id\, and a square function\nargument relying on the geometry of the parabo
loid.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/190/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fei Xu (Jilin University)
DTSTART:20240123T200000Z
DTEND:20240123T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/191
DESCRIPTION:Title: On the Nonlinear Schr\\"odinger Equation with Quasi-p
eriodic Initial Data\nby Fei Xu (Jilin University) as part of UCLA ana
lysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nThis talk d
iscusses the (derivative) NLS with quasi-periodic initial data. It is dedi
cated to the memory of Thomas Kappeler\, who proposed this problem in 2021
. We first discuss recent progress on the Deift conjecture and almost peri
odic functions. Then we consider the (derivative) nonlinear Schr\\"odinger
equation with quasi-periodic initial data. Under (exponential) polynomial
decay assumption in the Fourier space for the initial Fourier data\, this
Cauchy problem has a unique local-in-time solution that retains the spati
al quasi-periodicity. To this end\, we use a new combinatorial analysis me
thod and introduce the so-called Feynman diagram. Finally\, some remarks o
n the global problem are given.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/191/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theo Sturm (Bonn)
DTSTART:20240130T210000Z
DTEND:20240130T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/192
DESCRIPTION:Title: Wasserstein Diffusion on Multidimensional Spaces\
nby Theo Sturm (Bonn) as part of UCLA analysis and PDE seminar\n\nLecture
held in MS 6627.\n\nAbstract\nGiven any closed Riemannian manifold $M$\, w
e construct a reversible diffusion process\non the space $P(M)$ of probabi
lity measures on $M$ that is\n• reversible w.r.t. the entropic measure $
P^\\beta$ on $P(M)$\, heuristically given as\n$$ dP^\\beta(\\mu) = \\frac{
1}{Z} e^{-\\beta \\mathrm{Ent}(\\mu|m)}\\ dP^0(\\mu)\;$$\n• associated w
ith a regular Dirichlet form with carre du champ derived from the Wasserst
ein\ngradient in the sense of Otto calculus\n$$ E_W (f) = \\lim \\inf_{\\t
ilde f \\to f} \\frac{1}{2} \\int_{P(M)} \\| \\nabla_W \\tilde f\\|^2(\\mu
)\\ dP^\\beta(\\mu).$$\n• non-degenerate\, at least in the case of the n
-sphere and the n-torus.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/192/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noemi David (Lyon)
DTSTART:20240402T200000Z
DTEND:20240402T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/193
DESCRIPTION:Title: Convergence rates for the incompressible limit of non
linear diffusion equations\nby Noemi David (Lyon) as part of UCLA anal
ysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nNowadays a v
ast literature is available on the Hele-Shaw or incompressible limit for n
onlinear degenerate diffusion equations. This problem has attracted a lot
of attention due to its applications to tissue growth and crowd motion mod
elling as it constitutes a way to link soft congestion (or compressible) m
odels to hard congestion (or incompressible) descriptions. Nevertheless\,
little is known about the rate of convergence of this asymptotic. In this
talk\, I will address the question of estimating the rate in the presence
of external drifts. In a joint work with Tomasz Dębiec and Benoit Pertham
e\, we computed the rate in a negative Sobolev norm for generic bounded po
tentials\, while in a work in progress with Alpár Mészáros and Filippo
Santambrogio\, we provide improved results in the 2-Wasserstein distance w
hich are global in time thanks to the contractivity property that holds fo
r strictly convex potentials. I will present these two results\, which hol
d both for the barotropic pressure law (hence the porous medium equation)
and for a singular pressure law with density constraints.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/193/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ovidiu-Neculai Avadanei (Berkeley)
DTSTART:20240430T200000Z
DTEND:20240430T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/194
DESCRIPTION:Title: Low regularity well-posedness for the generalized sur
face quasi-geostrophic front equation\nby Ovidiu-Neculai Avadanei (Ber
keley) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 6627
.\n\nAbstract\nWe consider the well-posedness of the generalized surface q
uasi-geostrophic (gSQG) front equation. By making use of the null structur
e of the equation\, we carry out a paradifferential normal form analysis i
n order to obtain balanced energy estimates\, which allows us to prove the
local well-posedness of the g-SQG front equation in the non-periodic case
at a low level of regularity (in\nthe SQG case\, this is only one half of
a derivative above scaling).\n\nIn addition\, we establish global well-po
sedness for small and localized rough initial data\, as well as modified s
cattering\, by using the testing by wave packet approach of Ifrim-Tataru.\
n\nThis is joint work with Albert Ai.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/194/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Antoine Mellet (U. Maryland)
DTSTART:20240319T200000Z
DTEND:20240319T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/196
DESCRIPTION:Title: On the regularity of optimal transportation potential
s with discrete measures\nby Antoine Mellet (U. Maryland) as part of U
CLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nWe c
onsider a Kantorovich potential associated to an optimal transportation pr
oblem between measures that are not necessarily absolutely continuous with
respect to the Lebesgue measure\, but are comparable to the Lebesgue meas
ure when restricted to balls with radius greater than some $\\delta>0$. Su
ch a framework is very natural in the context of the numerical computation
s of optimal maps\, which often involves approximating "nice" measures by
sums of Dirac masses.\n\nWe will present some recent results (collaboratio
n with P.E. Jabin and M. Molina) which extend the classical regularity the
ory of optimal transportation to this framework. In particular\, we establ
ish both Hölder and Sobolev regularity results for Kantorovich potentials
up to some critical length scale depending on $\\delta$.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/196/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Hitrik (UCLA)
DTSTART:20240409T200000Z
DTEND:20240409T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/197
DESCRIPTION:Title: Magic angles and classically forbidden regions for tw
isted bilayer graphene\nby Michael Hitrik (UCLA) as part of UCLA analy
sis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nMagic angles
are a topic of current interest in condensed matter physics and refer to a
remarkable theoretical (Bistritzer--MacDonald\, 2011) and experimental (J
arillo-Herrero et al\, 2018) discovery: two sheets of graphene twisted by
a certain (magic) angle display unusual electronic properties\, such as su
perconductivity. In this talk\, we shall discuss a simple periodic Hamilto
nian describing the chiral limit of twisted bilayer graphene (Tarnopolsky-
Kruchkov-Vishwanath\, 2019)\, whose spectral properties are thought to det
ermine which angles are magical. We show that the corresponding eigenfunct
ions decay exponentially in suitable geometrically determined regions as t
he angle of twisting decreases\, which can be viewed as a form of semiclas
sical analytic hypoellipticity. This is joint work with Maciej Zworski.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/197/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shi Zhuo Looi (Caltech)
DTSTART:20240213T210000Z
DTEND:20240213T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/198
DESCRIPTION:Title: Fourier-based and physical approaches to late-time as
ymptotics of hyperbolic PDE\nby Shi Zhuo Looi (Caltech) as part of UCL
A analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nWe pre
sent an algorithm for deriving the precise and sharp asymptotics of linear
and non-linear wave equations on asymptotically flat spacetimes\, includi
ng non-stationary spacetimes without any spherical symmetry assumptions. S
ome features of our proofs include integrated local energy decay and a wei
ghted version thereof\, spectral-theoretic methods involving resolvent exp
ansions near zero energy\, and a method called geometric singular analysis
\, which distinguishes between different scales of the spacetime.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/198/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jani Virtanen (U. Reading)
DTSTART:20240312T200000Z
DTEND:20240312T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/199
DESCRIPTION:Title: Asymptotics of block Toeplitz determinants with piece
wise continuous symbols\nby Jani Virtanen (U. Reading) as part of UCLA
analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nA Toepl
itz matrix can be easily defined as a matrix constant along the parallels
to the main diagonal given by the Fourier coefficients of an integrable fu
nction (referred to as the symbol) on the unit circle. The study of the de
terminants of Toeplitz matrices dates back to Szegő\, who described their
asymptotic behavior for sufficiently smooth symbols in 1915 and 1952. The
latter result was generalized to the case of matrix-valued symbols by Wid
om in the 1970s using operator theoretic methods. In the scalar case\, the
asymptotic behavior of Toeplitz determinants with Fisher-Hartwig symbols\
, which allow for zeros\, (integrable) singularities\, discontinuities\, a
nd nonzero winding numbers\, was described completely by Deift\, Its\, and
Krasovsky in 2011 using the Riemann-Hilbert approach. In this talk\, I di
scuss the case of matrix-valued symbols that have finitely many discontinu
ities and some of their applications\, such as the study of entanglement e
ntropy in quantum spin chain models. The approach is largely based on oper
ator theoretic methods\, and it requires a new localization theorem for To
eplitz determinants and a new method of computing the Fredholm index of To
eplitz operators with piecewise continuous matrix-valued symbols. Joint wo
rk with Estelle Basor and Torsten Ehrhardt.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/199/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiajie Chen (Courant)
DTSTART:20240312T210000Z
DTEND:20240312T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/200
DESCRIPTION:Title: Nearly self-similar blowup of the slightly perturbed
homogeneous Landau equation with very soft potentials\nby Jiajie Chen
(Courant) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 6
627.\n\nAbstract\nWhether the Landau equation can develop a finite time si
ngularity is an important open problem in kinetic equations. In this talk\
, we will first discuss several similarities between the Landau equation a
nd some incompressible fluids equations. Then we will focus on the slightl
y perturbed homogeneous Landau equation with very soft potentials\, where
we increase the nonlinearity from $ c(f) f$ in the Landau equation to $\\a
lpha c(f) f$ with $\\alpha>1$. For $\\alpha > 1 $ and close to $1$\, we es
tablish finite time nearly self-similar blowup from some smooth non-negati
ve initial data\, which can be radially symmetric or non-radially symmetri
c. The blowup results are sharp as the homogeneous Landau equation $(\\alp
ha=1)$ is globally well-posed\, which was recently established by Guillen
and Silvestre. The proof builds on our previous framework on sharp blowup
results of the De Gregorio model with nearly self-similar singularity to o
vercome the diffusion. Our results shed light on potential singularity for
mation in the inhomogeneous setting\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/200/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandria Rose (ANU)
DTSTART:20240425T200000Z
DTEND:20240425T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/201
DESCRIPTION:Title: Lattice Covering Densities and Additive Combinatorics
\nby Alexandria Rose (ANU) as part of UCLA analysis and PDE seminar\n\
nLecture held in MS 6627.\n\nAbstract\nThe well-known Lattice Covering Pro
blem asks for the most optimal way to cover the space $\\mathbb{R}^n$\, $n
\\geq 2$\, by using copies of an Euclidean ball centered at points of a g
iven lattice. More precisely\, consider a closed Euclidean ball $B$ and a
lattice $L \\subset \\mathbb{R}^d$\, we say that $L$ is a covering lattice
for $B$ if\n$$ \\mathbb{R}^n = L + B \\tag{1}$$\nThe {\\emph{covering den
sity} $\\displaystyle \\Theta (L)$ of whole space $\\mathbb{R}^n$ is defin
ed as the minimal volume of a closed Euclidean Ball $B$ for which (1) hold
s. Define\n$$\\Theta_n := \\inf \\left \\{ \\Theta (L): L \\text{ is a
lattice in $\\mathbb{R}^n$ of covolume one} \\right \\} $$\nto be the mini
mal density of lattice coverings of $\\mathbb{R}^n$. Where the covolume of
$L$ is the volume of its fundamental parallepipeds (sometimes refer as th
e determinant of $L$). Thus the Lattice Covering Problem asks for the best
upper bound for $\\displaystyle \\Theta_n$.\nSo far\, this problem has on
ly been studied geometrically using Kakeya-type methods to obtain results
for convex bodies in place of balls. In this talk\, we make a connection b
etween lattice covering densities and additive combinatorics\, and conside
r the more general setting of approximate groups and sets with low doublin
g or high additive energy. This is joint work with Francisco Romero Acosta
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/201/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zane Li (NCSU)
DTSTART:20240507T210000Z
DTEND:20240507T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/202
DESCRIPTION:Title: Mixed norm decoupling for paraboloids\nby Zane Li
(NCSU) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 662
7.\n\nAbstract\nIn this talk we prove the sharp mixed norm (l^2\, L^{q}_{t
}L^{r}_{x}) decoupling estimates for the paraboloid in d + 1 dimensions. T
his is joint work with Shival Dasu\, Hongki Jung\, and José Madrid.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/202/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allen Wu (Oklahoma)
DTSTART:20240416T200000Z
DTEND:20240416T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/203
DESCRIPTION:Title: The Scattering Problem of the Intermediate Long Wave
Equation\nby Allen Wu (Oklahoma) as part of UCLA analysis and PDE semi
nar\n\nLecture held in MS 6627.\n\nAbstract\nThe Intermediate Long Wave eq
uation (ILW) describes long internal gravity waves in stratified fluids. K
odama\, Ablowitz and Satsuma discovered the formal complete integrability
of ILW and formulated inverse scattering transform solutions. If made rigo
rous\, the inverse scattering method will provide powerful tools for asymp
totic analysis of ILW. In this talk\, I will present some recent results o
n the ILW direct scattering problem. In particular\, a Lax pair formulatio
n is clarified\, and the spectral theory of the Lax operators can be studi
ed. Existence and uniqueness of scattering states are established for smal
l interaction potential. The scattering matrix can then be constructed fro
m the scattering states. The solution is related to the theory of analytic
functions on a strip. This is joint work with Peter Perry.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/203/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iván Moyano (Université Côte-d'Azur)
DTSTART:20240507T210000Z
DTEND:20240507T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/204
DESCRIPTION:Title: CANCELLED\nby Iván Moyano (Université Côte-d'A
zur) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\
nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/204/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Inwon Kim (UCLA)
DTSTART:20240521T200000Z
DTEND:20240521T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/205
DESCRIPTION:Title: Global existence for the supercooled Stefan problem\nby Inwon Kim (UCLA) as part of UCLA analysis and PDE seminar\n\nLectur
e held in MS 6627.\n\nAbstract\nWe will discuss the supercooled Stefan pro
blem in space dimensions higher than 1. We will show global-time existence
for general initial data\, the main ideas based on its variational formul
ation\, and open questions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/205/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaume de Dios (ETH Zurich)
DTSTART:20240509T190000Z
DTEND:20240509T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/206
DESCRIPTION:Title: Complexity lower bounds for log-concave sampling\
nby Jaume de Dios (ETH Zurich) as part of UCLA analysis and PDE seminar\n\
nLecture held in MS 5203.\n\nAbstract\nGiven a density rho(x)\, how does o
ne effectively generate samples from a random variable with this density r
ho? Variations of this question arise in most computational fields\, from
Statistics to Computer Science to computational Physics.\n \nSignificant e
ffort has been devoted to designing more and more efficient algorithms\, r
anging from relatively simple algorithms\, such as rejection sampling\, to
increasingly sophisticated such as langevin-based or diffusion based mode
ls.\n\nThis talk will focus on the model case in which log-density is a st
rongly concave smooth function. We will discuss some of the most widely us
ed algoritms\, and study fundamental limitations to the problem by finding
universal complexity bounds that no algorithm can beat. The construction
of thse tight bounds for fixed dimension will follow a modification of the
classical Perron's sprouting construction.\n \nBased on joint work with S
inho Chewi\, Jerry Li\, Chen Lu and Shyam Narayanan.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/206/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayla Gafni (U. Mississippi)
DTSTART:20240514T200000Z
DTEND:20240514T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/207
DESCRIPTION:Title: Counting number fields and polynomials\nby Ayla G
afni (U. Mississippi) as part of UCLA analysis and PDE seminar\n\nLecture
held in MS 6627.\n\nAbstract\nNumber fields are a central topic of number
theory\, and yet they are surprisingly difficult to count. We will discus
s the history of progress toward counting number fields\, and give a new b
ound on number fields of degree less than 94. The improved bound is achie
ved through a combination of harmonic analysis and modified sieve methods.
We'll also discuss how similar techniques have been useful in bounding t
he exceptional set in Hilbert's irreducibility theorem\; that is\, at coun
ting the number of irreducible polynomials without full Galois group.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/207/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaime Hernandez Palacios (U. Mississippi)
DTSTART:20240528T200000Z
DTEND:20240528T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/208
DESCRIPTION:Title: Gaps between zeros of zeta and L-functions of high de
gree\nby Jaime Hernandez Palacios (U. Mississippi) as part of UCLA ana
lysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nThere is a
great deal of evidence\, both theoretical and experimental\, that the dist
ribution of zeros of zeta and L-functions can be modeled using statistics
of eigenvalues of random matrices from classical compact groups. In partic
ular\, we expect that there are arbitrarily large and small normalized gap
s between the ordinates of (high) zeros zeta and L-functions. Previous res
ults are known for zeta and L-functions of degrees 1 and 2. We discuss som
e new results for higher degrees\, including Dedekind zeta-functions assoc
iated to Galois extensions of and principal automorphic L-functions.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/208/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stan Palasek (Princeton University)
DTSTART:20240502T190000Z
DTEND:20240502T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/209
DESCRIPTION:Title: Critical non-uniqueness in a shell model of the Navie
r-Stokes equations\nby Stan Palasek (Princeton University) as part of
UCLA analysis and PDE seminar\n\nLecture held in MS 5203.\n\nAbstract\nAn
outstanding question of the theory of the incompressible Navier-Stokes equ
ations is whether solutions are unique in the Leray class\, i.e.\, the wea
k solutions that dissipate energy. There is compelling numerical evidence
due to Jia and Sverak of a reflection symmetry-breaking phenomenon leading
to non-uniquness. In this talk we propose a new non-uniqueness scenario b
ased on breaking of the (discrete) scaling symmetry\, demonstrated in a sh
ell model of the Navier-Stokes equations first formulated by Obukhov. We c
onstruct data in a critical space that gives rise to distinct Leray soluti
ons which are approximately discretely self-similar and ``smooth'' (in the
sense of exponentially decaying energy spectrum) for positive times.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/209/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soonsik Kwon (KAIST)
DTSTART:20240529T000000Z
DTEND:20240529T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/210
DESCRIPTION:Title: Finite time blow-up construction of Calogero-Moser de
rivative nonlinear Schrodinger equations\nby Soonsik Kwon (KAIST) as p
art of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\nAbstract
: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/210/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrej Zlatos (UCSD)
DTSTART:20241022T200000Z
DTEND:20241022T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/211
DESCRIPTION:Title: Stable regime singularity for the Muskat problem\
nby Andrej Zlatos (UCSD) as part of UCLA analysis and PDE seminar\n\nLectu
re held in MS 6627.\n\nAbstract\nThe Muskat problem on the half-plane mode
ls motion of an interface between two fluids of distinct densities in a po
rous medium that sits atop an impermeable layer\, such as oil and water in
an aquifer above bedrock. We develop a local well-posedness theory for t
his model in the stable regime (lighter fluid above the heavier one)\, whi
ch includes considerably more general fluid interface geometries than even
existing whole plane results and allows the interface to touch the bottom
. The latter applies to the important scenario of the heavier fluid invad
ing a region occupied by the lighter fluid along the impermeable layer. W
e also show that finite time singularities do arise in this setting\, incl
uding from arbitrarily small smooth initial data\, by obtaining maximum pr
inciples for the height\, slope\, and potential energy of the fluid interf
ace.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/211/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Munoz (UCLA)
DTSTART:20241029T200000Z
DTEND:20241029T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/213
DESCRIPTION:Title: Free boundary regularity and support propagation in m
ean field games and optimal transport\nby Sebastian Munoz (UCLA) as p
art of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstra
ct\nIn this talk\, we present recent results on the regularity of first-or
der mean field games systems. We focus on systems where the initial densit
y is a compactly supported function on the real line. Our results show tha
t the solution is smooth in regions where the density is strictly positive
and that the density itself is globally continuous. Additionally\, the sp
eed of propagation is determined\n\nby the behavior of the cost function f
or small densities. When the coupling\nis entropic\, we demonstrate that t
he support of the density propagates with\ninfinite speed. On the other ha
nd\, when f(m) = m^θ with θ > 0\, we prove\nthat the speed of propagatio
n is finite. In this case\, we establish that under\na natural non-degener
acy assumption\, the free boundary is strictly convex and\nenjoys C^{1\,1}
regularity. We also establish sharp estimates on the speed of sup-\nport
propagation and the rate of long time decay for the density. Our methods a
re based on the analysis of an elliptic equation satisfied by the flow of
optimal trajectories. The results also apply to mean field planning proble
ms\, characterizing the structure of minimizers of a class of optimal tran
sport problems with congestion.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/213/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hugo Lavenant (Bocconi U.)
DTSTART:20241119T210000Z
DTEND:20241119T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/214
DESCRIPTION:Title: CANCELLED\nby Hugo Lavenant (Bocconi U.) as part
of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\n
CANCELLED\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/214/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Duvan Cardona Sanchez (U. Ghent)
DTSTART:20240910T200000Z
DTEND:20240910T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/215
DESCRIPTION:Title: Regularity properties of Fourier integral operators w
ith complex phases\nby Duvan Cardona Sanchez (U. Ghent) as part of UCL
A analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nIn thi
s talk we discuss the regularity properties of Fourier integral operators
with real-valued phases\, including\, the L^p result proved in the early 9
0´s by A. Seeger\, C. Sogge and E. Stein\, the weak (1\,1) estimate prove
d by T. Tao and the L^p regularity results due to M. Ruzhansky in the sett
ing of complex phases. We then discuss how these techniques can be combine
d to establish the weak (1\,1) inequality for Fourier integral operators w
ith complex phases.\nJoint work with Michael Ruzhansky.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/215/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Leng (UCLA)
DTSTART:20241008T200000Z
DTEND:20241008T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/216
DESCRIPTION:Title: Vinogradov's Theorem for primes with missing digits\nby James Leng (UCLA) as part of UCLA analysis and PDE seminar\n\nLectu
re held in MS 6627.\n\nAbstract\nIn the 1930s\, Vinogradov showed that all
sufficiently large odd numbers can be written as the sum of three primes.
In 2015\, Maynard showed that g is a large enough base and b is a digit\,
then there are infinitely many primes whose base g expansion doesn't cont
ain the digit b. In this talk\, we will discuss a synthesis of their resul
ts: that every sufficiently large odd number can be written as the sum of
three primes whose base g expansion doesn't contain the digit b. The proof
of this result is naturally a combination of the techniques of Vinogradov
and Maynard\, and we will discuss the crucial ingredients present in thes
e works\, and how it can be adapted to our problem. This is joint work wit
h Mehtaab Sawhney.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/216/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tongou Yang (UCLA)
DTSTART:20241015T200000Z
DTEND:20241015T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/217
DESCRIPTION:Title: Two principles of decoupling\nby Tongou Yang (UCL
A) as part of UCLA analysis and PDE seminar\n\n\nAbstract\nWe put forward
two principles of decoupling\, aiming to provide a new algebraic approach
of reducing decoupling for new manifolds to decoupling for known manifolds
.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/217/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Haoren Xiong (UCLA)
DTSTART:20241112T210000Z
DTEND:20241112T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/218
DESCRIPTION:Title: Semiclassical asymptotics for Bergman projections
\nby Haoren Xiong (UCLA) as part of UCLA analysis and PDE seminar\n\nLectu
re held in MS 6627.\n\nAbstract\nIn this talk\, we discuss the semiclassic
al asymptotics for Bergman kernels in exponentially weighted spaces of hol
omorphic functions. We will first review various approaches to the constru
ction of asymptotic Bergman projections\, for smooth weights and for real
analytic weights. We shall then explore the case of Gevrey weights\, which
can be thought of as the interpolating case between the real analytic and
smooth weights. In the case of Gevrey weights\, we show that Bergman kern
el can be approximated in certain Gevrey symbol class up to a Gevrey type
small error\, in the semiclassical limit. We will also introduce some micr
olocal analysis tools in the Gevrey setting\, including Borel's lemma for
symbols and complex stationary phase lemma. This talk is based on joint wo
rk with Hang Xu.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/218/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huynh\, Manh Khang (Georgia Tech)
DTSTART:20241105T210000Z
DTEND:20241105T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/219
DESCRIPTION:Title: Sparsity of Fourier mass of passively advected scalar
s in the Batchelor regime\nby Huynh\, Manh Khang (Georgia Tech) as par
t of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract
\nIn this paper we propose a general dynamical mechanism that can lead to
the failure of the Batchelor's mode-wise power spectrum law in passive sca
lar turbulence and hyperbolic dynamics\, while the cumulative law remains
true. Of technical interest\, we also employ a novel method of power spect
ral variance to establish an exponential radial shell law for the Batchelo
r power spectrum. An accessible explanation of the power spectrum laws via
harmonic analysis is also given.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/219/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Wu (UCLA)
DTSTART:20241126T210000Z
DTEND:20241126T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/220
DESCRIPTION:Title: Mean Field Limit for Congestion Dynamics in One Dimen
sion\nby Jeremy Wu (UCLA) as part of UCLA analysis and PDE seminar\n\n
Lecture held in MS 6627.\n\nAbstract\nIn this talk\, I will present recent
joint work with Inwon Kim and Antoine Mellet in which we derive a model f
or congested transport (a PDE at a macroscopic scale) from particle dynami
cs (a system of ODEs at the microscopic scale). Such PDEs appear very natu
rally in the description of crowd motion\, tumour growth\, and general agg
regation phenomena. We begin with a system where the particle trajectories
evolve according to a gradient flow constrained to some finite distance o
f separation from each other. This constraint leads to a Lagrange multipli
er which\, in the mean field limit (infinite number of particles)\, genera
tes a pressure variable to enforce the hard-congestion constraint. Our res
ults are confined to one spatial dimension wherein we rely on both the Eul
erian and Lagrangian perspectives for the continuum limit.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/220/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Javier Gomez-Serrano (Brown)
DTSTART:20250121T210000Z
DTEND:20250121T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/221
DESCRIPTION:Title: Machine Learning in PDE: Discovering new\, unstable s
olutions\nby Javier Gomez-Serrano (Brown) as part of UCLA analysis and
PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nIn this talk I will
explain several recent results combining machine learning techniques and m
ore traditional mathematics. The overarching theme is the interplay betwee
n modern (ML) and classical methods in order to discover new solutions of
certain PDE with low or very low numerical error. I will also outline how
to turn the numerical approximate solutions into a rigorous proof via comp
uter-assisted methods\, leading in some cases to singularity formation\, o
r to the existence of certain special solutions (e.g. traveling waves). In
particular\, unstable solutions are now amenable to be discovered. While
the motivation is originally coming from fluid mechanics' equations such a
s Euler or Navier-Stokes some of the methods can be tuned to other PDE.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/221/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shaoming Guo (Nankai University)
DTSTART:20250114T210000Z
DTEND:20250114T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/222
DESCRIPTION:Title: Oscillatory integral operators and related Kakeya pro
blems\nby Shaoming Guo (Nankai University) as part of UCLA analysis an
d PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nWe will discuss Hor
mander's oscillatory integral operators\, which can be viewed as perturbat
ions of the standard Fourier extension operators. We will also discuss osc
illatory integral operators on manifolds\, which are special cases of Horm
ander's operators. Sharp L^p bounds for these operators have interesting c
onnections to curvature properties of the underlying manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/222/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Yin (UCLA)
DTSTART:20250311T200000Z
DTEND:20250311T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/223
DESCRIPTION:Title: The delocalization conjecture for random band matrice
s\nby Jun Yin (UCLA) as part of UCLA analysis and PDE seminar\n\nLectu
re held in MS 6627.\n\nAbstract\nIn this talk\, we present a new proof of
the delocalization conjecture for random band matrices. The conjecture ass
erts that for an $N\\times N$ random matrix with bandwidth $W$\, the eigen
vectors in the bulk of the spectrum are delocalized provided that $W \\gg
N^{1/2}$. Moreover\, in this regime\, the eigenvalue distribution aligns w
ith that of Gaussian random ensembles (i.e.\, GOE or GUE). Our proof emplo
ys a novel loop hierarchy method and leverages the sum-zero property\, a c
oncept that was instrumental in the previous work on high-dimensional rand
om matrices. \n\nThis work is a joint collaboration with H.T. Yau.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/223/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patrick Flynn (UCLA)
DTSTART:20250128T210000Z
DTEND:20250128T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/224
DESCRIPTION:Title: Negative regularity mixing of passive scalars in stoc
hastic fluid mechanics\nby Patrick Flynn (UCLA) as part of UCLA analys
is and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nConsider a pas
sive scalar advected by a random vector field on a compact manifold\, such
as the solution to the 2D stochastic Navier-Stokes equation on a periodic
box. In this talk\, I discuss my work with J. Bedrossian and S. Punshon-S
mith\, where we show that if the passive scalar is initially in some negat
ive regularity Sobolev space\, then it will decay exponentially in the sam
e space (in expectation). We prove this result using techniques from dynam
ical systems theory and semiclassical analysis. Going forward\, we hope to
apply this result to the problem of turbulence.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/224/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianhui Li (Northwestern)
DTSTART:20250204T210000Z
DTEND:20250204T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/225
DESCRIPTION:Title: Weighted Fourier restriction estimates\nby Jianhu
i Li (Northwestern) as part of UCLA analysis and PDE seminar\n\nLecture he
ld in MS 6627.\n\nAbstract\nMotivated by the recent advances on Carleson
’s problem regarding the everywhere convergence of solutions of linear S
chr\\”odinger equation to its initial condition\, we study the L^2 to L^
p(B_R) maximal Schr\\”odinger estimates by establishing various weighted
Fourier restriction estimates. In this talk\, I will review of the method
of polynomial partitioning\, which has led to major progress to the unwei
ghted Fourier restriction conjecture introduced by Guth\, and outline how
we adapted the method to the weighted problem. This is joint work with Xiu
min Du and Terry Harris.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/225/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wojciech Ozanski (Florida State University)
DTSTART:20250204T220000Z
DTEND:20250204T230000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/226
DESCRIPTION:Title: Instantaneous continuous loss of regularity for the S
QG equation\nby Wojciech Ozanski (Florida State University) as part of
UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nTh
e issue of loss of regularity of unique solutions to the 3D incompressible
Euler equations is an important open question of fluid mechanics\, and is
closely related to the emergence of turbulence. We will discuss recent re
sults regarding loss of regularity of solutions of the 2D and 3D Euler equ
ations\, and of the surface quasi-geostrophic equations (SQG)\, which is a
well-established 2D model equation of the 3D Euler equations. We will dis
cuss a result of continuous-in-time loss of Sobolev regularity of solution
s to the SQG equation. Namely\, given $s\\in (3/2\,2)$ and $\\varepsilon
>0$\, we will describe a construction of a compactly supported initial dat
a $\\theta_0$ such that $\\| \\theta_0 \\|_{H^s}\\leq \\varepsilon$ and th
ere exist $T>0$\, $c>0$ and a local-in-time solution $\\theta$ of the SQG
equation such that $ \\theta (\\cdot \,t )$ belongs to ${H^{s/(1+ct)}}$ an
d does not belong to any other ${H^\\beta }$\, where $\\beta > s/(1+ct)$.
Moreover $\\theta$ is continuous and differentiable on $\\R^2\\times [0\,T
]$\, and is unique among all solutions with initial condition $\\theta_0$
which belong to $C([0\,T]\;H^{1+\\alpha })$ for any $\\alpha >0$.\nThis is
the first result of this kind in incompressible fluid mechanics. It is al
so the first ill-posedness result in the supercritical regime which has co
mpact support in space.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/226/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jincheng Yang (IAS)
DTSTART:20250415T200000Z
DTEND:20250415T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/227
DESCRIPTION:Title: Energy dissipation near the outflow boundary in the v
anishing viscosity limit\nby Jincheng Yang (IAS) as part of UCLA analy
sis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nWe consider t
he incompressible Navier-Stokes and Euler equations in a bounded domain wi
th non-characteristic boundary condition\, and study the energy dissipatio
n near the outflow boundary in the zero-viscosity limit. We show that in a
general setting\, the energy dissipation rate is proportional to $\\bar U
\\bar V ^2$\, where $\\bar U$ is the strength of the suction and $\\bar V
$ is the tangential component of the difference between Euler and Navier-S
tokes on the outflow boundary. Moreover\, we show that the enstrophy withi
n a layer of order $\\nu / \\bar U$ is comparable with the total enstrophy
. The rate of enstrophy production near the boundary is inversely proporti
onal to $\\nu$. This is based on joint work with Vincent Martinez\, Anna M
azzucato\, and Alexis Vasseur.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/227/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bjoern Bringmann (Princeton)
DTSTART:20250429T200000Z
DTEND:20250429T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/228
DESCRIPTION:Title: Global well-posedness of the stochastic Abelian-Higgs
equations in two dimensions\nby Bjoern Bringmann (Princeton) as part
of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\n
There has been much recent progress on the local solution theory for geome
tric singular SPDEs. However\, the global\ntheory is still largely open. I
n this talk\, we discuss the global well-posedness of the stochastic Abeli
an-Higgs model in two \ndimension\, which is a geometric singular SPDE ari
sing from gauge theory. The proof is based on a new covariant approach\, \
nwhich consists of two parts: First\, we introduce covariant stochastic ob
jects\, which are controlled using covariant heat kernel estimates. \nSeco
nd\, we control nonlinear remainders using a covariant monotonicity formul
a\, which is inspired by earlier work of Hamilton.\n\nThis is joint work w
ith S. Cao.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/228/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linfeng Li (UCLA)
DTSTART:20250225T210000Z
DTEND:20250225T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/229
DESCRIPTION:Title: Nodal set and observability estimate for Gevrey regul
ar parabolic equations\nby Linfeng Li (UCLA) as part of UCLA analysis
and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nIn this talk\, I
will discuss our recent work on the size of nodal set and an observability
estimate for Gevrey regular parabolic equations. We provide an upper boun
d of the nodal set as a function of time\, and the dependence agrees with
a sharp upper bound when the coefficients are analytic. Secondly\, we prov
e an observability inequality from measurable set for general Gevrey regul
ar functions. Moreover\, we provide applications to the sum of Laplace eig
enfunctions on a Riemannian manifold that belongs to the Gevrey class\, as
well as the solution of a one-dimensional parabolic equation of arbitrary
order with Gevrey coefficients.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/229/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Rybkin (U. Alaska Fairbanks)
DTSTART:20250318T210000Z
DTEND:20250318T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/230
DESCRIPTION:Title: Embedded bound states and bounded positon solutions o
f the Korteweg-de Vries equation\nby Alexei Rybkin (U. Alaska Fairbank
s) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\
nAbstract\nIn the context of the full line Schrodinger equation\, we revis
it the binary Darboux transformation (double commutation method) which ins
erts or removes any number of positive eigenvalues embedded into the absol
utely continuous spectrum without altering the rest of scattering data. We
then show that embedded eigenvalues produce an additional explicit term i
n the KdV solution. This term looks similar to multi-soliton solution and
describes waves traveling in the direction opposite to solitons. It also r
esembles the known formula for (singular) multi-positon solutions but rema
ins bounded\, which answers in the affirmative Matveev's question about ex
istence of bounded positons.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/230/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moon-Jin Kang (KAIST)
DTSTART:20250305T000000Z
DTEND:20250305T010000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/231
DESCRIPTION:Title: Well-posedness of small BV solutions to the isentropi
c Euler equations by Inviscid limits from Navier-Stokes (joint with USC)\nby Moon-Jin Kang (KAIST) as part of UCLA analysis and PDE seminar\n\nL
ecture held in MS 6627.\n\nAbstract\nIn the realm of mathematical fluid dy
namics\, a formidable challenge lies in establishing inviscid limits from
the Navier-Stokes equations to the Euler equations\, wherein physically ad
missible solutions can be discerned. The pursuit of solving this intricate
problem\, particularly concerning singular solutions\, persists in both c
ompressible and incompressible scenarios. For the compressible flows\, the
well-posedness of Cauchy problem for compressible Euler system from invis
cid limit of Navier-Stokes remains is a major open problem even within mon
o-dimensional framework with small BV initial data. In this talk\, I will
present a resolution of this problem in the 1D isentropic case. We will sh
ow the global well-posedness of entropy solutions with small BV initial da
ta within the broader class of inviscid limits of the associated Navier-St
okes equations with locally bounded energy initial values. The proof is ba
sed on the three main methodologies: the modified front tracking algorithm
\; the a-contraction with shifts\; the method of compensated compactness.
This is a joint work with Geng Chen (U. Kansas) and Alexis Vasseur (UT-Aus
tin).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/231/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Bedrossian (UCLA)
DTSTART:20250305T010000Z
DTEND:20250305T020000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/232
DESCRIPTION:Title: What is a mathematical theory of turbulence? (joint w
ith USC)\nby Jacob Bedrossian (UCLA) as part of UCLA analysis and PDE
seminar\n\nLecture held in MS 6627.\n\nAbstract\nStatistical theories of t
urbulence are of fundamental importance in many applications from engineer
ing to weather and climate prediction. However\, there is currently no pre
dictive theory which starts only from the 2D or 3D Navier-Stokes equations
and accurately matches the observations\, and certainly nothing of this k
ind which is mathematically rigorous. In this talk I will explain how to p
hrase the basic predictions of the statistical theories of turbulence such
as K41 theory as concise\, mathematically rigorous conjectures for statis
tically stationary solutions of the Navier-Stokes equations subjected to s
tochastic forcing in a periodic box. These remain far out of reach\, so I
will then discuss some work done by my collaborators on I on different rel
ated problems with the goal of building up tools and understanding related
phenomena in simpler contexts. I will discuss recent work with Alex Blume
nthal\, Keagan Callis and Kyle Liss on energy dissipation in degenerately
damped nonlinear SDEs and older work with Alex Blumenthal\, and Sam Punsho
n-Smith on Batchelor-regime passive scalar turbulence (where we *can* buil
d a mathematically rigorous theory that matches experiments and observatio
ns of temperature and salinity variations in the ocean).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/232/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kerrek Stinson (UCLA)
DTSTART:20250520T200000Z
DTEND:20250520T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/233
DESCRIPTION:Title: Variational methods in fracture mechanics\nby Ker
rek Stinson (UCLA) as part of UCLA analysis and PDE seminar\n\nLecture hel
d in MS 6627.\n\nAbstract\nThe Griffith criterion says that the energy to
crack a brittle elastic material is proportional to the length of the crac
k. Understanding minimizers of the energy requires unraveling the complex
interplay of bulk (elastic) and surface (crack) energies in the vectorial
setting of linear elasticity. We discuss a compactness result based on con
centration-compactness and existence of strong solutions. Further\, in dim
ension 2\, we prove that the crack of a minimizer is given by a Hölder s
urface outside of a singular set of points with dimension strictly less th
an 1\, analogous to results for the scalar-valued Mumford-Shah functional.
\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/233/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Max Xu (MSRI)
DTSTART:20250415T210000Z
DTEND:20250415T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/234
DESCRIPTION:Title: Recent progress in the study of random multiplicative
functions\nby Max Xu (MSRI) as part of UCLA analysis and PDE seminar\
n\nLecture held in MS 6627.\n\nAbstract\nRandom multiplicative functions (
RMF) are objects studied by both analytic number theorists and probabilist
s in recent years. They are probabilistic models motivated by arithmetic p
roblems in number theory. I will give an introduction and update on this r
apidly developing area. Part of the talk is based on a joint work with A.
Harper and K. Soundararajan.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/234/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhongkai Tao (Berkeley)
DTSTART:20250513T200000Z
DTEND:20250513T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/235
DESCRIPTION:Title: How to solve an undetermined PDE system?\nby Zhon
gkai Tao (Berkeley) as part of UCLA analysis and PDE seminar\n\nLecture he
ld in MS 6627.\n\nAbstract\nI will discuss a new method to solve underdete
rmined PDE systems. The motivation comes from the constraint equation in g
eneral relativity\, the scalar curvature equation in geometry\, and the di
vergence equation in fluid mechanics. If time permits\, I will discuss app
lications to the flexibility of initial data sets in general relativity. T
his talk is based on the joint work with Philip Isett\, Yuchen Mao\, and S
ung-Jin Oh.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/235/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nick Cook (Duke)
DTSTART:20250401T200000Z
DTEND:20250401T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/236
DESCRIPTION:Title: Branching Brownian motion and the Road-Field Model\nby Nick Cook (Duke) as part of UCLA analysis and PDE seminar\n\nLecture
held in MS 6627.\n\nAbstract\nThe Fisher-KPP equation was introduced in 1
937 to model the spread of an advantageous gene through a spatially distri
buted population. Remarkably precise information on the traveling front ha
s been obtained via a connection with branching Brownian motion\, beginnin
g with works of McKean and Bramson in the 70s. I will discuss an extension
of this probabilistic approach to the Road-Field Model: a reaction-diffus
ion PDE system introduced by H. Berestycki et al. to describe enhancement
of biological invasions by a line of fast diffusion\, such as a river or a
road. Based on joint work with Amir Dembo.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/236/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Beomjong Kwak (KAIST)
DTSTART:20250603T200000Z
DTEND:20250603T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/237
DESCRIPTION:Title: Global well-posedness of the cubic nonlinear Schrödi
nger equation on $\\mathbb{T}^2$\nby Beomjong Kwak (KAIST) as part of
UCLA analysis and PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nIn
this talk\, we present the global well-posedness for the cubic nonlinear S
chrödinger equation for periodic initial data in the mass-critical dimens
ion $d=2$ for large initial data in $H^s\,s>0$. The result is based on a n
ew inverse Strichartz inequality\, which is proved by using incidence geom
etry and additive combinatorics. In addition\, we construct an approximate
periodic solution showing ill-behavior of the flow map at the $L^2$ regul
arity. This is based on joint works with Sebastian Herr.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/237/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayla Gafni (U. Mississippi)
DTSTART:20250527T200000Z
DTEND:20250527T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/238
DESCRIPTION:Title: Exponential Sums Weighted by Additive Functions\n
by Ayla Gafni (U. Mississippi) as part of UCLA analysis and PDE seminar\n\
nLecture held in MS 6627.\n\nAbstract\nFor an arithmetic function $f$ and
a real number $\\alpha$\, consider the exponential sum\n\n$$ S_f(x\,\\alph
a) = \\sum_{n\\le x} f(n) e^{2\\pi i n \\alpha}. $$ \n\nThe growth of thes
e sums as $x$ increases plays an important role in many number theory tech
niques. We will discuss new bounds on these exponential sums for various
additive functions $f$\, including $\\omega(n)$ (the number of distinct pr
ime factors of $n$) and $\\Omega(n)$ (the total number of prime factors of
$n$). We will then apply these bounds to enumerate certain integer parti
tions and solutions to Diophantine equations.\n\nThis is joint work with N
icolas Robles.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/238/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qinghai Zhang (Zhejiang University)
DTSTART:20251007T200000Z
DTEND:20251007T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/239
DESCRIPTION:Title: Numerical PDEs on moving domains via algebraic topolo
gy\, differential geometry\, and AI\nby Qinghai Zhang (Zhejiang Univer
sity) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 5203.
\n\nAbstract\nThis talk is an overview of the framework of MARS (mapping a
nd adjusting regular semianalytic sets) for numerically solving the incomp
ressible Navier- Stokes equations (INSE) on moving domains with fourth-ord
er accuracy and structure-preserving capabilities. We first propose GePUP-
ES\, a fourth-order energy-stable adaptive projection method for solving I
NSE on a square box and then augment GePUP-ES to irregular and moving doma
ins. Different from current methods that avoid topology and geometry by co
nverting them into numerical ODEs/PDEs\, we tackle topological and geometr
ic problems with tools in topology and geometry. We show that the coupling
of numerical analysis with (even elementary) concepts in topology and geo
metry could be powerful for real world applications.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/239/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Otte Heinaevaara (Caltech)
DTSTART:20251007T210000Z
DTEND:20251007T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/240
DESCRIPTION:Title: Tracial joint spectral measures\nby Otte Heinaeva
ara (Caltech) as part of UCLA analysis and PDE seminar\n\nLecture held in
MS 5203.\n\nAbstract\nGiven two Hermitian matrices\, we introduce a new ty
pe of\nspectral measure\, a tracial joint spectral measure on the plane.\n
Existence of this measure implies that any two-dimensional subspace of\nth
e Schatten-p class is isometric to a subspace of L_p. We discuss\nsome app
lications\, limitations and generalisations of this result.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/240/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iqra Atlaf (UCLA)
DTSTART:20250930T200000Z
DTEND:20250930T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/241
DESCRIPTION:Title: On the modulus of continuity of functions whose image
has positive measure\, and metric embeddings into R^d without shrinking.<
/a>\nby Iqra Atlaf (UCLA) as part of UCLA analysis and PDE seminar\n\nLect
ure held in MS 5203.\n\nAbstract\nA generalization of the classical Sard t
heorem in the plane is the following. Let f be a function defined on a sub
set A ⊂ R^2. If f has modulus of continuity ω(r) ≲ r^2\, then f (A)
⊂ R has Lebesgue measure zero. Choquet claimed that this was a full char
acterization\, i.e. for every ω for which ω(r)/r^2 converges to ∞ as r
→ 0\, there is a counterexample. We disprove this by showing that the c
orrect characterization\, in R^d\, is integral_{0}^1 ω(r)^{−1/d} = ∞.
We obtain this as a special case of a more general result. We study which
spaces (X\, ρ) can be embedded into R^d without decreasing any of the di
stances in X.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/241/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jari Taskinen (Helsinki)
DTSTART:20251104T210000Z
DTEND:20251104T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/242
DESCRIPTION:Title: Spectra of Bergman-Toeplitz operators on periodic pla
nar do- mains\nby Jari Taskinen (Helsinki) as part of UCLA analysis an
d PDE seminar\n\nLecture held in MS 5203.\n\nAbstract\nAccording to [1]\,
Floquet-transform techniques can be applied to study\n\nBergman spaces\, B
ergman kernels and Toeplitz operators Ta on un-\nbounded periodic planar d
omains Π\, which are defined as the union of\n\ninfinitely many copies of
the translated\, bounded periodic cell π. The\nFloquet-transform yields
a connection between the Bergman projection\nPΠ : L\n2\n(Π) → A2\n\n(
Π) and a family of Bergman-type projections Pη in\n\nthe space L\n2\n(π
)\, where η ∈ [−π\, π] is the so-called Floquet variable.\nThis yie
lds an explicit formula for the corresponding kernels.\nThe article [2] co
ntains a study Toeplitz operators Ta : A2\n\n(Π) → A2\n(Π)\n\nwith per
iodic symbols. The Floquet-transform establishes a connec-\ntion of T with
family of Toepliz-type operators Ta\,η\, η ∈ [−π\, π]\, in\n\nthe
cell π. The main results of [2] include a proof for the ”spectral\nban
d formula” for Bergman-Toeplitz operators. The formula describes\nthe es
sential spectrum of Ta in terms of the spectra of the operators\nTa\,η\,
and it is well-known in the setting of Schr ̈odinger operators with\nperi
odic potentials.\nThe described methods are applied to construct new types
of examples\n\nof Toepliz operators with discontinuous essential spectra.
By using con-\nformal mappings\, the examples can be presented as Toeplit
z operators\n\non the standard Bergman-Hilbert space of the open unit disc
.\n\n[1] J. Taskinen\, On the Bergman projection and kernel in periodic pl
a-\nnar domains\, Proceedings of IWOTA 2022 Lancaster\, Springer (2023).\n
\n[2] J. Taskinen\, On Bergman-Toeplitz operators in periodic planar do-\n
mains\, Transactions London Math. Soc.\, 12 (2025).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/242/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Muha (University of Zagreb)
DTSTART:20251028T200000Z
DTEND:20251028T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/243
DESCRIPTION:Title: The Role of Dissipation in the Existence of Time-Peri
odic Solutions for PDE Systems\nby Boris Muha (University of Zagreb) a
s part of UCLA analysis and PDE seminar\n\nLecture held in MS 5203.\n\nAbs
tract\nIt is well known that in many conservative mechanical systems\, res
onance may occur: for certain time-periodic forces\, the system’s respon
se becomes unbounded. Classical examples of partial differential equations
exhibiting this behavior include the wave equation and the equations of l
inearized elasticity (the Lamé system). By contrast\, resonance does not
arise in strongly dissipative systems\, such as those governed by the heat
equation. In such cases\, one can show that for every time-periodic right
-hand side\, there exists a unique time-periodic solution.\n\nIn this lect
ure\, we address the question: how strong must dissipation be to prevent r
esonance? Our focus will be on periodic solutions of the so-called heat–
wave system\, in which the wave equation is coupled with the heat equation
through a common boundary. Dissipation is present only in the heat compon
ent\, and the model can be viewed as a simplified setting for fluid–stru
cture interaction. We show that in certain geometric configurations\, the
system admits a unique time-periodic solution for each time-periodic forci
ng term. The proof requires additional regularity of the right-hand side c
ompared with the Cauchy problem\, but we demonstrate that resonance is not
a consequence of low-regularity forcing.\n\nFinally\, we discuss the open
problem of whether this result extends to arbitrary geometries or whether
certain configurations could still lead to resonance. This is joint work
with Stanislav Mosný\, S. Schwarzacher\, and J. Webster\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/243/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Courtade (UC Berkeley)
DTSTART:20251202T210000Z
DTEND:20251202T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/244
DESCRIPTION:Title: On Entropy Comparisons and Functional Inequalities\nby Thomas Courtade (UC Berkeley) as part of UCLA analysis and PDE semin
ar\n\nLecture held in MS 5203.\n\nAbstract\nI'll present a concise isoperi
metric-like inequality enjoyed by Shannon entropy that unifies a variety o
f sharp inequalities in analysis and information theory (including\, for e
xample\, forward/reverse Brascamp--Lieb-type inequalities and entropy-powe
r-type inequalities). The proof is based on three separate ingredients: a
stochastic argument\, a generalization of Bennett\, Carbery\, Christ and
Tao's structural theory of Brascamp--Lieb inequalities\, and a certain geo
desic convexity enjoyed by sharp constants. Time permitting\, I'll descri
be some recent generalizations of the (functional) Blaschke--Santalo inequ
ality\, and highlight some intriguing contrasts and similarities with the
previous results.\n\nThis talk is based in part on joint work with Efe Ara
s (https://arxiv.org/pdf/2206.14182) and Edric Wang (https://arxiv.org/pdf
/2509.08998).\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/244/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yixuan Wang and Changhe Yang (Caltech)
DTSTART:20251118T210000Z
DTEND:20251118T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/245
DESCRIPTION:Title: Nonuniqueness of Leray--Hopf solutions to the unforce
d incompressible 3D Navier--Stokes Equation\nby Yixuan Wang and Changh
e Yang (Caltech) as part of UCLA analysis and PDE seminar\n\nLecture held
in MS 5203.\n\nAbstract\nThe nonuniqueness of Leray--Hopf solutions to the
unforced incompressible 3D Navier--Stokes equations is one of the central
open problems in mathematical fluid dynamics. Inspired by earlier works b
y Jia-Sverak\, Guillod-Sverak\, and Albritton-Brue-Colombo\, we construct
a Leray--Hopf solution in the self-similar setting and then establish the
existence of a second solution by analyzing the stability of the linearize
d operator around this profile\, showing that it corresponds to an unstabl
e perturbation. To achieve this\, we develop an innovative numerical metho
d that computes candidate solutions with high precision and propose a fram
ework for rigorously establishing exact solutions in a neighborhood of the
se candidates. A key step is to decompose the linearized operator into a c
oercive part plus a compact perturbation\, inspired by the works of Chen-H
ou\, which is further approximated by a finite-rank operator up to a small
error. The invertibility of the linearized operator restricted to the ima
ge of this finite-rank approximation is then rigorously verified using com
puter-assisted proofs.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/245/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mingfeng Chen (University of Wisconsin-Madison)
DTSTART:20251021T200000Z
DTEND:20251021T210000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/246
DESCRIPTION:Title: Towards a classification of Nikodym sets\nby Ming
feng Chen (University of Wisconsin-Madison) as part of UCLA analysis and P
DE seminar\n\nLecture held in MS 5203.\n\nAbstract\nI will discuss some re
cent works related to maximal operators and Nikodym sets in the plane. In
joint work with Shaoming Guo\, we present a complete classification of the
L^p-boundedness of Nikodym maximal operators associated with families of
curves in the plane.\n\nOur work introduces a strongly degenerate conditio
n. For curves satisfying this condition\, we construct explicit Nikodym-ty
pe sets to prove that the corresponding maximal operator is unbounded on L
^p for all p<∞. Conversely\, for curves that are not strongly degenerate
\, we establish a local smoothing estimate and determine the sharp range o
f p for which the maximal operator is bounded on L^p.\n\nI will also talk
about the multiparameter generalization.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/246/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cyril Imbert (CNRS et Univ. Paris Cité)
DTSTART:20251203T190000Z
DTEND:20251203T200000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/247
DESCRIPTION:Title: The Fisher information for the space-homogeneous Bolt
zmann equation\nby Cyril Imbert (CNRS et Univ. Paris Cité) as part of
UCLA analysis and PDE seminar\n\nLecture held in Boelter Hall 5436.\n\nAb
stract\nWe prove that the Fisher information is monotone decreasing in tim
e along solutions of the space-homogeneous Boltzmann equation for a large
class of collision kernels covering all classical interactions derived fro
m systems of particles. For general collision kernels\, a sufficient condi
tion for the monotonicity of the Fisher information along the flow is rela
ted to the best constant for an integro-differential inequality for functi
ons on the sphere\, which belongs in the family of the Log-Sobolev inequal
ities. As a consequence\, we establish the existence of global smooth solu
tions to the space-homogeneous Boltzmann equation in the main situation of
interest where this was not known\, namely the regime of very soft potent
ials. This is opening the path to the completion of both the classical pro
gram of qualitative study of space-homogeneous Boltzmann equation\, initia
ted by Carleman\, and the program of using the Fisher information in the s
tudy of the Boltzmann equation\, initiated by McKean. From the proofs and
discussion emerges a strengthened picture of the links between kinetic the
ory\, information theory and log-Sobolev inequalities. Joint work with Lui
s Silvestre and Cédric Villani.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/247/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bogdan Georgiev (Google Deepmind)
DTSTART:20251209T210000Z
DTEND:20251209T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/248
DESCRIPTION:Title: Machine Assisted Mathematics: Some recent progress\nby Bogdan Georgiev (Google Deepmind) as part of UCLA analysis and PDE s
eminar\n\nLecture held in MS 5203.\n\nAbstract\nWe give an overview and di
scuss some recent developments in using computer assistance for mathematic
al exploration and discovery. In particular\, we address the utility of ce
rtain machine learning techniques to detect patterns and discover extremal
or optimal mathematical constructions across a broad collection of mathem
atical problems.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/248/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmett Shear (Softmax)
DTSTART:20251210T193000Z
DTEND:20251210T203000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/249
DESCRIPTION:Title: From States To Fields And Back Again\nby Emmett S
hear (Softmax) as part of UCLA analysis and PDE seminar\n\nLecture held in
MS 5203.\n\nAbstract\nIf a state persists\, we are licensed to view it as
a learning system via the Free Energy Principle. We will present on the n
ature of such states\, their varieties and evolution\, and the way that a
cycle of self-directed renormalization enables them to "transcend" into in
creasingly complex forms. We aim to demonstrate that from this and a few o
ther relatively minimal assumptions we can unify a surprising amount of ma
th\, physics\, computer science\, and biology.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/249/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luccas Campos (UFMG)
DTSTART:20260120T210000Z
DTEND:20260120T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/250
DESCRIPTION:Title: Scattering for the k-DgBO\nby Luccas Campos (UFMG
) as part of UCLA analysis and PDE seminar\n\nLecture held in MS 5203.\n\n
Abstract\nIn this talk\, we consider the $k$-dispersion generalized Benjam
in–Ono equation\n\\[\n\\partial_t u + \\partial_x (D^\\alpha u + \\\, u
^{k+1}) = 0\, \\quad (t\,x) \\in \\mathbb R \\times \\mathbb R.\n\\]\nWe
prove that\, for any even integer $k\\geq 4$ and $\\alpha \\in (1\,2)$\, s
olutions with initial data in the energy space $H^{\\frac{\\alpha}{2}}(\\m
athbb R)$ exist globally in time and scatter. Our approach combines the co
ncentration–compactness–rigidity method introduced by Kenig and Merle
with monotonicity formulas developed by Tao for the KdV equation (\\textit
{cf.} Kim and Kwon for the Benjamin–Ono equation)\, together with the Ca
ffarelli-Silvestre fractional extension and commutator estimates. This is
based on joint works with F. Linares and T. S. R. Santos.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/250/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yvonne Alama Bronsard (MIT)
DTSTART:20260224T210000Z
DTEND:20260224T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/251
DESCRIPTION:Title: Numerical approximations to nonlinear dispersive equa
tions\, from short to long times\nby Yvonne Alama Bronsard (MIT) as pa
rt of UCLA analysis and PDE seminar\n\nLecture held in MS 5203.\n\nAbstrac
t\nThe first part of this talk deals with the numerical approximation to n
onlinear dispersive equations\, such as the prototypical nonlinear Schröd
inger or Korteweg-de Vries equations. We introduce integration techniques
allowing for the construction of schemes which preserve the geometric str
ucture and qualitative behavior of the equation on the discrete level. Hig
her order extensions will be presented\, following techniques based on dec
orated trees series inspired by singular stochastic PDEs via the theory of
regularity structures.\n\nIn the second part\, we introduce a new approac
h for designing and analyzing schemes for some nonlinear and nonlocal inte
grable PDEs. This work is based upon recent theoretical breakthroughs on e
xplicit formulas for nonlinear integrable equations. It opens the way for
studying the asymptotic behavior of the solutions\, with applications to t
he soliton resolution conjecture\, and in wave turbulence theory.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/251/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitrii Zakharov (MIT)
DTSTART:20260127T210000Z
DTEND:20260127T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/252
DESCRIPTION:Title: Furstenberg estimate for sticky sets of tubes in thre
e dimensions\nby Dimitrii Zakharov (MIT) as part of UCLA analysis and
PDE seminar\n\nLecture held in MS 6627.\n\nAbstract\nRecently\, there have
been two major breakthroughs in geometric measure theory. In 2023\, Ren a
nd Wang resolved the Furstenberg estimate in the plane. Their result answe
rs the following question: take a t-dimensional set of lines in the plane
and put an s-dimensional set of points on each line\, what is the smallest
dimension of the union of these sets? In 2025\, Wang and Zahl resolved th
e three dimensional case of the Kakeya conjecture. Namely\, they showed th
at a 2-dimensional family of lines in R^3 which satisfy a certain non-conc
entration assumption must cover a set of full Hausdorff dimension. Both pr
oofs proceed by reducing the problem to an important special case where tu
bes are "sticky" (aka almost AD-regular). The sticky cases of these two pr
oblems were established in earlier works by Orponen-Shmerkin-Wang and Wang
-Zahl\, respectively.\nIn 2024\, Wang and Wu posed a conjecture about inci
dences of tubes in R^3 which is a common generalization to the Kakeya and
Furstenberg problems. They moreover showed that a sufficiently strong vers
ion of their conjecture suffices to prove the Restriction conjecture. \nWe
resolve the sticky case of their conjecture. Our overall approach is simi
lar to the one taken by Wang and Zahl in their 2022 proof of the sticky ca
se of the Kakeya conjecture in R^3\, which follows the strategy proposed b
y Katz and Tao. However\, several new ideas are required to overcome new d
ifficulties in the Furstenberg setting and\, in particular\, when speciali
zed back to the case of sticky Kakeya in R^3\, our proof deviates signific
antly from Wang and Zahl's. We hope that our techniques will be applicable
more widely to "sticky" cases of other problems.\nJoint work with Hong Wa
ng.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/252/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arian Nadjimzadah (UCLA)
DTSTART:20260303T210000Z
DTEND:20260303T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/253
DESCRIPTION:Title: Bourgain’s condition\, sticky Kakeya\, and new exam
ples\nby Arian Nadjimzadah (UCLA) as part of UCLA analysis and PDE sem
inar\n\nLecture held in MS 6627.\n\nAbstract\nIn the 1970s\, Hörmander pr
oposed a generalization of the Fourier restriction conjecture to a much br
oader class of oscillatory integral operators. In 1991\, Bourgain showed t
hat Hörmander’s conjecture fails for generic operators. In 2023\, Guo\,
Wang\, and Zhang introduced the Hörmander dichotomy conjecture as a way
to patch the original conjecture: an operator satisfies the bounds in the
restriction conjecture if and only if it satisfies what they call Bourgain
’s condition. Just as classical Kakeya sets underlie the restriction con
jecture\, certain Kakeya sets of curves underlie the Hörmander dichotomy
conjecture. \n\nBuilding on the program which culminated in Wang and Zahl
’s resolution of the Kakeya conjecture in three dimensions\, we introduc
e a sticky Kakeya conjecture for curves corresponding to operators satisfy
ing Bourgain’s condition. We reduce our conjecture to the classical stic
ky Kakeya conjecture in all dimensions\, in particular proving it in dimen
sion 3. This relies on our new characterization of Bourgain’s condition
based on the structure of thin curved tubes in a thick curved tube. We als
o provide examples which show the characterization is essentially sharp\,
in particular providing the first operators satisfying Bourgain’s condit
ion for which there is no diffeomorphism taking the characteristic curves
to lines. This makes a “general to sticky” reduction in the spirit of
Wang and Zahl difficult.\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/253/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hagen Papenberg (UCLA)
DTSTART:20260428T210000Z
DTEND:20260428T220000Z
DTSTAMP:20260404T131155Z
UID:UCLAAnalysisSeminar/254
DESCRIPTION:Title: Nonlinear dispersive equations with quasi-periodic in
itial data\nby Hagen Papenberg (UCLA) as part of UCLA analysis and PDE
seminar\n\nInteractive livestream: https://ucla.zoom.us/j/9264073849\nLec
ture held in MS 6627.\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/UCLAAnalysisSeminar/254/
URL:https://ucla.zoom.us/j/9264073849
END:VEVENT
END:VCALENDAR