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BEGIN:VEVENT
SUMMARY:Yi Li (John Jay College\, CUNY)
DTSTART:20200515T180000Z
DTEND:20200515T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/1
DESCRIPTION:Title: Monotone properties of the eigenfunction of Neumann
problems\nby Yi Li (John Jay College\, CUNY) as part of CUNY Harmonic
Analysis and PDE's Seminar\n\n\nAbstract\nIn this talk\, we present recent
progress on the eigenvalue problem\n\n\n$\\Delta u + \\mu u = 0$ in $\\Om
ega$\, $\\frac{\\partial{u}}{\\partial{\\nu}} = 0$ on $\\partial{\\Omega}$
\, (1) \n\nwhere $\\Omega$ is a domain in $\\mathbb{R}^n$\, $\\frac{\\part
ial{u}}{\\partial{ν}} := \\nabla u \\cdot \\nu$\, $\\nu$ denotes the outw
ard unit normal vector on $\\partial{\\Omega}$. \n\n• “Hot spots” co
njecture given by Rauch: the second Neumann eigenfunction attains its maxi
mum and minimum values only on the boundary of the domain.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lacey (Georgia Institute of Technology)
DTSTART:20200522T180000Z
DTEND:20200522T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/2
DESCRIPTION:Title: Discrete improving inequalities\nby Michael Lace
y (Georgia Institute of Technology) as part of CUNY Harmonic Analysis and
PDE's Seminar\n\n\nAbstract\nAverages improve functions\, even if averagin
g over lower dimensional\nsurfaces\, most famously the sphere. Remarkably
there are discrete analogs\nof these inequalities. Their emerging theory c
omplements and extends\nthe more well known theories associated to discret
e maximal functions of\nBourgain\, and discrete Radon transforms of Stein\
, Wainger and Ionescu.\nI will survey some recent results\, and point out
some of the fascinating\ncomplications that arise in the proofs.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bingyang Hu (University of Wisconsin\, Madison)
DTSTART:20200529T180000Z
DTEND:20200529T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/3
DESCRIPTION:Title: On the general dyadic system in Euclidean spaces
\nby Bingyang Hu (University of Wisconsin\, Madison) as part of CUNY Harmo
nic Analysis and PDE's Seminar\n\n\nAbstract\nAdjacent dyadic systems are
pivotal in analysis and related fields to study continuous objects via col
lection of dyadic ones. In this talk\, we will first give a complete chara
cterization of the adjacent dyadic systems on the real line\, and then we
will generalize it to higher dimension. The first part of this talk (real
line case) is joint with Tess Anderson\, Liwei Jiang\, Cornor Olson and Ze
yu Wei under an REU project at Summer\, 2018 at UW-Madison\; while the sec
ond the part (general case) is a joint work with Tess Anderson.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joris Roos (University of Wisconsin-Madison)
DTSTART:20200724T180000Z
DTEND:20200724T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/4
DESCRIPTION:Title: Spherical maximal functions and fractal dimensions o
f dilation sets\nby Joris Roos (University of Wisconsin-Madison) as pa
rt of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThe talk is
about sharp $L^p$ improving properties of Euclidean spherical maximal oper
ators in two and higher dimensions with a supremum taken over a given set
of radii $E$ in $[1\,2]$. We will discuss a characterization of the closed
convex sets which can occur as closure of the sharp $L^p$ improving regio
n of this operator. The region depends not only on the Minkowski dimension
of $E$\, but also other properties of the fractal geometry such as the As
souad spectrum. This is joint work with Andreas Seeger\, extending earlier
joint work with Tess Anderson\, Kevin Hughes and Andreas Seeger.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura De Carli (Florida International University)
DTSTART:20200717T180000Z
DTEND:20200717T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/5
DESCRIPTION:Title: Sufficient conditions for the existence of exponenti
al bases on domains of $\\mathbb{R}^d$\nby Laura De Carli (Florida Int
ernational University) as part of CUNY Harmonic Analysis and PDE's Seminar
\n\n\nAbstract\nWe first consider the following general problem:\nGiven an
orthonormal basis $V$ in a separable Hilbert space and a set of unit vec
tors $B$\, we consider the sets $B_N$ obtained by replacing the first $N$
vectors of $V$ with the first $N$ vectors of $B$. We show necessary\nand s
ufficient conditions that ensure that the sets $B_N$ are Riesz bases of $H
$ and we estimate the frame constants of these bases. Then\, we show condi
tions that ensure that $B$ is a Riesz basis of $H$. We apply our results t
o prove sufficient conditions for the existence of exponential bases on d
omains of $\\mathbb{R}^d$. (joint work with Julian Edward\, FIU)\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Accomazzo Scotti (University of the Basque Country - Unive
rsity of British Columbia)
DTSTART:20200605T143000Z
DTEND:20200605T153000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/6
DESCRIPTION:Title: Maximal directional singular integrals\nby Natal
ia Accomazzo Scotti (University of the Basque Country - University of Brit
ish Columbia) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbs
tract\nMaximal directional operators are formed by taking a one-dimensiona
l operator acting along a line\, and then studying the maximal value as th
e line changes through a set of directions. One important example of this
type of operators is when we consider the one arising from the maximal fun
ction\, which we can find has been broadly studied in the literature. In t
his talk we will review a little bit the history of these operators and we
will give some new results on the operator that arises from a singular in
tegral.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bradley Currey (Saint Louis University)
DTSTART:20200703T180000Z
DTEND:20200703T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/7
DESCRIPTION:Title: Linear independence of translate systems\nby Bra
dley Currey (Saint Louis University) as part of CUNY Harmonic Analysis and
PDE's Seminar\n\n\nAbstract\nLet $G$ be a locally compact group and $f$ a
complex function on $G$. For $x ∈ G$ define $L_xf(y) = f(s^{−1}y)$. W
e say $f$ has independent translates if $\\{L_xf : x ∈ E\\}$ is linearly
independent for all finite subsets $E$ of $G$. The general problem is to
determine classes of functions on $G$ that have independent translates. We
recall known results for $G$ abelian\, and present a few new results for
the case where $G$ is the Heisenberg group.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/7
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theresa C. Anderson (Purdue University)
DTSTART:20200619T180000Z
DTEND:20200619T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/8
DESCRIPTION:Title: Discrete maximal functions over surfaces of higher c
odimension\nby Theresa C. Anderson (Purdue University) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nConsidering discrete (
integral) variants of continuous operators\, and\, separately\, continuous
operators that involve integration over surfaces of intermediate codimens
ion\, have been two challenging areas of investigation in analysis. Here
we unite these themes\, providing an interesting interplay of harmonic ana
lysis\, analytic number theory and discrete geometry. We will describe th
e key features of our technique as well as history and background to put t
his program into context. This is joint work with Eyvi Palsson and Angel
Kumchev.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/8
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bochen Liu (National Center for Theoretical Sciences\, Taiwan)
DTSTART:20200612T140000Z
DTEND:20200612T150000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/9
DESCRIPTION:Title: Fourier frames for surface-carried measures\nby
Bochen Liu (National Center for Theoretical Sciences\, Taiwan) as part of
CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWe show that the s
urface measure on a sphere does not admit a Fourier frame. On the other ha
nd\, surface measure on the boundary of a polytope always admits a Fourier
frame.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/9
/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Zhang (University of Rochester)
DTSTART:20200710T180000Z
DTEND:20200710T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/10
DESCRIPTION:Title: Sharp endpoint estimates for eigenfunctions restric
ted to submanifolds of codimension 2\nby Cheng Zhang (University of Ro
chester) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract
\nBurq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\\le p\\le \
\infty$) for the restriction of eigenfunctions to submanifolds. The estima
tes are sharp\, except for the log loss at the endpoint $L^2$ estimates fo
r submanifolds of codimension 2. It has long been believed that the log lo
ss at the endpoint can be removed in general\, while the problem is still
open. So in this talk we will talk about the study of sharp endpoint restr
iction estimates for eigenfunctions in this case. Recall that Chen-Sogge r
emoved the log loss for the geodesics on 3-dimensional manifolds. In a joi
nt work with Xing Wang\, we generalize their result to higher dimensions a
nd prove that the log loss can be removed for totally geodesic submanifold
s of codimension 2. Moreover\, on 3-dimensional manifolds\, we can remove
the log loss for curves with nonvanishing geodesic curvatures\, and more g
eneral finite type curves. The problem in 3D is essentially related to Hil
bert transforms along curves in the plane and a class of singular oscillat
ory integrals studied by Phong-Stein\, Ricci-Stein\, Pan\, Seeger\, Carber
y-Pérez. (Reference: arXiv:1606.09346v2)\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
0/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Krause (University of California\, Los Angeles)
DTSTART:20200731T180000Z
DTEND:20200731T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/11
DESCRIPTION:Title: Pointwise Ergodic Theorems for Non-conventional Bil
inear Polynomial Averages\nby Ben Krause (University of California\, L
os Angeles) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstr
act\nWe establish convergence in norm and pointwise almost everywhere for
the non-conventional bilinear polynomial ergodic averages\n\\[ A_N(f\,g)(x
) := \\frac{1}{N} \\sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\\]\nas $N \\to \\inf
ty$\, where $T \\colon X \\to X$ is a measure-preserving transformation of
a $\\sigma$-finite measure space $(X\,\\mu)$\, $P(n) \\in \\mathbb{Z}[n]$
is a polynomial of degree $d \\geq 2$\, and $f \\in L^{p_1}(X)\, \\ g \\i
n L^{p_2}(X)$ for some $p_1\,p_2 > 1$ with $\\frac{1}{p_1} + \\frac{1}{p_
2} \\leq 1$.\n\n(Joint with Mariusz Mirek (Rutgers)\, and Terry Tao (UCLA)
.)\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Máté Vizer (Alfréd Rényi Institute of Mathematics\, Hungary)
DTSTART:20200918T180000Z
DTEND:20200918T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/12
DESCRIPTION:Title: Recent developments in the discrete Fuglede conject
ure\nby Máté Vizer (Alfréd Rényi Institute of Mathematics\, Hungar
y) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nFugl
ede's conjecture was stated in 1974\, and connects an analytic with a geom
etric property of a given bounded measurable subset $T \\subset \\mathbf{R
}^d$. It states that $T$ accepts a complete orthogonal basis of exponentia
l functions if and only if it tiles $\\mathbf{R}^d$ by translations. This
conjecture has been disproved by Tao in 2004. The counterexample has been
achieved by lifting counterexamples of Fuglede's conjecture in finite abel
ian groups to counterexamples in Euclidean spaces.\n\n \nThis connection m
otivated researchers to pay more attention to the discrete version of Fugl
ede's conjecture. I would like to briefly overview recent progress on this
conjecture\, including some work of the speaker\, that is joint with Kiss
\, Malikiosis ans Somlai.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Negreira (University of the Republic (Uruguay))
DTSTART:20201211T190000Z
DTEND:20201211T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/13
DESCRIPTION:Title: Wavelets decompositions and its applications on reg
ular metric spaces\nby Felipe Negreira (University of the Republic (Ur
uguay)) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\
nIn this talk we will first survey the techniques used to reproduce the Li
ttlewood-Paley analysis and the subsequent wavelet expansions for Riemanni
an manifolds and then for the more general setting of regular metric space
s. Next\, within the same framework of regular metric spaces\, we will sho
w how one can use these decompositions to characterize various function sp
aces -in particular Besov- and obtain different results such as sampling i
nequalities\, trace theorems and multifractal analysis type estimates.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent R. Martinez (CUNY-Hunter College)
DTSTART:20201002T180000Z
DTEND:20201002T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/14
DESCRIPTION:Title: Data Assimilation & PDEs (Part I): An Overview of
Recent Rigorous Results\nby Vincent R. Martinez (CUNY-Hunter College)
as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThis ta
lk will be an introduction to a series of six talks in the study of Data A
ssimilation for PDEs. In this first talk\, we will introduce the concept o
f data assimilation\, survey recent results\, introduce some analytical to
ols and techniques involved in establishing these rigorous results\, and l
astly discuss open avenues to explore.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Alfréd Rényi Institute of Mathematics\, Hungary)
DTSTART:20201023T180000Z
DTEND:20201023T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/15
DESCRIPTION:Title: On the Fuglede conjecture for the product of elemen
tary abelian groups over prime fields\nby Gergely Kiss (Alfréd Rényi
Institute of Mathematics\, Hungary) as part of CUNY Harmonic Analysis and
PDE's Seminar\n\n\nAbstract\nFuglede in 1974 conjectured that a bounded d
omain $S \\subset \\mathbb{R}^d$\ntiles the $d$-dimensional\nEuclidean spa
ce if and only if the set of functions in $L^2(S)$ admits an orthogonal\nb
asis of exponential functions.\n\nIn my talk we will focus on the discrete
version of Fuglede’s conjecture that can be formulated as follows. Let
$G$ be a finite Abelian group $G$ and $\\widehat{G}$ the set of characters
of $G$\, indexed by the elements of $G$. Then $S\\subset G$ is spectra
l if and only if there exists a $\\Lambda\\in G$ such that ($\\chi_l)_
{l\\in \\Lambda}$ is an orthogonal base of complex valued functions define
d on $S$.\nFor a finite group $G$ and a subset $S$ of $G$ we say that $S$
is a tile of $G$ if there is a $T \\subset G$ such that $S+T=G$ and
$|S|\\cdot |T|=|G|$. The discrete version Fuglede conjecture states that
for an abelian group $G$ a subset $S$ is spectral if and only if $S$ is a
tile.\n\nI will talk about the Fuglede conjecture for the product of eleme
ntary abelian groups over prime fields. The importance of this particular
case can be illustrated with the fact that Fuglede's original conjecture w
ere disproved first by Tao and his proof is based on a counterexample for
the discrete Fuglede conjecture on elementary abelian $p$-groups.\n\nFirst
I will summarize the known results concerning the product of elementary a
belian groups over prime fields. In the second part of my talk I will pres
ent our recent result which shows that the discrete Fuglede conjecture hol
ds on\n$\\mathbb{Z}_p^2 \\times \\mathbb{Z}_q$\, where $p$ and $q$ are dif
ferent primes.\n\n(joint work with Gábor Somlai)\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael S. Jolly (Indiana University-Bloomington)
DTSTART:20201030T180000Z
DTEND:20201030T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/16
DESCRIPTION:Title: Data Assimilation & PDEs (Part IV): Data assimilati
on for the 2D Navier-Stokes equations using local observables\nby Mich
ael S. Jolly (Indiana University-Bloomington) as part of CUNY Harmonic Ana
lysis and PDE's Seminar\n\n\nAbstract\nWe will discuss an approximate\, gl
obal data assimilation/synchronization algorithm based on purely local obs
ervations for the two-dimensional Navier-Stokes equations on the torus. We
will present a rigorous result stating that\, for any error threshold\, i
f the reference flow is analytic with sufficiently large analyticity radiu
s\, then it can be recovered within that threshold. We will then show the
result of numerical tests of the effectiveness of this approach\, as well
as variants with data on moving subdomains. In particular\, computations d
emonstrate that machine precision synchronization is achieved for mobile d
ata collected from a small fraction of the domain. This is joint work with
Animikh Biswas and Zachary Bradshaw.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tural Sadigov (Hamilton College)
DTSTART:20201009T180000Z
DTEND:20201009T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/18
DESCRIPTION:Title: Data Assimilation & PDEs (Part II): A Computational
Study of a data assimilation algorithm for the damped\, driven Korteweg -
de Vries equation - Fourier Modes\, Nodes\, and Volume Elements cases
\nby Tural Sadigov (Hamilton College) as part of CUNY Harmonic Analysis an
d PDE's Seminar\n\n\nAbstract\nIn this talk\, we describe a continuous dat
a assimilation algorithm for damped\, driven Korteweg - de Vries equation\
, and summarize analytical results regarding the Fourier modes case. Then
we describe the numerical method we use to solve the Korteweg - de Vries e
quation and confirm these analytical results in the case of a particular f
orce and damping parameter that create an interesting chaotic solution. We
numerically show that\, even with a chaotic solution nearby a potential s
addle point in the attractor\, the data assimilation algorithm is robust e
nough to lock on to the reference solution with the right parameters in th
e assimilation algorithm. We also present various numerical results regard
ing two other cases in the context of the same equation: nodal case and fi
nite volume elements case. This work is sponsored by the U.S. Air Force un
der MOU FA8750-15-3-6000.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aseel Farhat (Florida State University)
DTSTART:20201016T180000Z
DTEND:20201016T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/19
DESCRIPTION:Title: Data Assimilation & PDEs (Part III): Data Assimilat
ion algorithms for the Rayleigh-Bénard Convection problem.\nby Aseel
Farhat (Florida State University) as part of CUNY Harmonic Analysis and PD
E's Seminar\n\n\nAbstract\nAnalyzing the validity and success of a data as
similation algorithm when some state variable observations are not availab
le is an important problem meteorology and engineering. In this talk\, we
will present various continuous data assimilation (downscaling) algorithms
for the Rayleigh-Bénard problem that do not require observations of all
evolving state variables of the system. For the 2D incompressible Bénard
convection problem\, for example\, our algorithm uses only velocity mea
surements (temperature measurements are not necessary). We rigorously
identify conditions that guarantee synchronization between the observed sy
stem and the model\, then confirm the applicability of these results via n
umerical simulations.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/1
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Collin Victor (University of Nebraska-Lincoln)
DTSTART:20201113T190000Z
DTEND:20201113T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/20
DESCRIPTION:Title: Data Assimilation & PDEs (Part VI): Continuous Data
Assimilation Enhanced by Mobile Observers\nby Collin Victor (Universi
ty of Nebraska-Lincoln) as part of CUNY Harmonic Analysis and PDE's Semina
r\n\n\nAbstract\nIn this talk we examine computationally a continuous data
assimilation algorithm using observers that move continuously in time. Sp
ecifically\, we look at using the Azouani-Olson-Titi data assimilation alg
orithm in the context of measurement devices which move in time\, such as
satellites or drones. We find that\, in the context of the 1D Allen-Cahn e
quations\, by moving the sampling points dynamically\, we can greatly redu
ce the number of sampling points required\, while achieving better accurac
y. Additionally we look at the adaptation of this algorithm to the 2D Inco
mpressible Navier-Stokes equations using observers that move according to
various regimes\, where we obtain similar results.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
0/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahya Ghandehari (University of Delaware)
DTSTART:20201120T190000Z
DTEND:20201120T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/21
DESCRIPTION:Title: Fourier algebras of the group of ${\\mathbb R}$-aff
ine transformations and a dual convolution\nby Mahya Ghandehari (Unive
rsity of Delaware) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n
\nAbstract\nA major trend in Non-commutative Harmonic Analysis is to inves
tigate function spaces related to Fourier analysis (and representation the
ory) of non-abelian groups.\nThe Fourier algebra\, which is associated wit
h the left regular representation of the ambient group\, is an important e
xample of such function spaces. This function algebra encodes the properti
es of the group in various ways\; for instance the existence of derivation
s on this algebra translates into information about the commutativity of t
he group itself. \n\nIn this talk\, we investigate the Fourier algebra of
the group of ${\\mathbb R}$-affine transformations. In particular\, we di
scuss the non-commutative Fourier transform for this group\, and provide
an explicit formula for the convolution product on the "dual side" of this
transform. As an application of this new dual convolution product\, we sh
ow an easy dual formulation for (the only known) symmetric derivative on t
he Fourier algebra of the group. \n\nThis talk is mainly based on joint a
rticles with Y. Choi.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A. Linnell (Virginia Tech)
DTSTART:20201218T190000Z
DTEND:20201218T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/22
DESCRIPTION:Title: The discrete Pompeiu problem\nby Peter A. Linne
ll (Virginia Tech) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n
\nAbstract\nLet K be a compact subset of R^n with nonzero Lebesgue measure
. The Pompeiu problem asks if f=0 is the only continuous function such th
at the integral of f over s(K) is 0 for all rigid motions s of R^n. We wi
ll consider a version of the Pompeiu problem for discrete groups. We shall
also describe U(G) and its role in this problem. This is joint work with
Mike Puls.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Luli (University of California\, Davis)
DTSTART:20201204T190000Z
DTEND:20201204T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/23
DESCRIPTION:Title: Smooth Nonnegative Interpolation\nby Kevin Luli
(University of California\, Davis) as part of CUNY Harmonic Analysis and
PDE's Seminar\n\n\nAbstract\nSuppose $E$ is an arbitrary subset of $R^n$.
Let $f: E \\rightarrow [0\, \\infty)$. How can we decide if $f$ extends
to a nonnegative $C^m$ function $F$ defined on all of $R^n$? Suppose $E$ i
s finite. Can we compute a nonnegative $C^m$ function $F$ on $R^n$ that ag
rees with $f$ on $E$ with the least possible $C^m$ norm? How many computer
operations does this take? In this talk\, I will explain recent results o
n these problems. Non-negativity is one of the most important shape preser
ving properties for interpolants. In real life applications\, the range of
the interpolant is imposed by nature. For example\, probability density\,
the amount of snow\, rain\, humidity\, chemical concentration are all non
negative quantities and are of interest in natural sciences. Even in one d
imension\, the existing techniques can only handle nonnegative interpolati
on under special assumptions on the data set. Our results work without any
assumptions on the data sets.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marcin Bownik (University of Oregon)
DTSTART:20210212T190000Z
DTEND:20210212T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/24
DESCRIPTION:Title: Parseval wavelet frames on Riemaniann manifolds
\nby Marcin Bownik (University of Oregon) as part of CUNY Harmonic Analysi
s and PDE's Seminar\n\n\nAbstract\nAbstract:\nIn this talk we discuss how
to construct Parseval wavelet frames in $L^2(M)$ for a general Riemannian
manifold $M$. We also show the existence of wavelet unconditional frames i
n $L^p(M)$ for $1 < p <\\infty$. This construction is made possible thanks
to smooth orthogonal projection decomposition of the identity operator on
$L^2(M)$\, which is an operator version of a smooth partition of unity. W
e also show some applications such as a characterization of Triebel-Lizork
in $F_{p\,q}^s(M)$ and Besov $B_{p\,q}^s(M)$ spaces on compact manifolds i
n terms of magnitudes of coefficients of Parseval wavelet frames. This tal
k is based on a joint work with Dziedziul and Kamont.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emily J. King (Colorado State University)
DTSTART:20210514T180000Z
DTEND:20210514T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/25
DESCRIPTION:Title: Mathematical analysis of neural networks\nby Em
ily J. King (Colorado State University) as part of CUNY Harmonic Analysis
and PDE's Seminar\n\n\nAbstract\nNeural networks have proven themselves to
be useful in a wide range of applications but operate more-or-less as bla
ck boxes. Mathematicians have an opportunity to crack the mystery. In this
talk\, after a gentle introduction to neural networks\, three approaches
to mathematically analyze neural networks will be presented. First\, sing
ular values will be generalized to better understand the inherent rank of
weight matrices in a certain type of neural network. Second\, a tool from
high-dimensional geometry\, Gaussian mean width\, will be shown empiricall
y to distinguish between correctly and incorrectly classified data as they
travel through a neural network. Finally\, we will analyze approximation
properties of spaces of neural networks of a fixed architecture.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20210226T190000Z
DTEND:20210226T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/26
DESCRIPTION:Title: Sharpness of Falconer-type estimates for dot produc
ts\nby Steven Senger (Missouri State University) as part of CUNY Harmo
nic Analysis and PDE's Seminar\n\n\nAbstract\nIn 1985\, Falconer conjectur
ed that if a subset has Hausdorff dimension sufficiently high with respect
to the ambient dimension\, then the Lebesgue measure of the set of distan
ces it determines should be positive. His first partial result toward this
end hinged on a lemma showing that if a measure satisfies an energy condi
tion related to the ambient dimension\, then its support must not have a h
igh concentration of points separated by any given distance. Since then\,
this lemma has been explored in other contexts\, and serves as a bit of a
litmus test on how much we can say about various functionals under similar
circumstances. We employ techniques from discrete geometry to construct a
family of sharpness examples showing that this energy condition is sharp
in the analogous problem for dot products.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Greenfeld (University of California\, Los Angeles)
DTSTART:20210312T190000Z
DTEND:20210312T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/27
DESCRIPTION:Title: Translational tilings in lattices\nby Rachel Gr
eenfeld (University of California\, Los Angeles) as part of CUNY Harmonic
Analysis and PDE's Seminar\n\n\nAbstract\nLet $F$ be a finite subset of $\
\mathbb{Z}^d$. We say that $F$ is a translational tile of $\\mathbb{Z}^d$
if it is possible to cover $\\mathbb{Z}^d$ by translates of $F$ without an
y overlaps.\nThe periodic tiling conjecture\, which is perhaps the most we
ll-known conjecture in the area\, suggests that any translational tile adm
its at least one periodic tiling. In the talk\, we will motivate and discu
ss the study of this conjecture. We will also present some new results\, j
oint with Terence Tao\, on the structure of translational tilings in latti
ces and introduce some applications.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pooja Rao (The State University of New York at Stony Brook)
DTSTART:20210319T180000Z
DTEND:20210319T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/28
DESCRIPTION:Title: Modeling mixing accurately in numerical simulations
of interfacial instabilities\nby Pooja Rao (The State University of N
ew York at Stony Brook) as part of CUNY Harmonic Analysis and PDE's Semina
r\n\n\nAbstract\nHydrodynamic instabilities\, such as the shock-driven Ric
htmyer-Meshkov instability (RMI) and the gravity-driven Rayleigh-Taylor in
stability (RTI)\, occur in unstable configurations where the density diff
ers between two fluids. The physical interface between two fluids is impe
rfect and the localized perturbations give rise to the growth of these ins
tabilities when accelerated either via shock or gravity.\n\nThese instabil
ities play a critical role in numerous applications ranging from performan
ce degradation in inertial confinement fusion (ICF) capsules to supernova
explosions. Oftentimes\, they occur in tandem\, such as in the ICF experim
ents.\n\nTo accurately model the instability growth requires special treat
ment of the discrete representation of the interface between the two fluid
s. Using one such numerical approach\, Front-tracking\, we investigate a s
implified representation of the instability growth in inertial confinement
experiments\, focusing on the growth profile of a Rayleigh-Taylor instabi
lity which is seeded by a Richtmyer-Meshkov instability.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Animikh Biswas (University of Maryland\, Baltimore County)
DTSTART:20210326T180000Z
DTEND:20210326T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/29
DESCRIPTION:Title: Determining Functionals and Maps\, Data Assimilatio
n and an Observable Regularity Criterion for Three-Dimensional Hydrodynami
cal Equations\nby Animikh Biswas (University of Maryland\, Baltimore C
ounty) as part of CUNY Harmonic Analysis and PDE's Seminar\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/2
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zachary Bradshaw (University of Arkansas)
DTSTART:20210409T180000Z
DTEND:20210409T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/30
DESCRIPTION:Title: Non-decaying solutions to the critical surface quas
i-geostrophic equations with symmetries\nby Zachary Bradshaw (Universi
ty of Arkansas) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nA
bstract\nWe develop a theory of self-similar solutions to the critical sur
face quasi-geostrophic equations. We construct self-similar solutions for
arbitrarily large data in various regularity classes and demonstrate\, in
the small data regime\, uniqueness and global asymptotic stability. These
solutions are non-decaying in space which leads to ambiguity in the drift
velocity. This ambiguity is corrected by imposing m-fold rotational symmet
ry. The self-similar solutions exhibited here lie just beyond the known we
ll-posedness theory and are expected to shed light on potential non-unique
ness\, due to symmetry-breaking bifurcations\, in analogy with work of Jia
and Sverak on the Navier-Stokes equations. This is joint work with Dallas
Albritton of Courant Institute\, NYU.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
0/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Sondak (Harvard University)
DTSTART:20210423T180000Z
DTEND:20210423T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/32
DESCRIPTION:Title: An Autoencoder and Reduced Basis for Dynamical Syst
ems\nby David Sondak (Harvard University) as part of CUNY Harmonic Ana
lysis and PDE's Seminar\n\n\nAbstract\nThe governing equations of nature a
re nearly always nonlinear and often exhibit chaos. Nevertheless\, scienti
sts and engineers must still make informative predictions based on solutio
ns to the governing equations. One promising approach for making predictio
ns in regimes of scientific interest is to develop reliable reduced models
of the physics. These models should be fast and easy to compute while res
pecting the\nunderlying dynamics of the system. The present work leverages
the expressibility of modern machine learning models to learn a basis for
the reduced space of dynamical systems. An autoencoder with a sparsity-pr
omoting latent space penalization is trained on data from the periodic Kur
amoto-Sivashinsky (K-S) equation and the damped and undamped KdV equations
. It is shown that the dimension of the learned reduced space is consisten
t with that of the inertial manifold for the K-S equation. From here\, a n
onlinear basis is determined for the K-S equation from the trained autoenc
oder model. For the KdV equations\, it is shown\, over a large range of da
mping coefficients\, that the autoencoder learns a reduced manifold and th
at the dimension of this manifold\ndecreases with increasing damping coeff
icient.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruxi Shi (Mathematical Institute of the Polish Academy of Sciences
)
DTSTART:20210219T190000Z
DTEND:20210219T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/33
DESCRIPTION:Title: Spectral sets and spectral measures on groups with
one prime factor\nby Ruxi Shi (Mathematical Institute of the Polish Ac
ademy of Sciences) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n
\nAbstract\nA Borel set $\\Omega$ (resp. a Borel probability measure $\\mu
$) on a locally compact group is called a spectral set (resp. a spectral m
easure) if there exists a subset of continuous group characters that forms
an orthogonal basis of the Hilbert space $L^2(\\Omega)$ (resp. $L^2(\\mu)
$). In this talk\, I will consider locally compact abelian groups with one
prime factor\, say $p$\, for example\, $\\mathbb{Z}/p^n\\mathbb{Z}$\, $(\
\mathbb{Z}/p\\mathbb{Z})^d$\, $\\mathbb{Q}_p$\, etc. I will discuss the pr
operties and characterization of spectral sets and spectral measures on th
ese groups.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Pakzad (Indiana University Bloomington)
DTSTART:20210430T180000Z
DTEND:20210430T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/34
DESCRIPTION:Title: The role of energy dissipation in modeling turbulen
ce\nby Ali Pakzad (Indiana University Bloomington) as part of CUNY Har
monic Analysis and PDE's Seminar\n\n\nAbstract\nA distinguishing feature o
f turbulent flows is the emergence of complicated chaotic structures invol
ving a wide range of length scales. Simulating all these scales is infeasi
ble for practical problems\, such as simulating storm fronts and hurricane
s using direct numerical simulation. To overcome this difficulty\, turbule
nce models\, which model the universal effect of small scales on large sca
les\, are introduced to account for sub-mesh scale effects on a coarse mes
h. Key to getting a good approximation for a turbulence model is to correc
tly calibrate the energy dissipation in the model on a coarse mesh. Energy
dissipation rates of various turbulence models have been analyzed but ass
uming a super-fine mesh.\n\nIn this talk\, I present a calculation of the
energy dissipation in a turbulence model discretized on a coarse mesh. Mot
ivated by this result\, I will show how the Smagorinsky model\, a common t
urbulence model used in Large Eddy Simulation\, can be modified for better
performance. If time allows\, I would describe some recent results in thi
s direction for shear driven turbulence with noise at the boundary.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Giorgini (Indiana University Bloomington)
DTSTART:20210507T180000Z
DTEND:20210507T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/35
DESCRIPTION:Title: Diffuse Interface modeling for two-phase flows: fro
m the model H to the AGG model\nby Andrea Giorgini (Indiana University
Bloomington) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbs
tract\nIn the last decades\, the Diffuse Interface theory (also known as P
hase Field theory) has made significant progresses in the description of m
ulti-phase flows from modeling to numerical simulations. A particularly ac
tive topic has been the development of thermodynamically consistent models
extending the well-known Model H in the case of unmatched fluid densities
. In this talk\, I will focus on the AGG model proposed by H. Abels\, H. G
arcke and G. Grün in 2012. The model consists of a Navier-Stokes-Cahn-Hil
liard system characterized by a concentration-dependent density and an add
itional flux term due to interface diffusion. Using the method of matched
asymptotic expansions\, the sharp interface limit of the AGG model corresp
onds to the two-phase Navier-Stokes equations. In the literature\, the ana
lysis of the AGG system has only been focused on the existence of weak sol
utions. During the seminar\, I will present the first results concerning t
he existence\, uniqueness and stability of strong solutions for the AGG mo
del in two dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anuj Kumar (Indiana University Bloomington)
DTSTART:20211001T170000Z
DTEND:20211001T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/36
DESCRIPTION:Title: On well-posedness and smoothing of solutions to the
generalized SQG equations in critical Sobolev spaces\nby Anuj Kumar (
Indiana University Bloomington) as part of CUNY Harmonic Analysis and PDE'
s Seminar\n\n\nAbstract\nThis talk is based on recent works in which we st
udy the dissipative generalized surface quasi-geostrophic equations in a s
upercritical regime where the order of the dissipation is small relative t
o order of the velocity\, and the velocities are less regular than the adv
ected scalar by up to one order of derivative. The existence and uniquenes
s theory of these equations in the borderline Sobolev spaces is addressed\
, as well as the instantaneous Gevrey class smoothing of their correspondi
ng solutions. These results appear to be the first of its kind for a quasi
linear parabolic equation whose coefficients are of higher order than its
linear term. The main tool is the use of an approximation scheme suitably
adapted to preserve the underlying commutator structure. We also study a f
amily of inviscid generalized SQG equations where the velocities have been
mildly regularized\, for instance\, logarithmically. The well-posedness o
f these equations in borderline Sobolev spaces is addressed.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bing-Ying Lu (University of Bremen)
DTSTART:20211015T170000Z
DTEND:20211015T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/37
DESCRIPTION:Title: Universality near the gradient catastrophe point in
the semiclassical sine-Gordon equation\nby Bing-Ying Lu (University o
f Bremen) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstrac
t\nWe study the semiclassical limit of the sine-Gordon (sG) equation with
below threshold pure impulse initial data of Klaus-Shaw type. The Whitham
averaged approximation of this system exhibits a gradient catastrophe in f
inite time. In accordance with a conjecture of Dubrovin\, Grava and Klein\
, we found that in an O(ε4/5) neighborhood near the gradient catastrophe
point\, the asymptotics of the sG solution are universally described by th
e Painlevé I tritronquée solution. A linear map can be explicitly made f
rom the tritronquée solution to this neighborhood. Under this map: away f
rom the tritronquée poles\, the first correction of sG is universally giv
en by the real part of the Hamiltonian of the tritronquée solution\; loca
lized defects appear at locations mapped from the poles of tritronquée so
lution\; the defects are proved universally to be a two parameter family o
f special localized solutions on a periodic background for the sG equation
. We are able to characterize the solution in detail. Our approach is the
rigorous steepest descent method for matrix Riemann--Hilbert problems\, su
bstantially generalizing Bertola and Tovbis's results on the nonlinear Sch
rödinger equation to establish universality beyond the context of solutio
ns of a single equation. This is joint work with Peter D. Miller.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suen Chun Kit Anthony (The Education University of Hong Kong)
DTSTART:20211022T170000Z
DTEND:20211022T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/38
DESCRIPTION:Title: Vanishing parameter limit for a class of active sca
lar equations\nby Suen Chun Kit Anthony (The Education University of H
ong Kong) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstrac
t\nIn this talk\, we study an abtract class of active scalar equations whi
ch depend on some viscosity parameters κ and ν. We examine the wellposed
ness of the equations in different scenarios and address the convergence o
f solutions as κ or ν vanishes. We further discuss some physical applica
tions of the general results obtained from such abtract class of active sc
alar equations.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hau-Tieng Wu (Duke University)
DTSTART:20211029T170000Z
DTEND:20211029T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/39
DESCRIPTION:Title: Some recent progress in diffusion based manifold le
arning\nby Hau-Tieng Wu (Duke University) as part of CUNY Harmonic Ana
lysis and PDE's Seminar\n\n\nAbstract\nDiffusion based manifold learning h
as been actively developed and applied in past decades. However\, there ar
e still many interesting practical and theoretical challenges. I will shar
e some recent progress in this direction\, particularly from the angle of
robustness and scalability and the associated theoretical support under th
e manifold setup. If time permits\, its clinical application will be discu
ssed.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/3
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent R. Martinez (CUNY-Hunter College)
DTSTART:20211008T170000Z
DTEND:20211008T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/40
DESCRIPTION:Title: On well-posedness and smoothing of solutions to the
generalized SQG equations in critical Sobolev spaces\, Part II\nby Vi
ncent R. Martinez (CUNY-Hunter College) as part of CUNY Harmonic Analysis
and PDE's Seminar\n\n\nAbstract\nWe will continue the discussion about the
issue of well-posedness at critical regularity for a family for active sc
alar equations with increasingly singular constitutive law.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
0/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deniz Bilman (University of Cincinnati)
DTSTART:20211105T170000Z
DTEND:20211105T181500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/41
DESCRIPTION:Title: High-Order Rogue Waves and Solitons\, and Solutions
Interpolating Between Them\nby Deniz Bilman (University of Cincinnati
) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nIt is
known from our recent work that both fundamental rogue wave solutions (wi
th Peter Miller and Liming Ling) and multi-pole soliton solutions (with R.
Buckingham) of the nonlinear Schrödinger (NLS) equation exhibit the same
asymptotic behavior in the limit of large order in a shrinking region nea
r the peak amplitude point\, despite the quite different boundary conditio
ns these solutions satisfy at infinity. We show how rogue waves and solito
ns of arbitrary orders can be placed within a common analytical framework
in which the ''order'' becomes a continuous parameter\, allowing one to tu
ne continuously between types of solutions satisfying different boundary c
onditions. In this scheme\, solitons and rogue waves of increasing integer
orders alternate as the continuous order parameter increases. We show tha
t 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. Howev
er\, in the unbounded complementary region one sees qualitatively differen
t asymptotic behavior along different sequences. In this talk we focus on
the behavior in this exterior region. The asymptotic behavior is most inte
resting for solutions that are neither rogue waves nor solitons. This is j
oint work with Peter Miller.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Götz Pfander (Katholische Universität Eichstätt-Ingolstadt)
DTSTART:20211119T180000Z
DTEND:20211119T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/42
DESCRIPTION:Title: Exponential bases for partitions of intervals\n
by Götz Pfander (Katholische Universität Eichstätt-Ingolstadt) as part
of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nFourier series
form a cornerstone of analysis\; it allows the expansion of a complex valu
ed 1-periodic function in the orthogonal basis of integer frequency expone
ntials (for the space of square integrable functions on the unit interval)
. A simple rescaling argument shows that by splitting the integers into ev
ens and odds\, we obtain orthogonal bases for functions defined on the fir
st\, respectively the second half of the unit interval. We shall generaliz
e this curiosity and show that\, given any finite partition of the unit in
terval into subintervals\, we can split the integers into subsets\, each o
f which forms a basis (not necessarily orthogonal) for functions on the re
spective subinterval. In addition\, novel fundamental results in the theor
y of Fourier series will be discussed.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tao Zhang (Guangzhou University)
DTSTART:20211203T180000Z
DTEND:20211203T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/43
DESCRIPTION:Title: Fuglede's conjecture holds in $\\mathbb{Z}_p\\times
\\mathbb{Z}_{p^n}$\nby Tao Zhang (Guangzhou University) as part of CUN
Y Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nFuglede's conjecture
states that for a subset $\\Omega$ of a locally compact abelian group $G$
with positive and finite Haar measure\, there exists a subset of the dual
group of $G$ which is an orthogonal basis of $L^2(\\Omega)$ if and only if
it tiles the group by translation. In this talk\, we consider the Fuglede
's conjecture in the group $\\mathbb{Z}_p\\times\\mathbb{Z}_{p^n}$. I will
talk about the main idea of our proof.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuan Pham (Brigham Young University)
DTSTART:20211210T180000Z
DTEND:20211210T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/44
DESCRIPTION:Title: Conservation of frequencies and applications to the
well-posedness problem of the Navier-Stokes equations\nby Tuan Pham (
Brigham Young University) as part of CUNY Harmonic Analysis and PDE's Semi
nar\n\n\nAbstract\nA well-known conserved quantity of the Navier-Stokes eq
uations is the total energy. This conservation law has been used extensive
ly in the local regularity theory\, especially since the groundbreaking wo
rk of Caffarelli-Kohn-Nirenberg in 1982. In the Fourier space\, the Navier
-Stokes equations are naturally associated with a stochastic cascade as no
ted by Le Jan and Sznitman in 1997. In the dynamic of this stochastic casc
ade\, the initial frequency is conserved. In this talk\, I will explain ho
w the conservation of frequencies can be used to study the well-posedness
problem of the Navier-Stokes equations. A notable application is that any
initial data in L2 whose Fourier transform is supported in the half-space
produces a unique global mild solution.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Balazs (Austrian Academy of Sciences)
DTSTART:20211217T180000Z
DTEND:20211217T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/45
DESCRIPTION:Title: Continuous Frames and Reproducing Kernels\nby P
eter Balazs (Austrian Academy of Sciences) as part of CUNY Harmonic Analys
is and PDE's Seminar\n\n\nAbstract\nFrame theory has become a tremendously
active research field\, with connection to many mathematical disciplines
but also applications. In short\, frames are a sequence of elements that a
llow stable representation of elements in a Hilbert space. One generalizat
ion of the original definition considers not a sequence indexed by a discr
ete set\, but a function indexed by a continuous set. This will be the top
ic of this talk\, in particular\, how closely this concept is intertwined
with reproducing kernel Hilbert spaces (RKHS). We start with a short motiv
ation why frame theory is important\, also for applications. We introduce
the basic definitions. We show recent developments which focus on the vari
ous facets of the interplay of continuous frames and RKHS. In particular\,
we analyze the structure of the reproducing kernel of a RKHS using frames
and reproducing pairs. We show that finite redundancy of a continuous fra
me implies atomic structure of the underlying measure space. This implies
that all the attempts to extend the notion of Riesz basis to general measu
re spaces are fruitless since every such family can be identified with a d
iscrete Riesz basis\, by using the RKHS structure of the range of the anal
ysis operator. This can also be used to formulate a result that classifies
all dual functions to a given continuous frame. Finally\, we will give ge
neral kernel theorems for operators acting between coorbit spaces\, which
are Banach spaces associated to continous frame representations and contai
n most of the usual function spaces (Besov spaces\, modulation spaces\, et
c.). This collects work with K. Gröchenig\, M. Speckbacher and N. Teofano
v.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Magsino (Ohio State University)
DTSTART:20220304T190000Z
DTEND:20220304T201500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/46
DESCRIPTION:Title: Singular Values of Random Subensembles of Frame Vec
tors\nby Mark Magsino (Ohio State University) as part of CUNY Harmonic
Analysis and PDE's Seminar\n\n\nAbstract\nFrame theory studies redundant
representations in a Hilbert space. In finite dimensions\, this is simply
a spanning set but there are many interesting and useful frames in these s
ettings. One application involves compressed sensing\, which is a method f
or efficient acquisition and reconstruction of signals using underrepresen
ted systems. However\, verifying the key property of compressed sensing fr
ames is NP-hard which makes constructing them difficult. One way around th
is is to examine random subensembles of these frames and try to control th
eir singular values. We will show that the singular values of random suben
sembles of so-called equiangular tight frames are closely linked to the Ke
sten-McKay distribution. This is joint work with Dustin Mixon and Hans Par
shall.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gareth Speight (University of Cincinnati)
DTSTART:20220311T190000Z
DTEND:20220311T201500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/47
DESCRIPTION:Title: Whitney Extension and Lusin Approximation in Carnot
Groups\nby Gareth Speight (University of Cincinnati) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nLusin's theorem states
that any measurable function can be approximated by a continuous function\
, except on a set of small measure. Analogous results for higher smoothnes
s give conditions under which a function may admit a Lusin type approximat
ion by C^m functions. Such results can often be obtained as a consequence
of a suitable Whitney extension theorem. We review what is known in the Eu
clidean setting then describe some recent extensions to Carnot groups\, a
family of non-Euclidean spaces that nevertheless have a rich geometric str
ucture. Based on joint work with Marco Capolli\, Andrea Pinamonti\, and Sc
ott Zimmerman.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weilin Li (New York University)
DTSTART:20220325T180000Z
DTEND:20220325T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/48
DESCRIPTION:Title: Function approximation with one-bit Bernstein and o
ne-bit neural networks\nby Weilin Li (New York University) as part of
CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThe celebrated uni
versal approximation theorems for neural networks (NNs) typically state th
at every sufficiently nice function can be arbitrarily well approximated b
y a neural network with carefully chosen real parameters. Motivated by rec
ent questions regarding NN compression\, we ask whether it is possible to
represent any reasonable function with a quantized NN -- a NN whose parame
ters are only allowed to be selected from a small set of allowable paramet
ers. We answer this question in the affirmative. Our main theorem shows th
at any continuously differentiable multivariate function can be approximat
ed by a one-bit quadratic NN (a NN with quadratic activation whose nonzero
weights and biases are only allowed to contain +1 or -1 entries) and the
rate of approximation of our scheme is able to exploit any additional smoo
thness of the target function. A key component of our work is a novel appr
oximation result by linear combinations of multivariate Bernstein polynomi
als\, with only +1 and -1 coefficients. Joint work with Sinan Gunturk.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Itay Londner (Weizmann Institute of Science)
DTSTART:20220401T180000Z
DTEND:20220401T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/49
DESCRIPTION:Title: Tiling the integers with translates of one tile: th
e Coven-Meyerowitz tiling conditions\nby Itay Londner (Weizmann Instit
ute of Science) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nA
bstract\nIt is well known that if a finite set of integers $A$ tiles the i
ntegers by translations\, then the translation set must be periodic\, so t
hat the tiling is equivalent to a factorization $A+B=\\Z_M$ of a finite cy
clic 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 Izabella
Laba (UBC)\, we proved that this is true for all sets tiling the integers
with period $M=(pqr)^2$. In my talk I will discuss this problem and introd
uce some ideas from the proof.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/4
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahaf Nitzan (Georgia Institute of Technology)
DTSTART:20220408T180000Z
DTEND:20220408T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/50
DESCRIPTION:Title: The uncertainty principle in finite dimensions\
nby Shahaf Nitzan (Georgia Institute of Technology) as part of CUNY Harmon
ic Analysis and PDE's Seminar\n\n\nAbstract\nI will give a survey of some
results related to the talks title\, and discuss a couple of new observati
ons in the area. The talk is based on joint work with Jan-Fredrik Olsen an
d Michael Northington.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
0/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weinan Wang (The University of Arizona)
DTSTART:20220429T180000Z
DTEND:20220429T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/51
DESCRIPTION:Title: Local well-posedness for the Boltzmann equation wit
h very soft potential and polynomially decaying initial data\nby Weina
n Wang (The University of Arizona) as part of CUNY Harmonic Analysis and P
DE's Seminar\n\n\nAbstract\nWe consider the local well-posedness of the sp
atially inhomogeneous non-cutoff Boltzmann equation when the initial data
decays polynomially in the velocity variable. We consider the case of very
soft potentials $\\gamma + 2s < 0$. Our main result completes the picture
for local well-posedness in this decay class by removing the restriction
$\\gamma + 2s > -3/2$ of previous works. It is based on the Carleman decom
position of the collision operator into a lower order term and an integro-
differential operator similar to the fractional Laplacian.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arie Israel (The University of Texas at Austin)
DTSTART:20220513T180000Z
DTEND:20220513T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/52
DESCRIPTION:Title: The norm of linear extension operators for $C^m(R^n
)$\nby Arie Israel (The University of Texas at Austin) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWe will describe a new
proof of the finiteness principle for Whitney's extension problem. As a b
yproduct\, we obtain the existence of linear extension operators with an i
mproved bound on the norm of the operator. We discuss connections to the a
lgorithmic problem of interpolation of data.\n\nThis is joint work with Ja
cob Carruth and Abraham Frei-Pearson.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura De Carli (Florida International University)
DTSTART:20220930T180000Z
DTEND:20220930T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/53
DESCRIPTION:Title: Weaving Riesz bases\, and piecewise weighted fram
es\nby Laura De Carli (Florida International University) as part of CU
NY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThis talk consists o
f two parts loosely connected to one another. In the first part we discuss
the properties of a family of Riesz bases on a separable Hilbert space $H
$ obtained in the following way: For every $N>1$ we let $B_N=\\{w_j\\}_{j
=1}^N\\bigcup\\{v_j\\}_{j=N+1}^\\infty$\, \nwhere $\\{v_j\\}_{j=1}^\\inft
y$ is a Riesz basis of $H$ and $B=\\{w_j\\}_{j=1}^\\infty$ is a set of uni
t vectors. We find necessary and sufficient conditions that ensure that th
e $B_N$ and $B$ are Riesz bases\, and we apply our results to the c
onstruction of exponential bases on domains of $L^2$.\n\nIn the second par
t of the talk we present results on weighted Riesz bases and frames in fin
ite or infinite-dimensional Hilbert spaces\, with piecewise constant weigh
ts. We use our results to construct tight frames in finite-dimensional Hi
lbert spaces.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Ginsberg (Princeton University)
DTSTART:20221104T180000Z
DTEND:20221104T191500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/54
DESCRIPTION:Title: The stability of model shocks and the Landau law of
decay\nby Daniel Ginsberg (Princeton University) as part of CUNY Harm
onic Analysis and PDE's Seminar\n\n\nAbstract\nIt is well-known that in th
ree space dimensions\, smooth solutions to the equations describing a comp
ressible gas can break down in finite time. One type of singularity which
can arise is known as a shock\, which is a hypersurface of discontinuity a
cross which the integral forms of conservation of mass and momentum hold a
nd through which there is nonzero mass flux. One can find approximate solu
tions to the equations of motion which describe expanding spherical shocks
. We use these model solutions to construct global-in-time solutions to th
e irrotational compressible Euler equations with shocks. This is joint wor
k with Igor Rodnianski.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleh Oleg Asipchuk (Florida International University)
DTSTART:20221111T190000Z
DTEND:20221111T201500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/55
DESCRIPTION:Title: Construction of exponential bases on split interval
s\nby Aleh Oleg Asipchuk (Florida International University) as part of
CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nLet $I$ be a unio
n of intervals of total length 1. It is well known that exponential bases
exist on $L^2(I)$\, but explicit expressions for such bases are only known
in special cases. In this work\, we construct exponential Riesz bases on
$L^2(I)$ with some mild assumptions on the gaps between the intervals. We
also generalize Kadec's stability theorem in some special and significant
cases.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Goluskin
DTSTART:20221118T200000Z
DTEND:20221118T211500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/56
DESCRIPTION:Title: Verifying global stability of fluid flows despite t
ransient growth of energy (Special Time)\nby David Goluskin as part of
CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nVerifying nonline
ar stability of a laminar fluid flow against all perturbations is a classi
c challenge in fluid dynamics. All past results rely on monotonic decrease
of a perturbation energy or a similar quadratic generalized energy. This
energy method cannot show global stability of any flow in which perturbati
on energy may grow transiently. For the many flows that allow transient en
ergy growth but seem to be globally stable (e.g. pipe flow and other paral
lel shear flows at certain Reynolds numbers) there has been no way to math
ematically verify global stability. After explaining why the energy method
was the only way to verify global stability of fluid flows for over 100 y
ears\, I will describe a different approach that is broadly applicable but
more technical. This approach\, proposed in 2012 by Goulart and Chernyshe
nko\, uses sum-of-squares polynomials to computationally construct non-qua
dratic Lyapunov functions that decrease monotonically for all flow perturb
ations. I will present a computational implementation of this approach for
the example of 2D plane Couette flow\, where we have verified global stab
ility at Reynolds numbers above the energy stability threshold. This energ
y stability result for 2D Couette flow had not been improved upon since be
ing found by Orr in 1907. The results I will present are the first verific
ation of global stability - for any fluid flow - that surpasses the energy
method. This is joint work with Federico Fuentes (Universidad Catolica de
Chile) and Sergei Chernyshenko (Imperial College London).\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Sagiv (Columbia University)
DTSTART:20221202T190000Z
DTEND:20221202T201500Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/57
DESCRIPTION:Title: Floquet Hamiltonians - effective gaps and resonant
decay\nby Amir Sagiv (Columbia University) as part of CUNY Harmonic An
alysis and PDE's Seminar\n\n\nAbstract\nFloquet topological insulators are
an emerging category of materials whose properties are transformed by tim
e-periodic forcing. Can their properties be understood from their first-pr
inciples continuum PDE models? Experimentally\, graphene is known to trans
form into an insulator under a time-periodic driving. A spectral gap\, how
ever\, is conjectured to not form. How do we reconcile these two facts? We
show that the original Schrodinger equation has an effective gap- a new a
nd physically-relevant relaxation of a spectral gap. Next\, we challenge t
he prevailing notion of Floquet edge modes\; due to resonance\, localized
modes in periodically-forced media are only metasable. Sufficiently rapid
forcing couples the localized mode to the bulk\, and so energy eventually
leaks away from the localized edge/defect\, in the spirit of the Fermi Gol
den Rule.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin D. Stubbs (Institute of Pure and Applied Mathematics\, UCLA)
DTSTART:20230203T190000Z
DTEND:20230203T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/59
DESCRIPTION:Title: A Mathematical Invitation to Wannier Functions\
nby Kevin D. Stubbs (Institute of Pure and Applied Mathematics\, UCLA) as
part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWannier fu
nctions\, first proposed in the 1930s\, have had a long history in computa
tional chemistry as a practical means to speed up calculations. Stated in
a mathematical language\, Wannier functions are an orthonormal basis for c
ertain types of spectral subspaces which are generated by the action of a
translation group. In the 1980s however\, it was realized that there is an
intimate connection between Wannier functions and topology. In particular
\, Wannier functions with fast spatial decay exist if and only if a certai
n vector bundle is topologically trivial. Materials with non-trivial topol
ogy host a number of remarkable properties which are robust to physical im
perfections. In this talk\, I will give a brief introduction to topologica
l materials and Wannier functions in periodic systems. I will then discuss
my work on extending these results to systems without any underlying peri
odicity.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/5
9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raghavendra Venkatraman (courant institute of mathematical science
s)
DTSTART:20230217T190000Z
DTEND:20230217T200000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/61
DESCRIPTION:Title: Homogenization questions inspired by machine learni
ng and the semi-supervised learning problem\nby Raghavendra Venkatrama
n (courant institute of mathematical sciences) as part of CUNY Harmonic An
alysis and PDE's Seminar\n\n\nAbstract\nThis talk comprises two parts. In
the first part\, we revisit the problem of pointwise semi-supervised learn
ing (SSL). Working on random geometric graphs (a.k.a point clouds) with fe
w "labeled points"\, our task is to propagate these labels to the rest of
the point cloud. Algorithms that are based on the graph Laplacian often pe
rform poorly in such pointwise learning tasks since minimizers develop loc
alized spikes near labeled data. We introduce a class of graph-based highe
r order fractional Sobolev spaces (H^s) and establish their consistency in
the large data limit\, along with applications to the SSL problem. A cruc
ial tool is recent convergence results for the spectrum of the graph Lapla
cian to that of the continuum.\nObtaining optimal convergence rates for su
ch spectra is an open question in stochastic homogenization. In the rest o
f the talk\, we'll discuss how to get state-of-the-art and optimal rates o
f convergence for the spectrum\, using tools from stochastic homogenizatio
n.\nThe first half is joint work with Dejan Slepcev (CMU)\, and the second
half is joint work with Scott Armstrong (Courant).\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fushuai Jiang (Univeristy of Maryland)
DTSTART:20230310T180000Z
DTEND:20230310T190000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/63
DESCRIPTION:Title: Quasi-optimal $C^2(R^n)$ Interpolation with Range R
estriction\nby Fushuai Jiang (Univeristy of Maryland) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nExperimental data often
have range or shape constraints imposed by nature. For example\, probabil
ity density or chemical concentration are non-negative quantities\, and th
e trajectory design through an obstacle course may need to avoid two bound
aries. In this talk\, we investigate the theory of multivariate smooth int
erpolation with range restriction from the perspective of Whitney Extensio
n Problems. Given a function defined on a finite set with no underlying ge
ometric assumption\, I will describe an $O(N(log N)^{-n})$ procedure to co
mpute a twice continuously differentiable interpolant that preserves a pre
scribed shape (e.g. nonnegativity) and whose second derivatives are as sma
ll as possible up to a constant factor (i.e.\, quasi-optimal). I will also
provide explicit numerical results in one dimension. This is based on the
joint works with Charles Fefferman (Princeton)\, Chen Liang (UC Davis)\,
Yutong Liang (former UC Davis)\, and Kevin Luli (UC Davis).\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur A. Danielyan (University of South Florida)
DTSTART:20230317T170000Z
DTEND:20230317T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/64
DESCRIPTION:Title: On a converse of Fatou's theorem\nby Arthur A.
Danielyan (University of South Florida) as part of CUNY Harmonic Analysis
and PDE's Seminar\n\n\nAbstract\nFatou's theorem states that a bounded ana
lytic function in the unit disc has radial limits a.e. on the unit circle
$T$. This talk presents the following new theorem in the converse directio
n. \n\nTheorem 1. Let $E$ be a subset on $T$. There exists a bounded analy
tic function in the open unit disc which has no radial limits on $E$ but h
as unrestricted limits at each point of $T\\backslash E$ if and only if $E
$ is an $F_\\sigma$ set of measure zero.\n\nThe sufficiency part of this t
heorem immediately implies a well-known theorem of Lohwater and Piranian t
he proof of which is complicated enough. However\, the proof of Theorem 1
only uses the Fatou's interpolation theorem\, for which too the author has
recently suggested a new simple proof.\n\nIt turns out that for the Blasc
hke products\, a well-known subclass of bounded analytic functions\, Theor
em 1 takes the following form. \n\nTheorem 2. Let $E$ be a subset on the u
nit circle $T$. There exists a Blaschke product which has no radial limits
on $E$ but has unrestricted limits at each point of $T\\backslash E$ if a
nd only if $E$ is a closed set of measure zero. \n\nThe proof of the neces
sity part of Theorem 2 is completely elementary\, but it still contains so
me methodological novelty. The proof of the sufficiency uses Theorem 1 as
well as some known results on Blaschke products. (Theorem 2 is a joint res
ult with Spyros Pasias.)\n\nReferences.\n\n1. A. A. Danielyan\, On Fatou's
theorem\, Anal. Math. Phys. V. 10\, Paper no. 28\, 2020. \n\n2. A. A. Dan
ielyan\, A proof of Fatou's interpolation theorem\, J. Fourier Anal. Appl.
\, V. 28\, Paper no. 45\, 2022. \n\n3. A. A. Danielyan and S. Pasias\, On
a boundary property of Blaschke products\, to appear in Anal. Mathematica\
n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael A. Perlmutter (University of California\, Los Angeles)
DTSTART:20230331T170000Z
DTEND:20230331T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/65
DESCRIPTION:Title: Geometric Scattering on Measure Spaces\nby Mich
ael A. Perlmutter (University of California\, Los Angeles) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nGeometric Deep Learnin
g is an emerging field of research that aims to extend the success of conv
olutional neural networks (CNNs) to data with non-Euclidean geometric stru
cture. Despite being in its relative infancy\, this field has already foun
d great success in many applications such as recommender systems\, compute
r graphics\, and traffic navigation. In order to improve our understanding
of the networks used in this new field\, several works have proposed nove
l versions of the scattering transform\, a wavelet-based model of CNNs for
graphs\, manifolds\, and more general measure spaces. In a similar spirit
to the original Euclidean scattering transform\, these geometric scatteri
ng transforms provide a mathematically rigorous framework for understandin
g the stability and invariance of the networks used in geometric deep lear
ning. Additionally\, they also have many interesting applications such as
drug discovery\, solving combinatorial optimization problems\, and predict
ing patient outcomes from single-cell data. In particular\, motivated by t
hese applications to single-cell data\, I will also discuss recent work pr
oposing a diffusion maps style algorithm with quantitative convergence gua
rantees for implementing the manifold scattering transform from finitely m
any samples of an unknown manifold.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nabil T. Fadai (University of Nottingham)
DTSTART:20230414T170000Z
DTEND:20230414T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/66
DESCRIPTION:Title: Semi-infinite travelling waves arising in moving-bo
undary reaction-diffusion equations\nby Nabil T. Fadai (University of
Nottingham) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstr
act\nTravelling waves arise in a wide variety of biological applications\,
from the healing of wounds to the migration of populations. Such biologic
al phenomena are often modelled mathematically via reaction-diffusion equa
tions\; however\, the resulting travelling wave fronts often lack the key
feature of a sharp edge. In this talk\, we will examine how the incorporat
ion of a moving boundary condition in reaction-diffusion models gives rise
to a variety of sharp-fronted travelling waves for a range of wave speeds
. In particular\, we will consider common reaction-diffusion models arisin
g in biology and explore the key qualitative features of the resulting tra
velling wave fronts.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geet Varma (RMIT University)
DTSTART:20230324T170000Z
DTEND:20230324T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/67
DESCRIPTION:Title: Weaving Frames Linked with Fractal Convolutions
\nby Geet Varma (RMIT University) as part of CUNY Harmonic Analysis and PD
E's Seminar\n\n\nAbstract\nWeaving frames have been introduced to deal wit
h some problems in signal processing and wireless sensor networks. More re
cently\, the notion of fractal operator and fractal convolutions have been
linked with perturbation theory of Schauder bases and frames. However\, t
he existing literature has established limited connections between the the
ory of fractals and frame expansions. In this paper we define Weaving fram
es generated via fractal operators combined with fractal convolutions. The
aim is to demonstrate how partial fractal convolutions are associated to
Riesz bases\, frames and the concept of Weaving frames. This current view
point deals with ones sided convolutions i.e both left and right partial f
ractal convolution operators on Lebesgue space $L^p$ for $1\\le p<\\infty$
. Some applications via partial fractal convolutions with null function h
ave been obtained for the perturbation theory of bases and weaving frames.
\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lu Zhang (Columbia University)
DTSTART:20230421T170000Z
DTEND:20230421T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/68
DESCRIPTION:Title: Coupling physics-deep learning inversion\nby Lu
Zhang (Columbia University) as part of CUNY Harmonic Analysis and PDE's S
eminar\n\n\nAbstract\nIn recent years\, there is an increasing interest in
applying deep learning to geophysical/medical data inversion. However\, d
irect application of end-to-end data-driven approaches to inversion have q
uickly shown limitations in the practical implementation. Indeed\, due to
the lack of prior knowledge on the objects of interest\, the trained deep
learning neural networks very often have limited generalization. In this t
alk\, we introduce a new methodology of coupling model-based inverse algor
ithms with deep learning for two typical types of inversion problems. In t
he first part\, we present an offline-online computational strategy of cou
pling classical least-squares based computational inversion with modern de
ep learning based approaches for full waveform inversion to achieve advant
ages that can not be achieved with only one of the components. In the seco
nd part\, we present an integrated data-driven and model-based iterative r
econstruction framework for joint inversion problems. The proposed method
couples the supplementary data with the partial differential equation mode
l to make the data-driven modeling process consistent with the model-based
reconstruction procedure. We also characterize the impact of learning unc
ertainty on the joint inversion results for one typical inverse problem.`1
\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chun-Kit Lai (San Francisco State University)
DTSTART:20230428T170000Z
DTEND:20230428T180000Z
DTSTAMP:20260404T111214Z
UID:HarmonicAnalysisandPDE/69
DESCRIPTION:Title: On measure and topological Erdos similarity problem
s\nby Chun-Kit Lai (San Francisco State University) as part of CUNY Ha
rmonic Analysis and PDE's Seminar\n\n\nAbstract\nA pattern is called unive
rsal in another collection of sets\, when every set in the collection cont
ains some linear and translated copy of the original pattern. Paul Erdős
proposed a conjecture that no infinite set is universal in the collection
of sets with positive measure.\n\nIn this talk\, we explore an analogous p
roblem in the topological setting. Instead of sets with positive measure\,
we investigate the collection of dense sets and in the collection of gen
eric sets (dense G-delta and complement has Lebesgue measure zero). We re
fer to such pattern as topologically universal and generically universal r
espectively. We will show that Cantor sets on $R^d$ are never topologicall
y universal and Cantor sets with positive Newhouse thickness on $R^1$ are
not generically universal. This gives a positive partial answer to a ques
tion by Svetic concerning the Erdős similarity problem on Cantor sets. Mo
reover\, we also obtain a higher dimensional generalization of the generic
universality problem.\n\nThis is a joint work with John Gallagher\, who w
as a Master student in SFSU\, and Eric Weber from Iowa State University.\n
LOCATION:https://stable.researchseminars.org/talk/HarmonicAnalysisandPDE/6
9/
END:VEVENT
END:VCALENDAR