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BEGIN:VEVENT
SUMMARY:William Beckner (University of Texas at Austin)
DTSTART:20200720T130000Z
DTEND:20200720T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/1
DESCRIPTION:Title: Symmetry in Fourier Analysis – Heisenberg to Stein-Weiss
\nby William Beckner (University of Texas at Austin) as part of Geomet
ric and functional inequalities and applications\n\n\nAbstract\nEmbedded s
ymmetry within the Heisenberg group is used to couple geometric insight an
d analytic calculation to obtain a new sharp Stein-Weiss inequality with m
ixed homogeneity on the line of duality. SL(2\,R) invariance and Riesz pot
entials define a natural bridge for encoded information that connects dist
inct geometric structures. Insight for Stein-Weiss integrals is gained fro
m vortex dynamics\, embedding on hyperbolic space\, and conformal geometry
. The intrinsic character of the Heisenberg group makes it the natural pla
ying field on which to explore the laws of symmetry and the interplay betw
een analysis and geometry on a manifold.\n\nZoom link: https://brown.zoom.
us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Malchiodi (Scuola Normale Superiore)
DTSTART:20200720T140000Z
DTEND:20200720T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/2
DESCRIPTION:Title: On the Sobolev quotient in sub-Riemannian geometry\nby
Andrea Malchiodi (Scuola Normale Superiore) as part of Geometric and func
tional inequalities and applications\n\n\nAbstract\nWe consider three-dime
nsional CR manifolds\, which are modelled on the Heisenberg group.\nWe int
roduce a natural concept of “mass” and prove its positivity under the
condition that\nthe scalar curvature is positive and in relation to their
(holomorphic) embeddability properties.\nWe apply this result to the CR Ya
mabe problem\, and we discuss extremality of Sobolev-type\nquotients\, giv
ing some counterexamples for “Rossi spheres”.\nThis is joint work with
J.H.Cheng and P.Yang.\n\nZoom link: https://brown.zoom.us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xavier Cabre (ICREA and UPC (Barcelona))
DTSTART:20200727T130000Z
DTEND:20200727T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/3
DESCRIPTION:Title: Stable solutions to semilinear elliptic equations are smoo
th up to dimension 9\nby Xavier Cabre (ICREA and UPC (Barcelona)) as p
art of Geometric and functional inequalities and applications\n\n\nAbstrac
t\nThe regularity of stable solutions to semilinear elliptic PDEs has been
studied since the 1970's. In dimensions 10 and higher\, there exist singu
lar stable energy solutions. In this talk I will describe a recent work in
collaboration with Figalli\, Ros-Oton\, and Serra\, where we prove that s
table solutions are smooth up to the optimal dimension 9. This answers to
an open problem posed by Brezis in the mid-nineties concerning the regular
ity of extremal solutions to Gelfand-type problems.\n\nZoom link: https://
brown.zoom.us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Jerison (MIT)
DTSTART:20201109T140000Z
DTEND:20201109T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/4
DESCRIPTION:Title: Rescheduled to Spring Semester 2021\nby David Jerison
(MIT) as part of Geometric and functional inequalities and applications\n\
nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiming Zhao (MIT)
DTSTART:20200803T130000Z
DTEND:20200803T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/5
DESCRIPTION:Title: Reconstruction of convex bodies via Gauss map\nby Yimi
ng Zhao (MIT) as part of Geometric and functional inequalities and applica
tions\n\n\nAbstract\nIn this talk\, we will discuss the Gauss image proble
m\, a problem that reconstructs the shape of a convex body using partial d
ata regarding its Gauss map. In the smooth category\, this problem reduces
to a Monge-Ampere type equation on the sphere. But\, we will use a variat
ional argument that works with generic convex bodies. This is joint work w
ith Károly Böröczky\, Erwin Lutwak\, Deane Yang\, and Gaoyong Zhang.\n\
nZoom link: https://brown.zoom.us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juncheng Wei (University of British Columbia)
DTSTART:20200817T140000Z
DTEND:20200817T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/6
DESCRIPTION:Title: Rigidity Results for Allen-Cahn Equation\nby Juncheng
Wei (University of British Columbia) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nI will discuss two recent rigi
dity results for Allen-Cahn: the first is Half Space Theorem which states
that if the nodal set lies above a half space then it must be one-dimensio
nal. The second result is the stability of Cabre-Terra saddle solutions in
R^8\, R^{10} and R^{12}.\n\nZoom link: https://brown.zoom.us/j/9168361286
2\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fengbo Hang (New York University)
DTSTART:20200810T140000Z
DTEND:20200810T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/7
DESCRIPTION:Title: Concentration compactness principle in critical dimensions
revisited\nby Fengbo Hang (New York University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nConcentration
compactness principle for functions in $W^{1\,n}_0$ on a\nn-dimensional do
main was introduced by Lions in 1985 with the\nMoser-Trudinger inequality
in mind. We will discuss some further\nrefinements after Cerny-Cianchi-Hen
cl's improvement in 2013. These\nrefinements unifiy the approach for n=2 a
nd n>2 cases and work for higher\norder or fractional order Sobolev spaces
as well. They are motivated by\nand closely related to the recent derivat
ion of Aubin's Moser-Trudinger\ninequality for functions with vanishing hi
gher order moments on the\nstandard 2-sphere.\n\nZoom link: https://brown.
zoom.us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (CalTech)
DTSTART:20200824T143000Z
DTEND:20200824T153000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/8
DESCRIPTION:Title: REVERSE HARDY–LITTLEWOOD–SOBOLEV INEQUALITIES\nby
Rupert Frank (CalTech) as part of Geometric and functional inequalities an
d applications\n\n\nAbstract\nWe are interested in a new family of reverse
Hardy–Littlewood–Sobolev inequalities which involve a power law kerne
l with positive exponent and a Lebesgue exponent <1. We characterize the r
ange of parameters for which the inequality holds and present results abou
t the existence of optimizers. A striking open question is the possibility
of concentration of a minimizing sequence.\n\nThis talk is based on joint
work with J. Carrillo\, M. Delgadino\, J. Dolbeault and F. Hoffmann.\n\nP
lease note the special time of this talk. \nFor Zoom link for each talk (f
uture links will not be posted here)\, please send an email to the organiz
ers at geometricinequalitiesandpdes@gmail.com\nZoom link: https://brown.zo
om.us/j/94525179475\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengfei Guan (McGill University)
DTSTART:20200831T140000Z
DTEND:20200831T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/9
DESCRIPTION:Title: A mean curvature type flow and isoperimetric problem in wa
rped product spaces\nby Pengfei Guan (McGill University) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nWe wil
l discuss a mean curvature type flow with the goal to solve isoperimetric
problem. The flow is induced from the variational properties associated to
conformal Killing fields. Such flow was first introduced in space forms i
n a previous joint work with Junfang Li\, where we provided a flow approac
h to the classical isoperimetric inequality in space form. Later\, jointly
with Junfang Li and Mu-Tao Wang\, we considered the similar flow in warpe
d product spaces with general base. Under some natural conditions\, the fl
ow preserves the volume of the bounded domain enclosed by a graphical hype
rsurface\, and monotonically decreases the hypersurface area. Furthermore\
, the regularity and convergence of the flow can be established\, thereby
the isoperimetric problem in warped product spaces can be solved. The flow
serves as an interesting way to achieve the optimal solution to the isope
rimetric problem.\n\nZoom link: https://brown.zoom.us/j/99054390401\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Hebey (Université de Cergy-Pontoise)
DTSTART:20201116T140000Z
DTEND:20201116T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/10
DESCRIPTION:Title: Schrödinger-Proca constructions in the closed setting\nby Emmanuel Hebey (Université de Cergy-Pontoise) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe discuss Sch
rödinger-Proca constructions in the context of closed manifolds leading \
nto the Bopp-Podolsky-Schrödinger-Proca and the Schrödinger-Poisson-Proc
a systems.\nThe goal is to present an introduction to these equations (how
we build them\, what do \nthey represent) and then to present the result
we got on these systems about the \nconvergence of (BPSP) to (SPP) as the
Bopp-Podolsky parameter goes to zero.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yi Wang (Johns Hopkins University)
DTSTART:20200907T130000Z
DTEND:20200907T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/11
DESCRIPTION:Title: Rigidity of local minimizers of the $\\sigma_k$ functiona
l\nby Yi Wang (Johns Hopkins University) as part of Geometric and func
tional inequalities and applications\n\n\nAbstract\nIn this talk\, I will
present a result on the rigidity of local minimizers of the functional $\\
int \\sigma_2+ \\oint H_2$ among all conformally flat metrics in the Eucli
dean (n + 1)-ball. We prove the metric is flat up to a conformal transform
ation in some (noncritical) dimensions. We also prove the analogous result
in the critical dimension n + 1 = 4. The main method is Frank-Lieb’s re
arrangement-free argument. If minimizers exist\, this implies a fully nonl
inear sharp Sobolev trace inequality. I will also discuss a nonsharp Sobol
ev trace inequality. This is joint work with Jeffrey Case.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dongmeng Xi (NYU)
DTSTART:20200803T140000Z
DTEND:20200803T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/12
DESCRIPTION:Title: An isoperimetric type inequality via a modified Steiner s
ymmetrization scheme\nby Dongmeng Xi (NYU) as part of Geometric and fu
nctional inequalities and applications\n\n\nAbstract\nWe establish an affi
ne isoperimetric inequality using a symmetrization scheme that involves a
total of 2n elaborately chosen Steiner symmetrizations at a time. The nece
ssity of this scheme\, as opposed to the usual Steiner symmetrization\, wi
ll be demonstrated with an example. This is a joint work with Dr. Yiming Z
hao.\n\nZoom link: https://brown.zoom.us/j/91683612862\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phan Thành Nam (LMU Munich)
DTSTART:20200921T130000Z
DTEND:20200921T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/13
DESCRIPTION:Title: Lieb-Thirring inequality with optimal constant and gradie
nt error term\nby Phan Thành Nam (LMU Munich) as part of Geometric an
d functional inequalities and applications\n\n\nAbstract\nIn 1975\, Lieb a
nd Thirring conjectured that the kinetic energy of fermions is not smaller
than its Thomas-Fermi (semiclassical) approximation\, at least in three o
r higher dimensions. I will discuss a rigorous lower bound with the sharp
semiclassical constant and a gradient error term which is normally of lowe
r order in applications. The proof is based on a microlocal analysis and
a variant of the Berezin-Li-Yau inequality. This approach can be extended
to derive an improved Lieb-Thirring inequality for interacting systems\,
where the Gagliardo-Nirenberg constant appears in the strong coupling limi
t.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Minicozzi (MIT)
DTSTART:20201026T140000Z
DTEND:20201026T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/14
DESCRIPTION:Title: Mean curvature flow in high codimension\nby William M
inicozzi (MIT) as part of Geometric and functional inequalities and applic
ations\n\n\nAbstract\nMean curvature flow (MCF) is a geometric heat equati
on where a\nsubmanifold evolves to minimize its area. A central problem i
s to\nunderstand the singularities that form and what these imply for the\
nflow. I will talk about joint work with Toby Colding on higher\ncodimens
ion MCF\, where the flow becomes a complicated system of\nequations and mu
ch less is known.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Saloff-Coste (Cornell University)
DTSTART:20201005T140000Z
DTEND:20201005T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/15
DESCRIPTION:Title: Heat kernel on manifolds with finitely many ends\nby
Laurent Saloff-Coste (Cornell University) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nFor over twenty years A.
Grigor'yan and the speaker have studied heat kernel estimates on manifolds
with a finite number of nice ends.\nDespite these efforts\, question rema
ins. In this talk\, after giving an overview of what the problem is and wh
at we know\, the main difficulty will be explained and recent progresses i
nvolving joint work with Grigor'yan and Ishiwata will be explained. They p
rovide results concerning Poincaré inequality in large central balls on s
uch manifold.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Dolbeault (Université Paris-Dauphine)
DTSTART:20200914T130000Z
DTEND:20200914T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/16
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg inequalities\nby Jea
n Dolbeault (Université Paris-Dauphine) as part of Geometric and function
al inequalities and applications\n\n\nAbstract\nOptimal constants and opti
mal functions are known in some functional inequalities. The next question
is the stability issue: is the difference of the two terms controlling a
distance to the set of optimal functions ? A famous example is provided by
Sobolev's inequalities: in 1991\, G. Bianchi and H. Egnell proved that th
e difference of the two terms is bounded from below by a distance to the m
anifold of the Aubin-Talenti functions. They argued by contradiction and g
ave a very elegant although not constructive proof. Since then\, estimatin
g the stability constant and giving a constructive proof has been a challe
nge. \n\nThis lecture will focus mostly on subcritical inequalities\, for
which explicit constants can be provided. The main tool is based on entrop
y methods and nonlinear flows. Proving stability amounts to establish\, un
der some constraints\, a version of the entropy - entropy production inequ
ality with an improved constant. In simple cases\, for instance on the sph
ere\, rather explicit results have been obtained by the « carré du champ
» method introduced by D. Bakry and M. Emery. In the Euclidean space\, r
esults based on constructive regularity estimates for the solutions of the
nonlinear flow and corresponding to a joint research project with Matteo
Bonforte\, Bruno Nazaret\, and Nikita Simonov will be presented.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungang Li (Brown University)
DTSTART:20200928T130000Z
DTEND:20200928T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/17
DESCRIPTION:Title: Higher order Brezis-Nirenberg problems on hyperbolic spac
es\nby Jungang Li (Brown University) as part of Geometric and function
al inequalities and applications\n\n\nAbstract\nThe Brezis-Nirenberg probl
em considers elliptic equations whose nonlinearity is associated with crit
ical Sobolev exponents. In this talk we will discuss a recent progress on
higher order Brezis-Nirenberg problem on hyperbolic spaces. The existence
of solutions relates closely to the study of higher order sharp Hardy-Sobo
lev-Maz'ya inequalities\, which is due to G. Lu and Q. Yang. On the other
hand\, we obtain a nonexistence result on star-shaped domains. In addition
\, with the help of Green's function estimates\, we apply moving plane met
hod to establish the symmetry of positive solutions. This is a joint work
with Guozhen Lu and Qiaohua Yang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Flynn (University of Connecticut)
DTSTART:20201102T150000Z
DTEND:20201102T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/18
DESCRIPTION:Title: Sharp Caffarelli-Kohn-Nirenberg Inequalities for Grushin
Vector Fields and Iwasawa Groups.\nby Joshua Flynn (University of Conn
ecticut) as part of Geometric and functional inequalities and applications
\n\n\nAbstract\nSharp Caffarelli-Kohn-Nirenberg inequalities are establish
ed for the Grushin vector fields and for Iwasawa groups (i.e.\, the bounda
ry group of a real rank one noncompact symmetric space). For all but one p
arameter case\, this is done by introducing a generalized Kelvin transform
which is shown to be an isometry of certain weighted Sobolev spaces. For
the exceptional parameter case\, the best constant is found for the Grushi
n vector fields by introducing Grushin cylindrical coordinates and studyin
g the transformed Euler-Lagrange equation.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristian Cazacu (University of Bucharest)
DTSTART:20201012T130000Z
DTEND:20201012T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/19
DESCRIPTION:Title: Optimal constants in Hardy and Hardy-Rellich type inequa
lities\nby Cristian Cazacu (University of Bucharest) as part of Geomet
ric and functional inequalities and applications\n\n\nAbstract\nIn this ta
lk we discuss Hardy and Hardy-Rellich type inequalities\, so important in
establishing useful properties for differential operators with singular po
tentials and their PDEs. We recall some well-known and recent results and
present some new extensions. We analyze singular potentials with one or va
rious singularities. The tools of our proofs are mainly based on the metho
d of supersolutions\, proper transformations and spherical harmonics decom
position. We also focus on the best constants and the existence/nonexisten
ce of minimizers in the energy space. This presentation is partially supp
orted by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuyi Zhu (Louisiana State University)
DTSTART:20201019T140000Z
DTEND:20201019T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/20
DESCRIPTION:Title: The bounds of nodal sets of eigenfunctions\nby Jiuyi
Zhu (Louisiana State University) as part of Geometric and functional inequ
alities and applications\n\n\nAbstract\nMotivated by Yau's conjecture\, th
e study of the measure of nodal sets (Zero level sets) for eigenfunctions
is interesting. We investigate the measure of nodal sets for Steklov\, Di
richlet and Neumann eigenfunctions in the domain and on the boundary of th
e domain. For Dirichlet or Neumann eigenfunctions in the analytic domains
\, we show some sharp upper bounds of nodal sets which touch the boundar
y. We will also discuss some upper bounds of nodal sets for eigenfunction
s of general eigenvalue problems. Furthermore\, some sharp doubling inequa
lities and vanishing order are obtained. Part of the talk is based on jo
int work with Fanghua Lin.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Almut Burchard (University of Toronto)
DTSTART:20201214T140000Z
DTEND:20201214T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/21
DESCRIPTION:Title: Rearrangement inequalities on spaces of bounded mean osci
llation\nby Almut Burchard (University of Toronto) as part of Geometri
c and functional inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giovanna Citti (University of Bologna)
DTSTART:20201207T140000Z
DTEND:20201207T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/22
DESCRIPTION:Title: Degree preserving variational formulas for submanifolds\nby Giovanna Citti (University of Bologna) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nI present a joint wor
k with M. Ritoré and G. Giovannardi related to an area functional for \ns
ubmanifolds of fixed degree immersed in a graded manifold. The expression
of this area functional \nstrictly depends on the degree of the manifold\,
so that\, while computing the first variation\, \nwe need to keep fixed i
ts degree. We will show that there are isolated surfaces\, \nfor which thi
s type of degree preserving variations do not exist: they can be consider
ed \nhigher dimensional extension of the subriemannian abnormal geodesics.
\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annalisa Baldi (University of Bologna)
DTSTART:20201130T140000Z
DTEND:20201130T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/23
DESCRIPTION:Title: Poincaré and Sobolev inequalities for differential forms
in Euclidean spaces and Heisenberg groups\nby Annalisa Baldi (Univers
ity of Bologna) as part of Geometric and functional inequalities and appli
cations\n\n\nAbstract\nIn this talk I present some recent results obtained
in collaboration with B. Franchi and P. Pansu about Poincaré and Sobolev
inequalities for differential forms in Heisenberg groups (some results ar
e new also for Euclidean spaces). For L^p\, p>1\, the estimates are conseq
uence of singular integral estimates. In the limiting case L^1\, the sin
gular integral estimates are replaced with inequalities which go back to B
ourgain-Brezis and Lanzani-Stein in Euclidean spaces\, and to Chanillo-Van
Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups. Also the case p
=Q (Q is the homogeneous dimension of the Heisenberg group ) is considered
.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Betsy Stovall (University of Wisconsin-Madison)
DTSTART:20201214T150000Z
DTEND:20201214T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/24
DESCRIPTION:Title: Fourier restriction to degenerate hypersurfaces\nby B
etsy Stovall (University of Wisconsin-Madison) as part of Geometric and fu
nctional inequalities and applications\n\n\nAbstract\nIn this talk\, we wi
ll describe various open questions and recent progress on the Fourier rest
riction problem associated to hypersurfaces with varying or vanishing curv
ature.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Gursky (University of Notre Dame)
DTSTART:20210118T140000Z
DTEND:20210118T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/25
DESCRIPTION:Title: Extremal Eigenvalues of the conformal laplacian\nby M
atthew Gursky (University of Notre Dame) as part of Geometric and function
al inequalities and applications\n\n\nAbstract\nI will report on joint wor
k with Samuel Perez-Ayala in which we consider the problem of extremizing
eigenvalues of the conformal laplacian in a fixed conformal class. This g
eneralizes the problem of extremizing the eigenvalues of the laplacian on
a compact surface. I will explain the connection of this problem to the e
xistence of harmonic maps\, and to nodal solutions of the Yamabe problem (
first noticed by Ammann-Humbert).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhen-Qing Chen (University of Washington)
DTSTART:20210125T150000Z
DTEND:20210125T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/26
DESCRIPTION:Title: Stability of Elliptic Harnack Inequality\nby Zhen-Qin
g Chen (University of Washington) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nHarnack inequality\, if it holds\
, is a useful tool in analysis and probability theory.\nIn this talk\, I w
ill discuss scale invariant elliptic Harnack inequality for general diffus
ions\, or equivalently\, for general differential operators on metric meas
ure spaces\, and show that it is stable under form-comparable perturbation
s for strongly local Dirichlet forms on complete locally compact separable
metric spaces that satisfy metric doubling property. \nBased on Joint wo
rk with Martin Barlow and Mathav Murugan.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilles Carron (University of Nantes)
DTSTART:20210111T140000Z
DTEND:20210111T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/27
DESCRIPTION:Title: Euclidean heat kernel rigidity\nby Gilles Carron (Uni
versity of Nantes) as part of Geometric and functional inequalities and ap
plications\n\n\nAbstract\nThis is joint work with David Tewodrose (Bruxel
les). I will explain that a metric measure space with Euclidean heat kerne
l are Euclidean. An almost rigidity result comes then for free\, and this
can be used to give another proof of Colding's almost rigidity for comple
te manifold with non negative Ricci curvature and almost Euclidean growth
.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yehuda Pinchover (Technion -Israel Institute of Technology)
DTSTART:20210215T140000Z
DTEND:20210215T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/28
DESCRIPTION:Title: On families of optimal Hardy-weights for linear second-or
der elliptic operators.\nby Yehuda Pinchover (Technion -Israel Institu
te of Technology) as part of Geometric and functional inequalities and app
lications\n\n\nAbstract\nWe construct families of optimal Hardy-weights fo
r a subcritical linear second-order elliptic operator using a one-dimensio
nal reduction. More precisely\, we first characterize all optimal Hardy-w
eights with respect to one-dimensional subcritical Sturm-Liouville operato
rs on $(a\,b)$\, $\\infty \\leq a < b \\leq \\infty$\, and then apply
this result to obtain families of optimal Hardy inequalities for general l
inear second-order elliptic operators in higher dimensions. This is a join
t work with Idan Versano.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ling Xiao (University of Connecticut)
DTSTART:20201109T140000Z
DTEND:20201109T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/29
DESCRIPTION:Title: Entire spacelike constant $\\sigma_{n-1}$ curvature in Mi
nkowski space\nby Ling Xiao (University of Connecticut) as part of Geo
metric and functional inequalities and applications\n\n\nAbstract\nWe prov
e that\, in the Minkowski space\, if a spacelike\, (n − 1)-convex hypers
urface M with constant $\\sigma_{n−1}$ curvature has bounded principal c
urvatures\, then M is convex. Moreover\, if M is not strictly convex\, aft
er an R^{n\,1} rigid motion\, M splits as a product $M^{n−1}\\times R.$
We also construct nontrivial examples of strictly convex\, spacelike hyper
surface M with constant $\\sigma_{n−1}$ curvature and bounded principal
curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Dindos (The University of Edinburgh)
DTSTART:20210208T140000Z
DTEND:20210208T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/30
DESCRIPTION:Title: On p-ellipticity and connections to solvability of ellipt
ic complex valued PDEs\nby Martin Dindos (The University of Edinburgh)
as part of Geometric and functional inequalities and applications\n\n\nAb
stract\nThe notion of an elliptic partial differential equation (PDE)\ngoe
s back at least to 1908\, when it appeared in a paper J. Hadamard. In\nth
is talk we present a recently discovered structural condition\, called\n$p
$-ellipticity\, which generalizes classical ellipticity. It was\nco-disco
vered independently by Carbonaro and Dragicevic on one hand\, and\nPipher
and myself on the other\, and plays a fundamental role in many\nseemingly
mutually unrelated aspects of the $L^p$ theory of elliptic\ncomplex-valued
PDE. So far\, $p$-ellipticity has proven to be the key\ncondition for:\n
\n(i) convexity of power functions (Bellman functions)\n(ii) dimension-fre
e bilinear embeddings\,\n(iii) $L^p$-contractivity and boundedness of semi
groups $(P_t^A)_{t>0}$\nassociated with elliptic operators\,\n(iv) holomor
phic functional calculus\,\n(v) multilinear analysis\,\n(vi) regularity th
eory of elliptic PDE with complex coefficients.\n\nDuring the talk\, I wil
l describe my contribution to this development\, in\nparticular to (vi).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Saikat Mazumdar (Indian Institute of Technology Bombay)
DTSTART:20210222T140000Z
DTEND:20210222T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/31
DESCRIPTION:Title: EXISTENCE RESULTS FOR THE HIGHER-ORDER $Q$-CURVATURE EQU
ATION\nby Saikat Mazumdar (Indian Institute of Technology Bombay) as p
art of Geometric and functional inequalities and applications\n\n\nAbstrac
t\nIn this talk\, we will obtain some existence results for the $Q$-curvat
ure equation\nof arbitrary $2k$-th order\, where $k \\geq 1$ is an integer
\, on a compact Riemannian\nmanifold of dimension $n \\geq 2k + 1$. This a
mounts to solving a nonlinear elliptic\nPDE involving the powers of Laplac
ian called the GJMS operator. The difficulty\nin determining the explicit
form of this GJMS operator together with a lack of\nmaximum principle comp
licates the issues of existence.\nThis is a joint work with Jérôme Véto
is (McGill University).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Del Pino (University of Bath)
DTSTART:20210301T140000Z
DTEND:20210301T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/33
DESCRIPTION:Title: Dynamics of concentrated vorticities in 2d and 3d Euler f
lows\nby Manuel Del Pino (University of Bath) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nA classical probl
em that traces back to Helmholtz and Kirchoff is the understanding \nof th
e dynamics of solutions to the 2d and 3d Euler equations of an inviscid in
compressible \nfluid\, when the vorticity of the solution is initially con
centrated near isolated points in 2d or \nvortex lines in 3d. We discuss s
ome recent result on existence and asymptotic behaviour of \nthese solutio
ns. We describe\, with precise asymptotics\, interacting vortices and tra
velling helices. We rigorously establish the law of of motion of of "leap
frogging vortex rings"\, originally conjectured by Helmholtz in 1858. Thi
s is joint work with Juan Davila\, Monica Musso and Juncheng Wei.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrique Zuazua (Friedrich-Alexander-Universität)
DTSTART:20210308T140000Z
DTEND:20210308T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/34
DESCRIPTION:Title: UNILATERAL BOUNDS FOR NONLINEAR SEMIGROUPS AND TIME-INVER
SION\nby Enrique Zuazua (Friedrich-Alexander-Universität) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nSome
classical nonlinear semigroups arising in mechanics induce unilateral bou
nds on solutions. \nHamilton--Jacobi equations and 1-d scalar conservatio
n laws are classical examples of such nonlinear effects: solutions spontan
eously develop one-sided Lipschitz or semi-concavity conditions.\n\nWhen t
his occurs the range of the semigroup is unilaterally bounded by a thresho
ld.\n\nOn the other hand\, in practical applications\, one is led to consi
der the problem of time-inversion\, so to identify the initial sources tha
t have led to the observed dynamics at the final time.\n\nIn this lecture
we shall discuss this problem answering to the following two questions: On
one hand\, to identify the range of the semigroup and\, given a target\,
to characterize and reconstruct the ensemble of initial data leading to it
.\n\nIllustrative numerical simulations will be presented\, and a complet
e geometric interpretation will also be provided.\n\nWe shall also present
a number of open problems arising in this area and the possible link with
reinforcement learning.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Street (University of Wisconsin-Madison)
DTSTART:20210329T130000Z
DTEND:20210329T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/35
DESCRIPTION:Title: Maximal Hypoellipticity\nby Brian Street (University
of Wisconsin-Madison) as part of Geometric and functional inequalities and
applications\n\n\nAbstract\nIn 1974\, Folland and Stein introduced a gene
ralization of ellipticity known as maximal hypoellipticity. This talk wil
l be an introduction to this concept and some of the ways it generalizes e
llipticity.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Kenig (University of Chicago)
DTSTART:20210426T140000Z
DTEND:20210426T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/36
DESCRIPTION:Title: Wave maps into the sphere\nby Carlos Kenig (Universit
y of Chicago) as part of Geometric and functional inequalities and applica
tions\n\n\nAbstract\nWe will introduce wave maps\, an important geometric
flow\, and\ndiscuss\, for the case when the target is the sphere\, the asy
mptotic\nbehavior near the ground state (without symmetry) and recent resu
lts in\nthe general case (under co-rotational symmetry) in joint work wit
h\nDuyckaerts\, Martel and Merle.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Man Wah Wong (York University)
DTSTART:20210315T140000Z
DTEND:20210315T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/37
DESCRIPTION:Title: Spectral Theory and Number Theory of the Twisted Bi-Lapla
cian\nby Man Wah Wong (York University) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nWe begin with the sub-L
aplacian on the Heisenberg group and then the twisted Laplacian by taking
its inverse Fourier transform with respect to the center of the group. The
eigenvalues and the eigenfunctions of the twisted Laplacian are computed
explicitly. Then we turn our attention to the product of the twisted Lapla
cian and its transpose\, thus obtaining a fourth order partial differentia
l operator dubbed the twisted bi-Laplacian. The connections between the sp
ectral analysis of the twisted bi-Laplacian and Dirichlet divisors\, the R
iemann zeta function and the Dixmier trace are explained.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yanyan Li (Rutgers University)
DTSTART:20210419T130000Z
DTEND:20210419T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/38
DESCRIPTION:Title: Regular solutions of the stationary Navier-Stokes equatio
ns on high dimensional Euclidean space\nby Yanyan Li (Rutgers Universi
ty) as part of Geometric and functional inequalities and applications\n\n\
nAbstract\nWe study the existence of regular solutions of the incompressib
le stationary Navier-Stokes equations in n-dimensional Euclidean space wit
h a given bounded external force of compact support. In dimensions $n\\le
5$\, the existence of such solutions was known. In this paper\, we extend
it to dimensions $n\\le 15$. This is a joint work with Zhuolun Yang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spring Recess (No Talk)
DTSTART:20210412T130000Z
DTEND:20210412T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/39
DESCRIPTION:by Spring Recess (No Talk) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenxiong Chen (Yeshiva University)
DTSTART:20210322T140000Z
DTEND:20210322T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/40
DESCRIPTION:Title: Asymptotic radial symmetry\, monotonicity\, non-existence
for solutions to fractional parabolic equations\nby Wenxiong Chen (Ye
shiva University) as part of Geometric and functional inequalities and app
lications\n\n\nAbstract\nIn this talk\, we will consider nonlinear parabol
ic fractional equations\n\nWe develop a systematical approach in applying
an asymptotic method\nof moving planes to investigate qualitative properti
es of positive solutions for\nfractional parabolic equations. To this end\
, we derive a series of needed key\ningredients such as narrow region prin
ciples\, and various asymptotic maximum and strong maximum principles for
antisymmetric functions in both bounded and unbounded domains. Then we ill
ustrate how these new methods can be employed to obtain asymptotic radial
symmetry and monotonicity\nof positive solutions in a unit ball and on the
whole space. Namely\, we show\nthat no matter what the initial data are\,
the solutions will eventually approach to radially symmetric functions. W
e will also consider the entire positive solutions on a half space\, in\nt
he whole space\, and with indefinite nonlinearity. Monotonicity and nonexi
stence of solutions are obtained. This is joint work with P. Wang\, Y. Niu
\, Y. Hu and L. Wu.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunfeng Zhang (University of Connecticut)
DTSTART:20210201T140000Z
DTEND:20210201T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/41
DESCRIPTION:Title: Schr\\"odinger equations on compact globally symmetric sp
aces\nby Yunfeng Zhang (University of Connecticut) as part of Geometri
c and functional inequalities and applications\n\n\nAbstract\nLet $M$ be a
compact manifold of dimension $d$. Scale-invariant Strichartz estimates o
f the form\n\n$$\\|e^{it\\Delta}f\\|_{L^p(I\\times M)}\\lesssim \\|f\\|_{H
^{d/2-(d+2)/p}(M)}$$\n\nhave only been proved for a few model cases of $M$
\, most of which are compact globally symmetric spaces.\n\nIn this talk\,
we report that the above estimate holds true on an arbitrary compact globa
lly symmetric space $M$ equipped with the canonical Killing metric\, for a
ll $p\\geq 2+8/r$\, where $r$ denotes the rank of $M$. As an immediate app
lication\, we provide local well-posedness results for nonlinear Schr\\"od
inger equations of polynomial nonlinearities of degree $\\beta\\geq 4$ on
any compact globally symmetric space of large enough rank\, in all subcrit
ical spaces.\n\nWe also discuss bilinear Strichartz estimates on compact g
lobally symmetric spaces\, and critical and subcritical local well-posedne
ss results for the cubic nonlinearity.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungang Li (Brown University)
DTSTART:20210503T140000Z
DTEND:20210503T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/42
DESCRIPTION:Title: Sharp critical and subcritical Moser-Trudinger inequaliti
es on complete and noncompact Riemannian manifolds\nby Jungang Li (Bro
wn University) as part of Geometric and functional inequalities and applic
ations\n\n\nAbstract\nMoser-Trudinger inequality is the borderline case of
the Sobolev inequality and has many applications in differential geometry
. In this talk\, I will report a recent progress of critical and subcritic
al Moser-Trudinger inequalities on complete noncompact Riemannian manifold
s. Classical results depend heavily on the validity of some rearrangement
inequalities\, which are unavailable on general manifolds. To overcome thi
s difficulty\, we applied a rearrangement-free approach to obtain those in
equalities on manifolds\, together with their sharp constants. This is a j
oint work with Guozhen Lu.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Struwe (ETH Zürich)
DTSTART:20210510T140000Z
DTEND:20210510T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/43
DESCRIPTION:Title: Normalized harmonic map flow\nby Michael Struwe (ETH
Zürich) as part of Geometric and functional inequalities and applications
\n\n\nAbstract\nFinding non-constant harmonic 3-spheres for a closed targe
t manifold N\nis a prototype of a super-critical variational problem. In f
act\, the\ndirect method fails\, as the infimum of Dirichlet energy in any
homotopy\nclass of maps from the 3-sphere to any closed N is zero\; moreo
ver\, the\nharmonic map heat flow may blow up in finite time\, and even th
e identity\nmap from the 3-sphere to itself is not stable under this flow.
\n\nTo overcome these difficulties\, we propose the normalized harmonic ma
p\nheat flow as a new tool\, and we show that for this flow the identity m
ap\nfrom the 3-sphere to itself now\, indeed\, is stable\; moreover\, the
flow\nconverges to a harmonic 3-sphere also when we perturb the target\nge
ometry. While our results are strongest in the perturbative setting\,\nwe
also outline a possible global theory.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No talk
DTSTART:20210517T140000Z
DTEND:20210517T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/44
DESCRIPTION:by No talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk (Memorial Day)
DTSTART:20210531T140000Z
DTEND:20210531T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/45
DESCRIPTION:by No Talk (Memorial Day) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaojun Huang (Rutgers University)
DTSTART:20210524T130000Z
DTEND:20210524T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/46
DESCRIPTION:Title: Revisit to a non-degeneracy property for extremal mapping
s\nby Xiaojun Huang (Rutgers University) as part of Geometric and func
tional inequalities and applications\n\n\nAbstract\nI will discuss a gener
alization of my previous result on the localization of extremal maps near
a strongly pseudo-convex point.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sun-Yung Alice Chang (Princeton University)
DTSTART:20210607T140000Z
DTEND:20210607T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/47
DESCRIPTION:Title: On bi-Lipschitz equivalence of a class of non-conformally
flat spheres\nby Sun-Yung Alice Chang (Princeton University) as part
of Geometric and functional inequalities and applications\n\n\nAbstract\nT
his is a report of some recent joint work with Eden Prywes and Paul Yang.
The main\nresult is a bi-Lipschitz equivalence of a class of metrics on 4-
shpere under curvature constraints. The proof involves two steps: first a
construction of quasiconformal maps between\ntwo conformally related metri
cs in a positive Yamabe class\, followed by the step of applying\nthe Ricc
i flow to establish the bi-Lipschitz equivalence from such a conformal cla
ss to the\nstandard conformal class on 4-sphere.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svitlana Mayboroda (University of Minnesota)
DTSTART:20210614T140000Z
DTEND:20210614T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/48
DESCRIPTION:Title: Green Function vs. Geometry\nby Svitlana Mayboroda (U
niversity of Minnesota) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nIn this talk we will discuss connections be
tween the geometric and PDE properties of sets. The emphasis is on quantif
iable\, global results which yield true equivalence between the geometric
and PDE notions in very rough scenarios\, including domains and equations
with singularities and structural complexity. The main result establishes
that in all dimensions $d < n$\, a $d$-dimensional set in $\\mathbb{R}^n$
is regular (rectifiable) if and only if the Green function for elliptic op
erators is well approximated by affine functions (distance to the hyperpla
nes). To the best of our knowledge\, this is the first free boundary resul
t of this type for lower dimensional sets and the first free boundary resu
lt in the classical case $d=n-1$ without restrictions on the coefficients
of the equation.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanna Terracini (Universitá di Torino)
DTSTART:20210628T130000Z
DTEND:20210628T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/49
DESCRIPTION:Title: Free boundaries in segregation problems\nby Susanna T
erracini (Universitá di Torino) as part of Geometric and functional inequ
alities and applications\n\n\nAbstract\nWe first consider classes of varia
tional problems for densities that repel each other at distance. Examples
are given by the minimizers of Dirichlet functional or the Rayleigh quotie
nt\n\\[\n D({\\bf u}) = \\sum_{i=1}^k \\int_{\\Omega} |\\nabla u_i|^2 \\q
uad \\text{or} \\quad R({\\bf u}) = \\sum_{i=1}^k \\frac{\\int_{\\Omega}
|\\nabla u_i|^2}{\\int_{\\Omega} u_i^2}\n\\]\nover the class of $H^1(\\Ome
ga\,\\R^k)$ functions attaining some boundary conditions on $\\partial \\O
mega$\, and subjected to the constraint \n\\[\n \\operatorname{dist} (\\{
u_i > 0\\}\, \\{u_j > 0\\}) \\ge 1 \\qquad \\forall i \\neq j.\n\\]\n\n\nA
s second class of problems\, we consider energy minimizers of Dirichlet e
nergies with different metrics\n\\[\n D({\\bf u}) = \\sum_{i=1}^k \\int_{
\\Omega} \\langle A_i\\nabla u_i\, \\nabla u_i\\rangle\n\\]\nwith constrai
nt\n\\[\n u_i(x)\\cdot u_j(x)=0\, \\qquad \\forall x\\in \\Omega\\\;\, \\f
orall i \\neq j.\n\\]\n\nFor these problems\, we investigate the optimal r
egularity of the solutions\, prove a free-boundary extremality condition\,
and derive some preliminary results characterising the emerging free boun
dary.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roger Moser (University of Bath)
DTSTART:20210621T130000Z
DTEND:20210621T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/50
DESCRIPTION:Title: The infinity-elastica problem\nby Roger Moser (Univer
sity of Bath) as part of Geometric and functional inequalities and applica
tions\n\n\nAbstract\nThe Euler elastica problem seeks to minimise the $L^2
$-norm of\nthe curvature of curves under certain boundary conditions. If w
e\nreplace the $L^2$-norm with the $L^\\infty$-norm\, then we obtain a\nva
riational problem with quite different properties. Nevertheless\, even\nth
ough the underlying functional is not differentiable\, it turns out\nthat
the solutions of the problem can still be described by\ndifferential equat
ions. An analysis of these equations then gives a\nclassification of the s
olutions.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jingzhi Tie (University of Georgia)
DTSTART:20210405T140000Z
DTEND:20210405T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/51
DESCRIPTION:Title: CR analogue of Yau’s Conjecture on pseudo harmonic func
tions of polynomial growth.\nby Jingzhi Tie (University of Georgia) as
part of Geometric and functional inequalities and applications\n\n\nAbstr
act\nCheng and Yau derived the well-known gradient estimate for positive h
armonic functions and obtained the classical Liouville theorem\, which sta
tes that any bounded harmonic function is constant in complete noncompact
Riemannian manifolds with nonnegative Ricci curvature. I will talk about t
he CR analogue of Yau’s conjecture. We need to derive the CR volume doub
ling property\, CR\\ Sobolev inequality\, and mean value inequality. Then
we can apply them to prove the CR analogue of Yau's conjecture on the spac
e consisting of all pseudoharmonic functions of polynomial growth of degre
e at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold.
As a by-product\, we obtain the CR analogue of volume growth estimate and
Gromov precompactness theorem.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lorenzo D'Ambrosio (Universita di Bari)
DTSTART:20210705T130000Z
DTEND:20210705T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/52
DESCRIPTION:Title: Liouville theorems for semilinear biharmonic equations an
d inequalities\nby Lorenzo D'Ambrosio (Universita di Bari) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nWe s
tudy nonexistence results for a coercive semilinear biharmonic equation on
the whole $R^N$. The analysis is made for general solutions without any a
ssumption on their sign nor on their behaviour at infinity. A relevant rol
e is played by some extensions of the Hardy-Rellich inequalities for gener
al functions (not necessarily compactly supported).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (University of Chicago)
DTSTART:20210913T140000Z
DTEND:20210913T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/53
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without c
utoff\nby Luis Silvestre (University of Chicago) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe study the r
egularization effect of the inhomogeneous Boltzmann equation without cutof
f. We obtain a priori estimates for all derivatives of the solution depend
ing only on bounds of its hydrodynamic quantities: mass density\, energy d
ensity and entropy density. As a consequence\, a classical solution to the
equation may fail to exist after a certain time T only if at least one of
these hydrodynamic quantities blows up. Our analysis applies to the case
of moderately soft and hard potentials. We use methods that originated in
the study of nonlocal elliptic and parabolic equations: a weak Harnack ine
quality in the style of De Giorgi\, and a Schauder-type estimate.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshikazu Giga (University of Tokyo)
DTSTART:20210927T130000Z
DTEND:20210927T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/54
DESCRIPTION:Title: On the Helmholtz decomposition of BMO spaces of vector fi
elds\nby Yoshikazu Giga (University of Tokyo) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nThe Helmholtz dec
omposition of vector fields is a fundamental tool for analysis of vector f
ields especially to analyze the Navier-Stokes equations in a domain. It gi
ves a unique decomposition of a (tangential) vector field defined in a dom
ain of an Euclidean space (or a riemannian maniford) into a sum of a gradi
ent field and a solenoidal field with supplemental condition like a bounda
ry condition.It is well-known that such decomposition gives an orthogonal
decomposition of the space of L^2 vector fields in an arbitrary domain an
d known as the Weyl decomposition. It is also well-studied that in various
domains including the half space\, smooth bounded and exterior domain\, i
t gives a topological direct sum decomposition of the space of L^p vector
fields for 1 < p < ∞. The extension to the case p=∞ (or p=1) is impo
ssible because otherwise it would imply the boundedness of the Riesz type
operator in L^∞ (or L^1) which is absurd.\n In this talk\, we extend th
e Hemlholtz decomposition in a space of vector fields with bounded mean os
cillations (BMO) when the domain of vector field is a smooth bounded domai
n in an Euclidean space. There are several possible definitions of\na BMO
space of vector fields. However\, to have a topological direct sum decompo
sition\, it turns out that components of normal and tangential to the bou
ndary should be handled separately.\n This decomposition problem is equiv
alent to solve the Poisson equation with the divergence of the original ve
ctor field v as a data with the Neumann data with the normal trace of v. T
he desired gradient field is the gradient of the solution of this Poisson
equation. To solve this problem we construct a kind of volume potential so
that the problem is reduced to the Neumann problem for the Laplace equati
on. Unfortunately\, taking usual Newton potential causes a problem to esti
mate necessary norm so we construct another volume potential based on norm
al coordinate.We need a trace theorem to control L^∞ norm of the normal
trace. This is of independent interest. Finally\,we solve the Neumann prob
lem with L^∞ data in a necessary space. The Helmholtz decomposition fo
r BMO vector fields is previously known only in the whole Euclidean space
or the half space so this seems to be the first result for a domain with a
curved boundary. This is a joint work with my student Z.Gu (University of
Tokyo).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jill Pipher (Brown University)
DTSTART:20210920T140000Z
DTEND:20210920T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/55
DESCRIPTION:Title: Boundary value problems for p-elliptic operators\nby
Jill Pipher (Brown University) as part of Geometric and functional inequal
ities and applications\n\n\nAbstract\nWe give some background about the re
gularity of solutions to real and complex elliptic operators\, motivating
a new algebraic condition (p-ellipticity). We introduce this condition in
order to solve new boundary value problems for operators with complex coef
ficients. Results with M. Dindos\, and with M. Dindos and J. Li\, are disc
ussed.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sundaram Thangavelu (INDIAN INSTITUTE OF SCIENCE)
DTSTART:20211004T130000Z
DTEND:20211004T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/56
DESCRIPTION:Title: On the extension problem for the sublaplacian on the Heis
enberg group\nby Sundaram Thangavelu (INDIAN INSTITUTE OF SCIENCE) as
part of Geometric and functional inequalities and applications\n\n\nAbstra
ct\nIn this talk we plan to describe some results on the extension problem
associated to the sublaplacian $ \\mathcal{L} $ on the Heisenberg group $
\\H^n .$ The Dirichlet to Neumann map induced by this problem leads to co
nformally invariant fractional powers of $ \\mathcal{L}.$ We use the resul
ts to prove a version of Hardy's inequality for such fractional powers. Th
ese results are based on my joint work with Luz Roncal.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriele Grillo (Politecnico di Milano)
DTSTART:20211011T130000Z
DTEND:20211011T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/57
DESCRIPTION:Title: Nonlinear characterizations of stochastic completeness\nby Gabriele Grillo (Politecnico di Milano) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nA manifold is said to
be stochastically complete if the free heat semigroup preserves probabili
ty. It is well-known that this property is equivalent to nonexistence of n
onnegative\, bounded solutions to certain (linear) elliptic problems\, and
to uniqueness of solutions to the heat equation corresponding to bounded
initial data. We prove that stochastic completeness is also equivalent to
similar properties for certain nonlinear elliptic and parabolic problems.
This fact\, and the known analytic-geometric characterizations of stochast
ic completeness\, allow to give new explicit criteria for existence/nonexi
stence of solutions to certain nonlinear elliptic equations on manifolds\,
and for uniqueness/nonuniqueness of solutions to certain nonlinear diffus
ions on manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (University of Washington)
DTSTART:20211025T140000Z
DTEND:20211025T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/58
DESCRIPTION:Title: Mean-Value Inequalities for Convex Domains\nby Stefan
Steinerberger (University of Washington) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nThe Mean Value Theorem im
plies that the average value of a subharmonic\nfunction in a disk can be b
ounded from above by the average value on the boundary. \nWhat happens if
we replace the disk by another domain? Maybe surprisingly\, the problem
\nhas a relatively clean answer -- we discuss a whole range of mean value
inequalities for \nconvex domains in IR^n. The extremal domain remains a
mystery for most of them. \nThe techniques are an amusing mixture of class
ical potential theory\, complex analysis\,\na little bit of elliptic PDEs
and\, surprisingly\, the theory of solids from the 1850s.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Manfredi (University of Pittsburgh)
DTSTART:20211129T140000Z
DTEND:20211129T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/62
DESCRIPTION:Title: NATURAL $p$-MEANS FOR THE $p$-LAPLACIAN IN EUCLIDEAN SPAC
E AND THE HEISENBERG GROUP\nby Juan Manfredi (University of Pittsburg
h) as part of Geometric and functional inequalities and applications\n\n\n
Abstract\nWe consider semi-discrete approximations to $p$-harmonic functio
ns based on the natural\n$p$-means of Ishiwata\, Magnanini\, and Wadade in
2017 (CVPDE 2017)\, who proved their local convergence. In the Euclidean
case we prove uniform convergence in bounded Lipschitz domains. We also co
nsider adapted semi-discrete approximations in the Heisenberg group $\\mat
hbb{H}$ and prove uniform convergence in bounded $C^{1\,1}$-domains.\n\nTh
is talk is based in joint work with András Domokos and Diego Ricciotti (S
acramento)\nand Bianca Stroffolini (Naples)\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guofang Wang (University of Freiburg)
DTSTART:20211101T140000Z
DTEND:20211101T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/63
DESCRIPTION:Title: Geometric inequalities in the hyperbolic space and their
applications.\nby Guofang Wang (University of Freiburg) as part of Geo
metric and functional inequalities and applications\n\n\nAbstract\nWe will
talk about Alexandrov-Fenchel type inequalities in the hyperbolic space a
nd their applications in a higher order mass of asymptotically hyperbolic
manifolds. The talk is based on a series of work joint with Yuxin Ge\, Jie
Wu and Chao Xia\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Yang (Princeton University)
DTSTART:20220207T150000Z
DTEND:20220207T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/64
DESCRIPTION:Title: Sturm comparison for Jacobi vector fields and application
s\nby Paul Yang (Princeton University) as part of Geometric and functi
onal inequalities and applications\n\n\nAbstract\nFor CR manifolds of real
dimension three\, we study the Jacobi field equation. Under the condition
that the torsion be parallel\, we obtain comparison results against a fam
ily of homogeneous CR structures. As application\, we describe the singula
rities of contact forms on the the homogeneous structures with finite tota
l Q-prime curvature. This is ongoing joint work with Sagun Chanillo.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Chen (University of California at Berkeley)
DTSTART:20211108T150000Z
DTEND:20211108T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/65
DESCRIPTION:Title: Integral curvature pinching and sphere theorems via the R
icci flow\nby Eric Chen (University of California at Berkeley) as part
of Geometric and functional inequalities and applications\n\n\nAbstract\n
I will discuss how uniform Sobolev inequalities obtained from the monotoni
city of Perelman's W-functional can be used to prove curvature pinching th
eorems on Riemannian manifolds. These are based on scale-invariant integr
al norms and generalize some earlier pointwise and supercritical integral
pinching statements. This is joint work with Guofang Wei and Rugang Ye.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qing Han (University of Notre Dame)
DTSTART:20211206T150000Z
DTEND:20211206T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/67
DESCRIPTION:Title: A Concise Boundary Regularity for the Loewner-Nirenberg P
roblem\nby Qing Han (University of Notre Dame) as part of Geometric an
d functional inequalities and applications\n\n\nAbstract\nLoewner and Nire
nberg discussed complete metrics conformal to the Euclidean metric and wit
h a constant scalar curvature in bounded domains in the Euclidean space. T
he conformal factors blow up on boundary. The asymptotic behaviors of the
conformal factors near boundary are known in smooth and sufficiently smoot
h domains. In this talk\, we introduce the logarithm of the distance to bo
undary as an additional independent self-variable and establish a concise
boundary regularity.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jyotshana Prajapat (University of Mumbai)
DTSTART:20211122T140000Z
DTEND:20211122T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/68
DESCRIPTION:Title: Geodetically convex sets in Heisenberg group $H^n$\nb
y Jyotshana Prajapat (University of Mumbai) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nA classification of geo
detically convex subsets of Heisenberg group of homogeneous dimension 4 wa
s proved by Monti-Rickly. We extend their result to a higher dimension
Heisenberg group. This is ongoing work with my PhD student Anoop Varghese.
\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McCann (University of Toronto)
DTSTART:20211220T150000Z
DTEND:20211220T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/69
DESCRIPTION:Title: Inscribed radius bounds for lower Ricci bounded metric me
asure spaces with mean convex boundary\nby Robert McCann (University o
f Toronto) as part of Geometric and functional inequalities and applicatio
ns\n\n\nAbstract\nConsider an essentially nonbranching metric measure spac
e with the measure contraction property of Ohta and Sturm. We prove a shar
p upper bound on the inscribed radius of any subset whose boundary has a s
uitably signed lower bound on its generalized mean curvature. This provide
s a nonsmooth analog of results dating back to Kasue (1983) and subsequent
authors. We prove a stability statement concerning such bounds and --- in
the Riemannian curvature-dimension (RCD) setting --- characterize the cas
es of equality. This represents joint work with Annegret Burtscher\, Chris
tian Ketterer and Eric Woolgar.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitali Kapovitch (University of Toronto)
DTSTART:20220131T140000Z
DTEND:20220131T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/70
DESCRIPTION:Title: Mixed curvature almost flat manifolds\nby Vitali Kapo
vitch (University of Toronto) as part of Geometric and functional inequali
ties and applications\n\n\nAbstract\nA celebrated theorem of Gromov says t
hat given $n>1$ there is an $\\epsilon(n)>0$ such that if a closed Riemann
ian manifold $M^n$ satisfies $-\\epsilon < sec_M < \\epsilon\, diam(M) < 1
$ then $M$ is diffeomorphic to an infranilmanifold. I will show that the l
ower sectional curvature bound in Gromov’s theorem can be weakened to th
e lower Bakry-Emery Ricci curvature bound. I will also discuss the relatio
n of this result to the study of manifolds with Ricci curvature bounded be
low.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Vega (Basque Center for Applied Mathematics)
DTSTART:20220124T150000Z
DTEND:20220124T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/71
DESCRIPTION:Title: New Conservation Laws and Energy Cascade for 1d Cubic NLS
\nby Luis Vega (Basque Center for Applied Mathematics) as part of Geom
etric and functional inequalities and applications\n\n\nAbstract\nI’ll p
resent some recent results concerning the IVP of 1d cubic NLS at the crit
ical level of regularity. I’ll also exhibit a cascade of energy for the
1D Schrödinger map which is related to NLS through the so called Hasimoto
transformation. For higher regularity these two equations are completely
integrable systems and therefore no cascade of energy is possible.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Mondino (University of Oxford)
DTSTART:20220509T130000Z
DTEND:20220509T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/72
DESCRIPTION:Title: Optimal transport and quantitative geometric inequalities
\nby Andrea Mondino (University of Oxford) as part of Geometric and fu
nctional inequalities and applications\n\n\nAbstract\nThe goal of the talk
is to discuss a quantitative version of the Levy-Gromov isoperimetric ine
quality (joint with Cavalletti and Maggi) as well as a quantitative form o
f Obata's rigidity theorem (joint with Cavalletti and Semola). Given a clo
sed Riemannian manifold with strictly positive Ricci tensor\, one estimate
s the measure of the symmetric difference of a set with a metric ball with
the deficit in the Levy-Gromov inequality. The results are obtained via a
quantitative analysis based on the localisation method via L1-optimal tra
nsport. For simplicity of presentation\, the talk will present the results
in case of smooth Riemannian manifolds with Ricci Curvature bounded below
\; moreover it will not require previous knowledge of optimal transport th
eory.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linhan Li (University of Minnesota)
DTSTART:20211213T150000Z
DTEND:20211213T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/73
DESCRIPTION:Title: Comparison between the Green function and smooth distance
s\nby Linhan Li (University of Minnesota) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nIn the upper half-spa
ce\, the distance function to the boundary is a positive solution to Lapla
ce's equation that vanishes on the boundary\, which can be interpreted as
the Green function with pole at infinity for the Laplacian. We are interes
ted in understanding the exact relations between the behavior of the Green
function\, the structure of the underlying operator\, and the geometry of
the domain. In joint work with G. David and S. Mayboroda\, we obtain a pr
ecise and quantitative control of the proximity of the Green function and
the distance function on the upper half-space by the oscillation of the co
efficients of the operator. The class of the operators that we consider is
of the nature of the best possible for the Green function to behave like
a distance function. More recently\, together with J. Feneuil and S. Maybo
roda\, we obtain analogous results for domains with uniformly rectifiable
boundaries.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaodan Zhou (Okinawa Institute of Science and Technology)
DTSTART:20220221T140000Z
DTEND:20220221T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/74
DESCRIPTION:Title: Quasiconvex envelope in the Heisenberg group\nby Xiao
dan Zhou (Okinawa Institute of Science and Technology) as part of Geometri
c and functional inequalities and applications\n\n\nAbstract\nVarious noti
ons of convexity of sets and functions in the Heisenberg group have been s
tudied in the past two decades. In this talk\, we focus on the horizontall
y quasiconvex ($h$-quasiconvex) functions in the Heisenberg group. Inspire
d by the first-order characterization and construction of quasiconvex enve
lope by Barron\, Goebel and Jensen in the Euclidean space\, we obtain a PD
E approach to construct the $h$-quasiconvex envelope for a given function
$f$ in the Heisenberg group. In particular\, we show the uniqueness and ex
istence of viscosity solutions to a non-local Hamilton-Jacobi equation and
iterate the equation to obtain the $h$-quasiconvex envelope. Relations be
tween $h$-convex hull of a set and the $h$-quasiconvex envelopes are also
investigated. This is joint work with Antoni Kijowski (OIST) and Qing Liu
(Fukuoka University/OIST).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jian Song (Rutgers University)
DTSTART:20220214T150000Z
DTEND:20220214T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/75
DESCRIPTION:Title: Positivity conditions for complex Hessian equations\n
by Jian Song (Rutgers University) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nIn this talk\, we will discuss th
e relation between complex Hessian equation and positivity of algebraic nu
merical conditions. In particular\, we will prove a Naki-Moishezon criteri
on for Donaldson's J-equation.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enno Lenzmann (University of Basel)
DTSTART:20220228T140000Z
DTEND:20220228T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/76
DESCRIPTION:Title: Symmetry and symmetry-breaking for solutions of PDEs via
Fourier methods\nby Enno Lenzmann (University of Basel) as part of Geo
metric and functional inequalities and applications\n\n\nAbstract\nIn this
talk\, I will review recent results on symmetry and symmetry-breaking for
optimizing solutions of a general class of nonlinear elliptic PDEs. On on
e hand\, I will discuss a novel approach to prove symmetry by using the so
-called Fourier rearrangements\, which can be applied to PDEs of arbitrary
order (where classical method such as the moving plane method or the Poly
a-Szegö principle fail short). On the other hand\, I will discuss recent
results on symmetry-breaking for optimizers by using Fourier methods and t
he Stein-Tomas inequality. This talk is based on joint work with Tobias We
th and Jeremy Sok.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carolyn Gordon (Dartmouth College)
DTSTART:20220328T130000Z
DTEND:20220328T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/77
DESCRIPTION:Title: Inverse spectral problems on compact Riemannian orbifolds
\nby Carolyn Gordon (Dartmouth College) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nOrbifolds are a general
ization of manifolds in which various types of singularities may occur.
After reviewing the notion of Riemannian orbifolds and their Hodge Laplaci
ans\, we will address the question: Does the spectrum of the Hodge Laplaci
an on p-forms detect the presence of singularities? This question remains
open in the case of the Laplace-Beltrami operator (i.e.\, the case p=0)\,
although many partial results are known. We will show that the spectra of
the Hodge Laplacians on functions and 1-forms together suffice to distingu
ish manifolds from orbifolds with sufficiently large singular set. In par
ticular\, these spectra always distinguish low-dimensional orbifolds (dime
nsion at most 3) with singularities from smooth manifolds. We also obtain
weaker affirmative results for the spectrum on 1-forms alone and show via
counterexamples that some of these results are sharp.\n\n(This is based o
n recent joint work with Katie Gittins\, Magda Khalile\, Ingrid Membrillo
Solis\, Mary Sandoval\, and Elizabeth Stanhope and work in progress with t
he same co-authors along with Juan Pablo Rossetti.) \n\nTime permitting\,
we will also make a few remarks concerning the Steklov spectrum on Riemann
ian orbifolds with boundary. The Steklov spectrum is the spectrum of the
Dirichlet-to-Neumann operator\, which maps Dirichlet boundary values of h
armonic functions to their Neumann boundary values.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jérôme Vétois (McGill University)
DTSTART:20220307T140000Z
DTEND:20220307T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/78
DESCRIPTION:Title: Stability and instability results for sign-changing solut
ions to second-order critical elliptic equations\nby Jérôme Vétois
(McGill University) as part of Geometric and functional inequalities and a
pplications\n\n\nAbstract\nIn this talk\, we will consider a question of s
tability (i.e. compactness of solutions to perturbed equations) for sign-c
hanging solutions to second-order critical elliptic equations on a closed
Riemannian manifold. I will present a stability result obtained in the cas
e of dimensions greater than or equal to 7. I will then discuss the optima
lity of this result by constructing counterexamples in every dimension. Th
is is a joint work with Bruno Premoselli (Université Libre de Bruxelles\,
Belgium).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Loss (Georgia Institute of Technology)
DTSTART:20220404T130000Z
DTEND:20220404T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/79
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Michae
l Loss (Georgia Institute of Technology) as part of Geometric and function
al inequalities and applications\n\n\nAbstract\nI present some results con
cerning the size of magnetic fields that support zero modes for the three
dimensional Dirac equation and related problems for spinor equations. The
critical quantity\, is the $3/2$ norm of the magnetic field $B$. The point
is that the spinor structure enters the analysis in a crucial way. This i
s joint work with Rupert Frank at LMU Munich.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fritz Gesztesy (Baylor University)
DTSTART:20220516T140000Z
DTEND:20220516T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/80
DESCRIPTION:Title: Continuity properties of the spectral shift function for
massless Dirac operators and an application to the Witten index\nby Fr
itz Gesztesy (Baylor University) as part of Geometric and functional inequ
alities and applications\n\n\nAbstract\nWe report on recent results regard
ing the limiting absorption principle for multi-dimensional\, massless Dir
ac-type operators (implying absence of singularly continuous spectrum) and
continuity properties of the associated spectral shift function.\n\nWe wi
ll motivate our interest in this circle of ideas by briefly describing the
connection to the notion of the Witten index for a certain class of non-F
redholm operators.\n\nThis is based on various joint work with A. Carey\,
J. Kaad\, G. Levitina\, R. Nichols\, D. Potapov\, F. Sukochev\, and D. Zan
in.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Carlen (Rutgers University)
DTSTART:20220411T130000Z
DTEND:20220411T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/83
DESCRIPTION:Title: Some trace inequalities related to quantum entropy\nb
y Eric Carlen (Rutgers University) as part of Geometric and functional ine
qualities and applications\n\n\nAbstract\nMany inequalities for trace func
tional are formulated as concavity/convexity theorems. These generally hav
e an equivalent monotonicity version asserting monotonicity of the functio
nal under some class of completely positive maps. The monotonicty formulat
ion has advantages: (1) Often this has a direct physical interpretation. (
2) Often a direct proof of the monotonicity version is simpler than a dire
ct proof of the concavity/convexity version\, and the later is always reco
vered using a simple partial trace argument. (3) Often the monotonicty the
orem holds for a broader class of maps\, not\, necessarily completely posi
tive\, and is thus a strictly stronger result. We discus significant exam
ples\, some coming from recent joint work with Alexander Mueller-Hermes.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiaping Wang (University of Minnesota)
DTSTART:20220418T140000Z
DTEND:20220418T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/84
DESCRIPTION:Title: Spectrum of complete manifolds\nby Jiaping Wang (Univ
ersity of Minnesota) as part of Geometric and functional inequalities and
applications\n\n\nAbstract\nSpectrum of Laplacian is an important set of g
eometric invariants. The talk\, largely based on\njoint work with Peter Li
and Ovidiu Munteanu\, concerns its structure and size on complete manifol
ds under various curvature conditions. The focus is on sharp estimates of
the bottom spectrum in terms of either Ricci or scalar curvature lower bou
nd.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Galia Dafni (Concordia University)
DTSTART:20220523T140000Z
DTEND:20220523T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/85
DESCRIPTION:Title: Locally uniform domains and extension of nonhomogeneous B
MO spaces\nby Galia Dafni (Concordia University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nIn joint work
with Almaz Butaev (Cincinnati)\, we study local versions of uniform domain
s\, which can be identified with the epsilon-delta domains used by Jones t
o extend Sobolev spaces. We show that a domain is locally uniform if and o
nly if it is an extension domain for the nonhomogeneous (also known as "lo
cal") space of functions of bounded mean oscillation introduced by Goldber
g\, and denoted by bmo. We also prove analogous results for functions of
vanishing mean oscillation.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Stancu (Concordia University)
DTSTART:20220425T130000Z
DTEND:20220425T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/86
DESCRIPTION:Title: On the fundamental gap of convex sets in hyperbolic space
\nby Alina Stancu (Concordia University) as part of Geometric and func
tional inequalities and applications\n\n\nAbstract\nThe difference between
the first two eigenvalues of the Dirichlet Laplacian on convex sets of R^
n and\, respectively S^n\, satisfies the same strictly positive lower boun
d depending on the diameter of the domain. In work with collaborators\, we
have found that the gap of the hyperbolic space on convex sets behaves st
rikingly different even if a stronger notion of convexity is employed. Thi
s is very interesting as many other features of first two eigenvalues beha
ve in the same way on all three spaces of constant sectional curvature.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pei-Yong Wang (Wayne State University)
DTSTART:20220502T140000Z
DTEND:20220502T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/88
DESCRIPTION:Title: A Bifurcation Phenomenon Of The Perturbed Two-Phase Trans
ition Problem\nby Pei-Yong Wang (Wayne State University) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nThis t
alk presents a joint work with F. Charro\, A. Haj Ali\, M. Raihen\, and\nM
. Torres on a bifurcation phenomenon in a two-phase\, singularly perturbed
\, free\nboundary problem of phase transition. We show that the uniqueness
of the solution\nfor the two-phase problem breaks down as the boundary da
ta decreases through\na threshold value. For boundary values below the thr
eshold\, there are at least\nthree solutions\, namely\, the harmonic solut
ion which is treated as a trivial solution\nin the absence of a free bound
ary\, a nontrivial minimizer of the functional under\nconsideration\, and
a third solution of the mountain-pass type. We classify these\nsolutions a
ccording to the stability through evolution. The evolution with initial\nd
ata near a stable solution\, such as the trivial harmonic solution or a mi
nimizer of\nthe functional\, converges to the stable solution. On the othe
r hand\, the evolution\ndeviates away from a non-minimal solution of the f
ree boundary problem.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Stroffolini (Universit`a degli Studi di NAPOLI ”Federico
II”)
DTSTART:20220919T140000Z
DTEND:20220919T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/89
DESCRIPTION:Title: Taylor formula and regularity properties for degenerate K
olmogorov equations with Dini continuous coefficients\nby Bianca Strof
folini (Universit`a degli Studi di NAPOLI ”Federico II”) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nWe stu
dy the regularity properties of the second order linear operator in $\\mat
hbb{R}^{N+1}$:\n\\[\n\\mathscr{L}u:= \\sum_{j\,k=1}^{m} a_{jk} \\partial^{
2}_{x_j x_k} u + \\sum_{j\,k=1}^N b_{jk} x_k \\partial_{x_j} u - \\partial
_t u\,\n\\]\nwhere $A = (a_{jk})_{j\,k=1\,\\ldots m}$\, $B = (b_{jk})_{j\,
k=1\,\\ldots N}$ are real valued matrices with constant coefficients\, wit
h $A$ symmetric and strictly positive. We prove that\, if the operator $\\
mathscr{L}$ satisfies Hörmander's hypoellipticity condition\, and $f$ is
a Dini continuous function\, then the second order derivatives of the solu
tion $u$ to the equation $\\mathscr{L}u = f$ are Dini continuous functions
as well. We also consider the case of Dini continuous coefficients $a_{jk
}$'s. A key step in our proof is a Taylor formula for classical solutions
to $\\mathscr{L}u=f$ that we establish under minimal regularity assumption
s on $u$.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mimi Dai (University of Illinois at Chicago)
DTSTART:20221003T140000Z
DTEND:20221003T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/91
DESCRIPTION:Title: Navier-Stokes equation: determining wavenumber\, Kolmogor
ov’s dissipation number\, and Kraichnan’s number\nby Mimi Dai (Uni
versity of Illinois at Chicago) as part of Geometric and functional inequa
lities and applications\n\n\nAbstract\nWe show the existence of determinin
g wavenumber for the Naiver-Stokes equation in both 3D and 2D. Estimates o
n the determining wavenumber are established in term of the phenomenologic
al Kolmogorov’s dissipation number (3D) and Kraichnan’s number (2D). T
he results rigorously justify the criticality of Kolmogorov’s dissipatio
n number and Kraichnan’s number.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias König (Goethe-Universität Frankfurt)
DTSTART:20221010T140000Z
DTEND:20221010T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/92
DESCRIPTION:Title: Multibubble blow-up analysis for the Brezis-Nirenberg pro
blem in three dimensions\nby Tobias König (Goethe-Universität Frankf
urt) as part of Geometric and functional inequalities and applications\n\n
\nAbstract\nIn this talk\, I will present a recent result about blow-up as
ymptotics in the three-dimensional Brezis-Nirenberg problem. More precise
ly\, for a smooth bounded domain $\\Omega \\subset \\R^3$ and smooth funct
ions $a$ and $V$\, consider a sequence of positive solutions $u_\\epsilon$
to $-\\Delta u_\\epsilon + (a+\\epsilon V) u_\\epsilon = u_\\epsilon^5$ o
n $\\Omega$ with zero Dirichlet boundary conditions\, which blows up as $\
\epsilon \\to 0$. We derive the sharp blow-up rate and characterize the lo
cation of concentration points in the general case of multiple blow-up\, t
hereby obtaining a complete picture of blow-up phenomena in the framework
of the Brezis-Peletier conjecture in dimension $N=3$. I will also indicate
a forthcoming new result parallel to this one for dimension $N \\geq 4$.\
n\nThis is joint work with Paul Laurain (IMJ-PRG Paris and ENS Paris).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianxiong Wang (University of Connecticut)
DTSTART:20221017T140000Z
DTEND:20221017T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/93
DESCRIPTION:Title: Symmetry of solutions to higher and fractional order semi
linear equations on hyperbolic spaces\nby Jianxiong Wang (University o
f Connecticut) as part of Geometric and functional inequalities and applic
ations\n\n\nAbstract\nWe show that nontrivial solutions to higher and frac
tional order equations with certain nonlinearity are radially symmetric an
d nonincreasing on geodesic balls in the hyperbolic space $\\mathbb{H}^n$
as well as on the entire space $\\mathbb{H}^n$ . Applying Helgason-Fourier
analysis techniques on $\\mathbb{H}^n$ \, we developed a moving plane app
roach for integral equations on $\\mathbb{H}^n$. We also established the s
ymmetry to solutions of certain equations with singular terms on Euclidean
spaces. Moreover\, we obtained symmetry to solutions of some semilinear e
quations involving fractional order derivatives.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Flynn (Centre de Recherches Mathématiques)
DTSTART:20221024T140000Z
DTEND:20221024T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/94
DESCRIPTION:Title: Sharp Uncertainty Principles for Physical Vector Fields a
nd Second Order Derivatives\nby Joshua Flynn (Centre de Recherches Mat
hématiques) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nThe Heisenberg uncertainty principle is a fundamental
result in quantum mechanics. Related inequalities are the hydrogen and Har
dy uncertainty principles and all three belong to the family of geometric
inequalities known as the Caffarelli-Kohn-Nirenberg inequalities. In this
talk\, we present our recent results pertaining to uncertainty principles
and CKN inequalities with a particular focus on higher order derivatives a
nd vector-valued cases. Presented works were done jointly with G. Lu\, N.
Lam and C. Cazacu.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandru Kristaly (Babes-Bolyai University)
DTSTART:20221031T140000Z
DTEND:20221031T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/95
DESCRIPTION:Title: Lord Rayleigh’s conjecture for clamped plates in curved
spaces\nby Alexandru Kristaly (Babes-Bolyai University) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nThis t
alk is focused on the vibrating clamped plate problem\, initially formulat
ed by Lord Rayleigh in 1877\, and solved by M. Ashbaugh & R. Benguria (Duk
e Math. J.\, 1995) and N. Nadirashvili (ARMA\, 1995) in 2 and 3 dimensiona
l euclidean spaces. We consider the same problem on both negatively and po
sitively curved spaces\, and provide various answers depending on the curv
ature\, dimension and the width/size of the clamped plate.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaolong Han (California State University)
DTSTART:20221107T150000Z
DTEND:20221107T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/96
DESCRIPTION:Title: Fractal uncertainty principle for discrete Fourier transf
orm and random Cantor sets\nby Xiaolong Han (California State Universi
ty) as part of Geometric and functional inequalities and applications\n\n\
nAbstract\nThe Fourier uncertainty principle describes a fundamental pheno
menon that a function and its Fourier transform cannot simultaneously loca
lize. Dyatlov and his collaborators (Zahl\, Bourgain\, Jin\, Nonnenmacher)
recently introduced a concept of Fractal Uncertainty Principle (FUP). It
is a mathematical formulation concerning the limit of localization of a fu
nction and its Fourier transform on sets with certain fractal structure. \
n\nThe FUP has quickly become an emerging topic in Fourier analysis and al
so has important applications to other fields such as wave decay in obstac
le scattering. In this talk\, we consider the discrete Fourier transform a
nd the fractal sets are given by discrete Cantor sets. We present the FUP
in this discrete setting with a much more favorable estimate than the one
known before\, when the Cantor sets are constructed by a random procedure.
This is a joint work with Suresh Eswarathasan.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Reiter (University of Technology at Chemnitz)
DTSTART:20221114T150000Z
DTEND:20221114T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/97
DESCRIPTION:Title: Elasticity models with self-contact\nby Philipp Reite
r (University of Technology at Chemnitz) as part of Geometric and function
al inequalities and applications\n\n\nAbstract\nMaintaining the topology o
f objects undergoing deformations is a crucial\naspect of elasticity model
s. In this talk we consider two different\nsettings in which impermeabilit
y is implemented via regularization by a\nsuitable nonlocal functional.\n\
nThe behavior of long slender objects may be characterized by the classic\
nKirchhoff model of elastic rods. Phenomena like supercoiling which play a
n\nessential role in molecular biology can only be observed if\nself-penet
rations are precluded. This can be achieved by adding a\nself-repulsive fu
nctional such as the tangent-point energy. We discuss the\ndiscretization
of this approach and present some numerical simulations.\n\nIn case of ela
stic solids whose shape is described by the image of a\nreference domain u
nder a deformation map\, self-interpenetrations can be\nruled out by claim
ing global invertibility. Given a suitable stored energy\ndensity\, the la
tter is ensured by the Ciarlet–Nečas condition which\,\nhowever\, is di
fficult to handle numerically in an efficient way. This\nmotivates approxi
mating the latter by adding a self-repulsive functional\nwhich formally co
rresponds to a suitable Sobolev–Slobodeckiĭ seminorm of\nthe inverse de
formation.\n\nThis is joint work with Sören Bartels (Freiburg) and Stefan
Krömer\n(Prague).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Falocchi (Politecnico di Milano)
DTSTART:20221121T150000Z
DTEND:20221121T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/100
DESCRIPTION:Title: Some results on the 3D Stokes eigenvalue problem under N
avier boundary conditions\nby Alessio Falocchi (Politecnico di Milano)
as part of Geometric and functional inequalities and applications\n\n\nAb
stract\nWe study the Stokes eigenvalue problem under Navier boundary condi
tions in $C^{1\,1}$-domains $\\Omega\\subset \\mathbb{R}^3$. Differently f
rom the Dirichlet boundary conditions\, zero may be the least eigenvalue.
We fully characterize the domains where this happens\, showing the related
validity/failure of a suitable Poincar\\'{e}-type inequality.\n\nAs appli
cation we prove regularity results for the solution of the evolution Navi
er-Stokes equations under Navier boundary conditions in a class of merely
{\\em Lipschitz domains} of physical interest\, that we call {\\em sectors
}.\n\nThis is a joint work with Filippo Gazzola\, Politecnico di Milano.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Schlein (University of Zurich)
DTSTART:20230306T150000Z
DTEND:20230306T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/101
DESCRIPTION:Title: Gross-Pitaevskii and Bogoliubov theory for trapped Bose-
Einstein condensates.\nby Benjamin Schlein (University of Zurich) as p
art of Geometric and functional inequalities and applications\n\n\nAbstrac
t\nWe consider a quantum system consisting of N bosons (particles describe
d by a permutation symmetric wave function) trapped in a volume of order o
ne and interacting through a short range potential\, with scattering leng
th of the order 1/N (this is known as the Gross-Pitaevskii regime). First\
, we will show how non-linear Gross-Pitaevskii theory describes\, to leadi
ng order\, the ground state energy of the gas and the time-evolution resul
ting from a change of the external fields. In the second part of the talk\
, I will then explain how Bogoliubov theory predicts the next order correc
tions.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk (Spring Break)
DTSTART:20230313T130000Z
DTEND:20230313T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/103
DESCRIPTION:by No Talk (Spring Break) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre-Damien Thizy (University of Lyon 1 (Claude Bernard))
DTSTART:20230410T140000Z
DTEND:20230410T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/104
DESCRIPTION:Title: Large blow-up sets for Q-curvature equations.\nby Pi
erre-Damien Thizy (University of Lyon 1 (Claude Bernard)) as part of Geome
tric and functional inequalities and applications\n\n\nAbstract\nOn a boun
ded domain of the Euclidean space $\\mathbb{R}^{2m}$\, $m>1$\, Adimurthi\,
Robert and Struwe pointed out that\, even assuming a volume bound $\\int
e^{2mu} dx \\leq C$\, some blow-up solutions for prescribed Q-curvature eq
uations $(-\\Delta)^m u= Q e^{2m u}$ without boundary conditions may blow-
up not only at points\, but also on the zero set of some nonpositive nontr
ivial polyharmonic function. This is in striking contrast with the two dim
ensional case ($m=1$). During this talk\, starting from a work in progress
with Ali Hyder and Luca Martinazzi\, we will discuss the construction of
such solutions which involves (possible generalizations of) the Walsh-Lebe
sgue theorem and some issues about elliptic problems with measure data.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deping Ye (Memorial University of Newfoundland)
DTSTART:20230123T150000Z
DTEND:20230123T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/105
DESCRIPTION:Title: The $L_p$ surface area measure and related Minkowski pro
blem for log-concave functions\nby Deping Ye (Memorial University of N
ewfoundland) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nThe Minkowski type problems for convex bodies are fund
amental\nin convex geometry\, and have found many important connections an
d\napplications in analysis\, partial differential equations\, etc. It is\
nwell-known that the log-concave functions behave rather similar to\nconve
x bodies in many aspects\, for example the famous Prékopa–Leindler\nine
quality to the (dimension free) Brunn-Minkowski inequality.\n\nIn this tal
k\, I will present an $L_p$ theory for the log-concave\nfunctions\, which
is analogous to the $L_p$ Brunn-Minkowski theory of\nconvex bodies. In par
ticular\, I will explain how to define the $L_p$ sum\nof log-concave funct
ions\, present a variational formula related to the\n$L_p$ addition\, and
talk about the corresponding $L_p$ Minkowski\nproblems as well as their so
lutions.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Strauss (Brown University)
DTSTART:20230227T150000Z
DTEND:20230227T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/106
DESCRIPTION:Title: Instability of Water Waves (even small ones)\nby Wal
ter Strauss (Brown University) as part of Geometric and functional inequal
ities and applications\n\n\nAbstract\nAfter a gentle introduction on water
waves\, I will present an exposition of joint work with Huy Quang Nguyen.
We prove rigorously that the classical (small-amplitude irrotational stea
dy periodic) water waves are unstable with respect to long-wave perturbati
ons. That is\, the perturbations grow exponentially in time. This instab
ility was first observed heuristically more than half a century ago by Ben
jamin and Feir. However\, a rigorous proof was never found except in the c
ase of finite depth. We provide a completely different and self-contained
proof of both the finite and infinite depth cases that retains the physica
l variables. The proof reduces to an analysis of the spectrum of an expli
cit operator. The growth is obtained by means of a Liapunov-Schmidt reduc
tion that more or less reduces the analysis to four dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonard Gross (Cornell University)
DTSTART:20230911T140000Z
DTEND:20230911T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/107
DESCRIPTION:Title: Invariance of intrinsic hypercontractivity under perturb
ation of Schrodinger operators.\nby Leonard Gross (Cornell University)
as part of Geometric and functional inequalities and applications\n\n\nAb
stract\nA Schrodinger operator that is bounded below and has a unique\npos
itive ground state can be transformed into a Dirichlet form operator\nby t
he ground state transformation. If the resulting Dirichlet form\noperator
is hypercontractive\, Davies and Simon call the Schrodinger\noperator ``in
trinsically hypercontractive”. I will show that if one adds a\nsuitable
potential onto an intrinsically hypercontractive Schrodinger\noperator it
remains intrinsically hypercontractive. All bounds are dimension independe
nt. I will show how to use this theorem in two examples.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monica Visan (University of California at Los Angeles)
DTSTART:20230220T150000Z
DTEND:20230220T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/108
DESCRIPTION:Title: The derivative nonlinear Schrodinger equation\nby Mo
nica Visan (University of California at Los Angeles) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nI will discuss
the derivative nonlinear Schrodinger equation\, how some inherent instabi
lities have hindered the study of this equation\, and how we were able to
demonstrate global well-posedness in the natural scale-invariant space. Th
is is joint work with Ben Harrop-Griffiths\, Rowan Killip\, and Maria Ntek
oume.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rowan Killip (University of California at Los Angles)
DTSTART:20230320T140000Z
DTEND:20230320T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/109
DESCRIPTION:Title: From Optics to the Deift Conjecture\nby Rowan Killip
(University of California at Los Angles) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nAfter providing a mathema
tical background for some curious\noptical experiments in the 19th century
\, we will then describe how\nthese ideas inform our understanding of the
Deift conjecture for the\nKorteweg--de Vries equation. Specifically\, the
y allow us to show that the\nevolution of almost-periodic initial data nee
d not remain almost\nperiodic. This is joint work with Andreia Chapouto a
nd Monica Visan.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Silvia Cingolani (Università degli Studi di Bari Aldo Moro)
DTSTART:20230508T140000Z
DTEND:20230508T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/111
DESCRIPTION:Title: On the planar Schrödinger-Newton system\nby Silvia
Cingolani (Università degli Studi di Bari Aldo Moro) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nI present som
e existence results for nonlocal interaction energy functionals\nin two di
mension. I also characterize critical rates for some inequalities with\nlo
garithmic kernels. The seminar is based on joint papers with Tobias Weth\,
\nGoethe-Universität Frankfurt (Germany).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk
DTSTART:20230327T130000Z
DTEND:20230327T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/112
DESCRIPTION:by No Talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenchuan Tian (UC Santa Barbara)
DTSTART:20230130T150000Z
DTEND:20230130T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/113
DESCRIPTION:Title: On a family of integral operators on the ball\nby We
nchuan Tian (UC Santa Barbara) as part of Geometric and functional inequal
ities and applications\n\n\nAbstract\nIn this work\, we prove an extension
inequality in the hyperbolic space. The inequality involves the hyperbol
ic harmonic extension of a function on the boundary and the Fefferman-Grah
am compactification of the hyperbolic metric. We offer an interpretation o
f the extension inequality as a conformally invariant generalization of Ca
rleman's inequality to higher dimensions. \nIn addition to that\, we class
ify all the solutions to the Euler-Lagrange equation of the extension ineq
uality. The proof uses the moving sphere method and relies on the properti
es of the Fefferman-Graham compactification of the hyperbolic metric.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Roysdon (Brown University)
DTSTART:20230213T150000Z
DTEND:20230213T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/114
DESCRIPTION:Title: Intersection Functions\nby Michael Roysdon (Brown Un
iversity) as part of Geometric and functional inequalities and application
s\n\n\nAbstract\nThe classical Busemann-Petty Problem from the 1950s asked
the following tomographic question:\n\nAssuming you have two origin-symme
tric convex bodies $K$ and $L$ in the $n$-dimensional Euclidean space sati
sfying the following volume inequality:\n\n\n$$|K \\cap \\theta^{\\perp}|
\\leq |L \\cap \\theta^{\\perp}| for all \\theta \\in S^{n-1}\,$$\n\ndoes
it follow that $|K| \\leq |L|$? The answer is affirmative for $n \\leq 4$
and negative whenever $n >5$. However\, if $K$ belongs to a certain clas
s of convex bodies\, the intersection bodies\, then the answer to the Buse
mann-Petty problem is affirmative in all dimension. Several extensions of
this result have been shown in the case of measures on convex bodies\, an
d isomorphic results of the same type have been established. Moreover\, t
he isomorphic Busemann-Petty problem is actually equivalent to the isomorp
hic slicing problem of Bourgain (1986)\, which remains open to this day.
\n\nIn this talk\, we will introduce the notion of an intersection functio
n\, provide a Fourier analytic characterization for such functions\, and s
how some versions of the Busemann-Petty problem in this setting. In parti
cular\, we will show that if you have a pair of continuous\, even\, integr
able functions $f\,g \\colon \\R^n \\to \\R_+$ which satisfy $[Rf] \\leq [
Rg]$\, where $R$ denotes the Radon transform\, then one has that $|f|_{L^2
} \\leq |g|_{L^2}$ provided that the function $f$ is an intersection funct
ion. \n\nThis is based on a joint work with Alexander Koldobsky and Artem
Zvavitch\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luca Capogna (Smith College)
DTSTART:20230417T140000Z
DTEND:20230417T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/115
DESCRIPTION:Title: The Neumann problem and the fractional p-Laplacian in me
asure metric spaces\nby Luca Capogna (Smith College) as part of Geomet
ric and functional inequalities and applications\n\n\nAbstract\nIn this ta
lk we will report on some recent joint work with Josh Kline\, Riikka Korte
\, Marie Snipes and Nages Shanmugalingam\, concerning the Neumann problem
in PI spaces\, and a new definition of fractional p-Laplacians in arbitrar
y doubling measure metric space. Following ideas of Caffarelli and Silvest
re in and using recent progress in hyperbolic fillings\, we define fractio
nal p-Laplacians on any compact\, doubling metric measure space\, and prov
e existence\, regularity\, Harnack inequality and stability for the corr
esponding non-homogeneous non-local equation. These results\, in turn\, re
st on the new existence\, global Hölder regularity and stability theorem
s that we prove for the Neumann problem for p-Laplacians in bounded domai
ns of measure metric spaces endowed with a doubling measure that supports
a Poincaré inequality. Our work also includes as special cases many of t
he previous results by other authors in the Euclidean\, Riemannian and Car
not group settings. Unlike other recent contributions in the metric measur
e space context\, our work does not rely on the assumption that the space
supports a Poincare inequality.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Qing (UC Santa Cruz)
DTSTART:20230501T140000Z
DTEND:20230501T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/116
DESCRIPTION:Title: Potential theory in conformal geometry\nby Jie Qing
(UC Santa Cruz) as part of Geometric and functional inequalities and appli
cations\n\n\nAbstract\nIn this talk we would like to report my recent join
t work with Shiguang Ma\non the application of the potential theory in con
formal geometry. We will mention\nmany interesting equations in conformal
geometry and the potential-theoretic\napproach to study them. In particula
r\, we present the recent work on the extensions\nof Huber type theorem in
higher dimensions under integral conditions of various\ncurvature. We wil
l demonstrate main ideas via the outlines of a proof of Huber Theorem.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lu Wang (Yale University)
DTSTART:20231009T140000Z
DTEND:20231009T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/117
DESCRIPTION:Title: A mean curvature flow approach to density of minimal con
es\nby Lu Wang (Yale University) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nMinimal cones are models for s
ingularities in minimal submanifolds\, as well as stationary solutions to
the mean curvature flow. In this talk\, I will explain how to utilize mean
curvature flow to yield near optimal estimates on density of topologicall
y nontrivial minimal cones. This is joint with Jacob Bernstein.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:María del Mar González (Universidad Autónoma de Madrid)
DTSTART:20230424T140000Z
DTEND:20230424T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/118
DESCRIPTION:Title: Spectral properties of Levy Fokker-Planck equations\
nby María del Mar González (Universidad Autónoma de Madrid) as part
of Geometric and functional inequalities and applications\n\n\nAbstract\n
We study the spectrum of a fractional Laplacian equation with drift in sui
table weighted spaces. This operator arises when studying the fractional h
eat equation in self-similar variables. We show\, in the radially symmetri
c case\, compactness\, and then calculate the eigenfunctions in terms of L
aguerre polynomials. The proofs involve Mellin transform and complex analy
sis methods. This is joint work with H. Chan\, M. Fontelos and J. Wei.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irene Fonseca (Carnegie Mellon University)
DTSTART:20230925T151000Z
DTEND:20230925T161000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/119
DESCRIPTION:Title: From Phase Separation in Heterogeneous Media to Learning
Training Schemes for Image Denoising\nby Irene Fonseca (Carnegie Mell
on University) as part of Geometric and functional inequalities and applic
ations\n\n\nAbstract\nWhat do these two themes have in common? Both are tr
eated variationally\, both deal with\n\nenergies of different dimensionali
ties\, and concepts of geometric measure theory prevail in both.\n\n\nP
hase Separation in Heterogeneous Media: Modern technologies and biolog
ical systems\, such as temperature-responsive polymers and lipid rafts\, t
ake advantage of engineered inclusions\, or natural heterogeneities of the
medium\, to obtain novel composite materials with specific physical prope
rties. To model such situations using a variational approach based on the
gradient theory of phase transitions\, the potential and the wells may hav
e to depend on the spatial position\, even in a discontinuous way\, and di
fferent regimes should be considered.\n\n\nIn the critical case case where
the scale of the small heterogeneities is of the same order of the scale
governing the phase transition and the wells are fixed\, the interaction b
etween homogenization and the phase transitions process leads to an anisot
ropic interfacial energy. The supercritical case for fixed wells is also a
ddressed\, now leading to an isotropic interfacial energy. In the subcriti
cal case with moving wells\, where the heterogeneities of the material are
of a larger scale than that of the diffuse interface between different ph
ases\, it is observed that there is no macroscopic phase separation and th
at thermal fluctuations play a role in the formation of nanodomains.\n\n\n
This is joint work with Riccardo Cristoferi (Radboud University\, The Neth
erlands) and Likhit Ganedi (Aachen University\, Germany)\, USA)\, based on
previous results also obtained with Adrian Hagerty (USA) and Cristina Pop
ovici (USA).\n\n\n\nLearning Training Schemes for Image Denoising:
Due to their ability to handle discontinuous images while having a well-un
derstood behavior\, regularizations with total variation (TV) and total ge
neralized variation (TGV) are some of the best known methods in image deno
ising. However\, like other variational models including a fidelity term\,
they crucially depend on the choice of their tuning parameters. A remedy
is to choose these automatically through multilevel approaches\, for examp
le by optimizing performance on noisy/clean image training pairs. Such met
hods with space-dependent parameters which are piecewise constant on dyadi
c grids are considered\, with the grid itself being part of the minimizati
on. Existence of minimizers for discontinuous parameters is established\,
and it is shown that box constraints for the values of the parameters lead
to existence of finite optimal partitions. Improved performance on some r
epresentative test images when compared with constant optimized parameters
is demonstrated.\n\n\n\nThis is joint work with Elisa Davoli (TU Wien\, A
ustria)\, Jose Iglesias (U. Twente\, The Netherlands) and Rita Ferreira (K
AUST\, Saudi Arabia)\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Malabika Pramanik (University of British Columbia)
DTSTART:20231204T150000Z
DTEND:20231204T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/120
DESCRIPTION:Title: Points and distances\nby Malabika Pramanik (Universi
ty of British Columbia) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nThe Pythagorean theorem\, dating back to be
fore 500 BC\, gives a\nformula for computing the Euclidean distance betwee
n two points. It is\nsimply astounding that a concept so simple and classi
cal has continued to\nfascinate mathematicians over the ages\, and remains
a tantalizing source of\nopen problems to this day.\n\nGiven a set E\, it
s distance set consists of numbers representing distances\nbetween points
of E. If E is large\, how large is its distance set? How does\nthe structu
re of a set influence the structure of distances in the set?\nSuch questio
ns play an important role in many areas of mathematics and\nbeyond. The ta
lk will mention a few research problems involving Euclidean\ndistances bet
ween points and some landmark results associated with them.\n\nThe present
ation is intended to be an introduction to a vibrant research\narea\; no a
dvanced mathematical background will be assumed.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Donatella Danielli (Arizona State University)
DTSTART:20230918T140000Z
DTEND:20230918T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/121
DESCRIPTION:Title: Obstacle problems for fractional powers of the Laplacian
\nby Donatella Danielli (Arizona State University) as part of Geometri
c and functional inequalities and applications\n\n\nAbstract\nIn this talk
we will discuss two models of obstacle-type problems associated with the
fractional Laplacian $(−\\Delta)^s$ \, for $1 < s < 2$. Our goals are to
establish regularity properties of the solution and to describe the struc
ture of the free boundary. To this end\, we combine classical techniques f
rom potential theory and the calculus of variations with more modern metho
ds\, such as the localization of the operator and monotonicity formulas. T
his is joint work with A. Haj Ali (Arizona State University)\, A. Petrosya
n (Purdue University)\, and M. Talluri (University of Pisa).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (New York University)
DTSTART:20231002T140000Z
DTEND:20231002T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/122
DESCRIPTION:Title: Furstenberg sets estimate in the plane\nby Hong Wang
(New York University) as part of Geometric and functional inequalities an
d applications\n\n\nAbstract\nA $(s\,t)$-Furstenberg set is a set $E$ in t
he plane with the following property: there exists a $t$-dim family of lin
es such that each line intersects $E$ in a $\\geq s$--dimensional set. An
unpublished conjecture of Furstenberg states that any $(s\,1)$-Furstenberg
set has dimensions at least $(3s+1)/2$. Furstenberg set problem can be v
iewed as a natural generalization of Davies's result that a Kakeya set in
the plane (a set that contains a line segment in any direction) has dimens
ion 2. \n\nWe will survey a sequence of results by Orponen\, Shmerkin\, a
nd a recent result by Ren and myself that leads to the solution of Fursten
berg sets conjecture in the plane: any $(s\,t)$-Furstenberg set has dimens
ion at least $\\min \\{s+t\, (3s+t)/2\, s+1\\}$.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lei Ni (UC San Diego)
DTSTART:20231023T140000Z
DTEND:20231023T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/123
DESCRIPTION:Title: Complex Monge-Ampère equations\, Grauert tube and estim
ate of Betti numbers\nby Lei Ni (UC San Diego) as part of Geometric an
d functional inequalities and applications\n\n\nAbstract\nIn this talk I s
hall explain a geometric construction motivated by the study of complex Mo
nge-Ampère equations\, namely the so-called Grauert tube\, and its applic
ation in obtaining an effective estimate on the Betti numbers of the loop
space of a compact Riemannian manifold whose sectional curvature is bound
ed from below.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wilhelm Schlag (Yale University)
DTSTART:20231120T150000Z
DTEND:20231120T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/124
DESCRIPTION:Title: On continuous time bubbling for the harmonic map heat fl
ow in two dimensions\nby Wilhelm Schlag (Yale University) as part of G
eometric and functional inequalities and applications\n\n\nAbstract\nI wil
l describe recent work with Jacek Jendrej (CNRS\, Paris Nord) and Andrew L
awrie (MIT) on harmonic maps of finite energy from the plane to the two sp
here\, without making any symmetry assumptions. While it has been known si
nce the 1990s that bubbling occurs along a carefully chosen sequence of ti
mes via an elliptic Palais-Smale mechanism\, we show that this continues t
o hold continuously in time. The key notion is that of the “minimal coll
ision energy” which appears in the soliton resolution result by Jendrej
and Lawrie on critical equivariant wave maps.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihaela Ifrim (University of Wisconsin – Madison)
DTSTART:20240122T150000Z
DTEND:20240122T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/127
DESCRIPTION:Title: The small data global well-posedness conjecture for 1D d
efocusing dispersive flows\nby Mihaela Ifrim (University of Wisconsin
– Madison) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nI will present a very recent conjecture which broadly
asserts that small data should yield global solutions for 1D defocusing d
ispersive flows with cubic nonlinearities\, in both semilinear and quasili
near settings. This conjecture was recently proved in several settings in
joint work with Daniel Tataru.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yannick Sire (Johns Hopkins University)
DTSTART:20240205T150000Z
DTEND:20240205T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/128
DESCRIPTION:Title: Eigenvalue estimates and a conjecture of Yau\nby Yan
nick Sire (Johns Hopkins University) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nI will describe various upper
and lower bounds on the spectrum of the Laplace-Beltrami on Riemannian man
ifolds. The upper bounds led to some important results in spectral geometr
y establishing a link between the so-called conformal spectrum and branche
d minimal immersions into Euclidean spheres. I will then move to describe
a conjecture by Yau on the first eigenvalue on minimal submanifolds of the
sphere\, which is known only for some examples. I will then present some
recent results where we improve quantitatively the best known lower bound
(in the general case) of Choi and Wang of the mid 80’s. I will address
some open problems and possible generalizations of our argument.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yumeng Ou (University of Pennsylvania)
DTSTART:20240129T150000Z
DTEND:20240129T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/129
DESCRIPTION:Title: New improvement to Falconer’s distance set conjecture
in higher dimensions\nby Yumeng Ou (University of Pennsylvania) as par
t of Geometric and functional inequalities and applications\n\n\nAbstract\
nFalconer’s distance set conjecture says that if a compact set in $\\mat
hbb{R}^d$ has Hausdorff dimension larger than $d/2$\, then its distance se
t must have positive measure. The conjecture is currently open in all dime
nsions. In this talk\, I’ll discuss some recent progress towards it in d
imension three and higher\, which involves new techniques from the theory
of radial projections and decoupling. This is based on joint works with Xi
umin Du\, Kevin Ren\, and Ruixiang Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yifeng Yu (University of California-Irvine)
DTSTART:20240212T150000Z
DTEND:20240212T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/130
DESCRIPTION:Title: Existence and nonexistence of effective burning velocity
under the curvature G-equation model\nby Yifeng Yu (University of Cal
ifornia-Irvine) as part of Geometric and functional inequalities and appli
cations\n\n\nAbstract\nG-equation is a well known level set model in turbu
lent combustion\, and becomes an advective mean curvature type evolution e
quation when the curvature effect is considered:\n$$\nG_t + \\left(1-d\\\,
\\Div{\\frac{DG}{|DG|}}\\right)_+|DG|+V(x)\\cdot DG=0.\n$$\n In 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\,
which can be extended to more general two dimensional incompressible perio
dic flows. Our proof combines PDE methods with a dynamical analysis of th
e Kohn-Serfaty deterministic game characterization of the curvature G-equa
tion based on the two dimensional structures. In three dimensions\, the
effective burning velocity will cease to exist even for simple periodic sh
ear flows when the flow intensity surpasses a bifurcation value.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debdip Ganguly (Indian Institute of Technology)
DTSTART:20240219T150000Z
DTEND:20240219T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/131
DESCRIPTION:Title: Sharp Quantitative stability of Poincar\\'e-Sobolev ineq
uality in the hyperbolic space\nby Debdip Ganguly (Indian Institute of
Technology) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nThe talk is devoted to the sharp stability of\nPoincar
\\'e-Sobolev inequalities in the hyperbolic space. To begin with\,\nI shal
l formulate the question of the stability of the classical Sobolev\ninequa
lity in the Euclidean space and recall some of the seminal results\nof Bia
nchi-Egnelland Ciraolo\, Figalli and Maggi and many others. Then I\nshall
deduce the (sharp) quantitative gradient stability of the\nPoincar\\'e-Sob
olev inequalities in the hyperbolic space and the\ncorresponding Euler-Lag
range equation locally around a bubble (and\npossibly at a higher energy l
evel!) if time permits. This is joint work\nwith M.~Bhakta\, D~Karmakar\,
and S.~Mazumdar.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiqin Lu (UC Irvine)
DTSTART:20240226T150000Z
DTEND:20240226T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/132
DESCRIPTION:Title: The Spectrum of Laplacian on forms over open manifolds\nby Zhiqin Lu (UC Irvine) as part of Geometric and functional inequalit
ies and applications\n\n\nAbstract\nWe computed the $L^p$ spectrum of Lapl
acians on $k$-forms on hyperbolic spaces. Moreover\, we proved the $L^p$ b
oundedness of certain resolvent of Laplacians by assuming the Ricci lower
bound and manifold volume growth. This generalized a result of M. Taylor\,
in which bounded geometry of the manifold is assumed. This is a joint wor
k of N. Charalambous.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Maggi (UT Austin)
DTSTART:20240304T150000Z
DTEND:20240304T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/134
DESCRIPTION:Title: Plateau's laws for soap films\, the Allen--Cahn equation
\, and a hierarchy of Plateau-type problems\nby Francesco Maggi (UT Au
stin) as part of Geometric and functional inequalities and applications\n\
n\nAbstract\nThe incompatibility between Plateau's laws and stable solutio
ns to the Allen--Cahn equation is resolved by the formulation and analysis
of a new model for soap films as small volume regions with diffused inter
faces. As a result\, Plateau-type singularities are approximated by stable
solutions to free boundary problems for modified Allen--Cahn equations. U
nderlying our approach is the study of a hierarchy of Plateau problems tha
t showcases the newly introduced diffused interface model at the top\, a s
oap film capillarity model with sharp interfaces and bulk spanning at the
intermediate level\, and the classical Plateau model at the bottom. Centra
l to our analysis is a measure-theoretic revision of the topological notio
n of homotopic spanning that has been behind much recent progress on the c
lassical Plateau problem.\n\nThis is joint work with Michael Novack (CMU P
ittsburgh) and Daniel Restrepo (Johns Hopkins University).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaodong Wang (Michigan State University)
DTSTART:20240429T140000Z
DTEND:20240429T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/135
DESCRIPTION:Title: Geometric inequalities on asymptotically Poincaré-Einst
ein manifolds\nby Xiaodong Wang (Michigan State University) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nPoi
ncaré-Einstein manifolds are a class of noncompact Riemannian manifolds w
ith a well-defined boundary at infinity. They appear as the framework of A
dS/CFT correspondence in string theory and have been studied intensively.
I will discuss some recent results relating the Yamabe invariant of the bo
undary and that of the interior. This is based on joint work with Zhixin W
ang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniela De Silva (Columbia University)
DTSTART:20240415T140000Z
DTEND:20240415T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/136
DESCRIPTION:Title: An energy model for harmonic graphs with junctions\n
by Daniela De Silva (Columbia University) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nWe consider an energy mod
el for $N$ ordered elastic membranes subject to forcing and boundary condi
tions. The heights of the membranes are described by real functions $u_1\,
u_2\,...\,u_N$\, which minimize an energy functional involving the Dirich
let integral and a potential term depending on the cardinality of the set
$\\{u_1\,..\,u_N\\}$. The potential term corresponds to the physical situa
tion when consecutive membranes are glued to each other on their coinciden
ce set.\nThe problem can be understood as a system of $N-1$ coupled one-ph
ase free boundary problems with interacting free boundaries.\nI will revie
w the known results in the scalar case\, and discuss the free boundary reg
ularity when dealing with 3 or more membranes (joint with O. Savin).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Tataru (UC Berkeley)
DTSTART:20240318T140000Z
DTEND:20240318T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/137
DESCRIPTION:Title: Free boundary problems for Euler type flows\nby Dani
el Tataru (UC Berkeley) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nFree boundary problems are very interesting
but also very challenging problems in fluid dynamics\, where the boundary
of the fluid is also freely moving along with the fluid flow. I will di
scuss two such models\, governed by the compressible\, respectively the in
compressible Euler equations. This is joint work with Mihaela Ifrim\, and
in part with Benjamin Pineau and Mitchell Taylor.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeremy Tyson (UIUC and NSF)
DTSTART:20240506T140000Z
DTEND:20240506T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/139
DESCRIPTION:Title: Horizontal polar coordinates and H-type stability of ste
p two Carnot groups\nby Jeremy Tyson (UIUC and NSF) as part of Geometr
ic and functional inequalities and applications\n\n\nAbstract\nThe sub-Rie
mannian Heisenberg group is foliated (a.e.) by a smooth family of horizont
al curves equipped with a transversal `spherical’ measure\, with respect
to which an analog of the classical Euclidean polar coordinate integratio
n formula holds. Such `horizontal polar coordinates’ were first construc
ted by Kor\\’anyi and Reimann (1987)\, who used them to compute explicit
expressions for the conformal moduli of spherical ring domains with appli
cations to the regularity of quasiconformal mappings. We motivate the stud
y of rectifiable polar coordinates in metric measure spaces\, and especial
ly horizontal polar coordinates in sub-Riemannian Carnot groups\, via thre
e ostensibly different topics: (i) existence of coherent global fundamenta
l solutions for the $p$-Laplace operators\, (ii) optimal H\\”older regul
arity of quasiconformal mappings\, and (iii) existence and uniqueness of l
imits at infinity for homogeneous Sobolev functions. Horizontal polar coor
dinates exist in a restricted class of sub-Riemannian Carnot groups\, whic
h includes the Heisenberg-type groups. We discuss geometric and analytic c
onsequences of such integration formulas as well as examples and non-examp
les of `polarizable’ Carnot groups. It is conjectured that the only pola
rizable groups are the groups of Heisenberg type (H-type). We conclude wit
h an overview of some recent and ongoing work addressing this conjecture\,
based on the notion of the H-type deviation of a step two Carnot group.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alberto Setti (UNIVERSITÀ DEGLI STUDI DELL'INSUBRIA)
DTSTART:20240408T140000Z
DTEND:20240408T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/140
DESCRIPTION:Title: ${\\bf L^{p}}$-parabolicity of Riemannian manifolds\
nby Alberto Setti (UNIVERSITÀ DEGLI STUDI DELL'INSUBRIA) as part of Geome
tric and functional inequalities and applications\n\n\nAbstract\nUsing non
-linear $L^p$ capacities\, we define a notion of\n$L^p$-parabolicity of Ri
emannian manifolds which extends the usual\nparabolicity\, corresponding t
o $p=1$\, to the whole range $1\\leq p\\leq\n\\infty$. $L^p$-parabolicity
turns out to be equivalent to the\n$L^q$-Liouville property for positive s
uperharmonic functions\, where $p$\nand $q$ are H\\"older conjugate expone
nts\, and\, when $p=2$ it coincides\nwith the biparabolicity as defined b
y S.Faraji and A. Grigor'yan. We\nalso provide a new capacitary characte
rization of the $L^1$-Liouville\nproperty. Finally we obtain an almost opt
imal volume growth conditions implying $L^p$-parabolicity for $1< p\\leq
2$ as well as a sharp volume condition valid for all $1< p <\\infty$
in the case of model manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiumin Du (Northwestern University)
DTSTART:20240909T140000Z
DTEND:20240909T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/141
DESCRIPTION:Title: Weighted refined decoupling and Falconer distance set pr
oblem\nby Xiumin Du (Northwestern University) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nIn this talk\, I
’ll discuss a refinement of Bourgain--Demeter’s decoupling inequality
in the weighted setting and the case that the directions of wave packets a
re in a small neighborhood of a subspace. Such inequalities arise from the
study of Falconer distance set problem. Combining weighted refined decoup
ling and new radial projection estimates by Ren\, we proved the following
result: if a compact set $E\\subset \\mathbb{R}^d$ has Hausdorff dimension
larger than $\\frac{d}{2}+\\frac{1}{4}-\\frac{1}{8d+4}$\, where $d\\geq 4
$\, then there is a point $x\\in E$ such that the pinned distance set $\\D
elta_x(E):=\\{|x-y|:y\\in E\\}$ has positive Lebesgue measure. The result
also holds for dimension $d=3$\, but it requires more geometric input. Joi
nt work with Yumeng Ou\, Kevin Ren\, and Ruixiang Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ovidiu Savin (Columbia University)
DTSTART:20240923T140000Z
DTEND:20240923T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/142
DESCRIPTION:Title: Non C^1 solutions to the special Lagrangian equation
\nby Ovidiu Savin (Columbia University) as part of Geometric and functiona
l inequalities and applications\n\n\nAbstract\nThe special Lagrangian equa
tion (SLE) was introduced by Harvey and Lawson in the context of calibrate
d geometries. We will talk about the construction of singular viscosity so
lutions to SLE that are Lipschitz but not C^1\, and have non-minimal gradi
ent graphs. We also discuss certain degenerate Bellman equations that appe
ar in the study of this type of singularities. This is a joint work with C
. Mooney.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allan Greenleaf (University of Rochester)
DTSTART:20240916T130000Z
DTEND:20240916T140000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/143
DESCRIPTION:Title: Partition Optimization: Multilinear Estimates from Linea
r Bounds\nby Allan Greenleaf (University of Rochester) as part of Geom
etric and functional inequalities and applications\n\n\nAbstract\nThe boun
dedness of a linear operator between two function spaces is equivalent to
boundedness of\nthe corresponding bilinear form\, and this extends to mult
ilinear operators and forms. I will discuss how\,\nin some situations\, on
e can use this principle to leverage boundedness of linear operators in mo
re variables \nto obtain nontrivial and useful bounds for multilinear ones
. After discussing a general framework\, we will\nfocus on multilinear Rad
on transforms\, particularly ones arising in configuration set problems.\n
This is joint work with Alex Iosevich and Krystal Taylor.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isabel Fernandez (University of Seville)
DTSTART:20241014T140000Z
DTEND:20241014T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/144
DESCRIPTION:Title: Free Boundary Minimal and CMC Annuli in the Ball\nby
Isabel Fernandez (University of Seville) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nFree boundary minimal/CMC
surfaces in a 3-manifold with boundary appear as critical points for the
area functional for surfaces whose boundary varies freely in the boundary
of the ambient manifold. \n\n\n\nIn this talk we will show the existence
of free boundary minimal annuli immersed in the unit ball of Euclidean 3-s
pace\, the first such examples other than the critical catenoid. We will a
lso construct embedded free boundary CMC annuli and embedded capillary min
imal annuli in the unit ball that are not rotational. \n\n\n\nIn the minim
al case\, this construction answers in the negative a problem of the theor
y that dates back to Nitsche in 1985\, who claimed that such annuli could
not exist. In the CMC case\, the existence of these embedded annuli is see
mingly unexpected\, and solves a problem by Wente (1995). Joint work with
Mira-Hauswirth (for the minimal case) and Cerezo-Mira (for the CMC case).\
n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francois Golse (École Polytechnique)
DTSTART:20241007T140000Z
DTEND:20241007T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/145
DESCRIPTION:Title: Quantum Wasserstein and Observability\nby Francois G
olse (École Polytechnique) as part of Geometric and functional inequaliti
es and applications\n\n\nAbstract\nThis talk discusses the observation pro
blem in quantum dynamics (what kind of \npartial information is needed on
the solution of the quantum Liouville equation in \norder to determine the
solution completely?) An observation inequality is obtained\nfor this pro
blem which is based on \n(1) a geometric condition due to Bardos\, Lebeau
and Rauch\, used in the control\nof solutions to the wave equation\, and\n
(2) a quantum analogue of the optimal transport metric known as the Wasser
stein\,\nor Monge-Kantorovich distance of order 2\, introduced in an earli
er collaboration\nwith T. Paul.\nThe observation inequality so obtained in
volves only effective constants\, which\ncan be computed explicitly in ter
ms of the various data involved.\n\n(Joint work with T. Paul)\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Camillo De Lellis (Institute for the Advanced Study)
DTSTART:20241104T150000Z
DTEND:20241104T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/146
DESCRIPTION:Title: Boundary regularity of minimal surfaces\nby Camillo
De Lellis (Institute for the Advanced Study) as part of Geometric and func
tional inequalities and applications\n\n\nAbstract\nThe critical points of
the area functional\, usually called minimal surfaces\, have a long histo
ry in mathematics. Perhaps the most famous examples are the solutions of t
he so-called Plateau's problem\, i.e. surfaces which minimize the area amo
ng the ones spanning a given contour. It is known since long that area min
imiziers can form singularities and several concepts of generalized soluti
ons\, which serve different purposes\, have been introduced in the literat
ure since the first decades of the last century. A wide field of study is
the regularity of the latter objects. While there is a quite good understa
nding of the size of singularities away from the boundary in very many sit
uations\, the same cannot be said for the case of boundary singularities\,
for which we have very satisfactory theorems only in relatively few\, alb
eit important\, cases. I will review some results of the last decade which
touched for the first time a category of problems in the area\, I will th
en explain a recent joint work with Stefano Nardulli and Simone Steinbr\\"
uchel which gives a first positive answer to a question of Allard and Whit
e and finally\, if time allows\, mention recent further developments in th
e same direction by Fleschler and Resende.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xueying Yu (Oregon State University)
DTSTART:20240930T140000Z
DTEND:20240930T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/147
DESCRIPTION:Title: Some unique continuation results for Schrödinger equati
ons\nby Xueying Yu (Oregon State University) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nThis talk focuses
on a fundamental concept in the field of partial differential equations
— unique continuation principles. Such a principle describes the propaga
tion of the zeros of solutions to PDEs. Specifically\, it answers the ques
tion: what condition is required to guarantee that if a solution to a PDE
vanishes on a certain subset of the spatial domain\, then it must also van
ish on a larger subset of the domain. Motivated by Hardy’s uncertainty p
rinciple\, Escauriaza\, Kenig\, Ponce\, and Vega were able to show in a se
ries of papers that if a linear Schrödinger solution decays sufficiently
fast at two different times\, the solution must be trivial. In this talk\,
we will discuss unique continuation properties of solutions to higher-ord
er Schrödinger equations and variable-coefficient Schrödinger equations\
, and extend the classical Escauriaza-Kenig-Ponce-Vega type of result to t
hese models. This is based on joint works with S. Federico-Z. Li\, and Z.
Lee.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugenia Malinnikova (Stanford University)
DTSTART:20241021T140000Z
DTEND:20241021T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/148
DESCRIPTION:by Eugenia Malinnikova (Stanford University) as part of Geomet
ric and functional inequalities and applications\n\n\nAbstract\nFollowing
the classical work of Donnelly and Fefferman\, we consider eigenfunctions
of the Laplace-Beltrami operators on compact Riemannian manifolds and show
that they behave as polynomials of bounded degree\, governed by the eigen
value. In particular we give a sharp version of the Bernstein inequality a
nd a sharp version of the Remez inequality for the eigenfunctions. We will
describe some applications of these inequalities if times permits.\n\n \n
\nThe talk is based on a joint work with Decio and Nazarov\, and a joint w
ork with Logunov.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zheng-Chao Han (Rutgers University)
DTSTART:20241111T150000Z
DTEND:20241111T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/149
DESCRIPTION:Title: Asymptotic Behavior and Symmetry of Singular Solutions t
o Certain Geometric PDEs\nby Zheng-Chao Han (Rutgers University) as pa
rt of Geometric and functional inequalities and applications\n\n\nAbstract
\nWe will discuss some results on the asymptotic behavior or symmetry of s
ingular solutions to \ncertain geometric PDEs. A prototype problem is the
singular Yamabe problem\, where the solution \nbecomes singular on approac
h to a singular set of dimension $k$\, and the central question is whether
\nthere is a universal growth rate of the solution on approach to the sin
gular set\; whether\, when the domain of solution is the round sphere wit
h a lower dimensional sphere removed\, which exhibits a lot of symmetry\,
then the singular solution also exhibits corresponding symmetry\; and whet
her\, in certain cases\, one can get more precise asymptotic behavior of t
he singular solution?\n\nI will discuss some results\, joint with Alice
Chang and Paul Yang\, on the growth rate and symmetry of complete\, locall
y conformally flat metrics on canonical domains of the round sphere with c
onstant Q-curvature\, which in some sense relate to earlier work of Löwn
er-Nirenberg\, Schoen\, Delanoè\, and Finn-McOwen. \n\nI will also discus
s joint work with Jingang Xiong (Beijing Normal University) and\nLei Zhang
(University of Florida)\, which proves that any positive solution of the
Yamabe equation on an asymptotically flat $n$-dimensional manifold of flat
ness order at least (n-2)/2\nand n no greater than 24 must converge at inf
inity either to a fundamental solution of the Laplace operator on the Eucl
idean space or to a radial Fowler solution defined on the entire Euclidean
space. \nThe flatness order (n-2)/2 is the minimal flatness order require
d to define ADM mass in general relativity\; the dimension 24 is the divid
ing dimension of the validity of compactness of solutions to the Yamabe pr
oblem. We also prove such alternatives for bounded solutions when n>24.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiangdong Xie (Bowling Green State University)
DTSTART:20241118T150000Z
DTEND:20241118T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/150
DESCRIPTION:Title: Uniform quasiconformal groups of nilpotent Lie groups\nby Xiangdong Xie (Bowling Green State University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nA group of qu
asiconformal maps of a metric space X is called a uniform quasiconformal g
roup if there is some constant\n K such that each element of the group is
K-quasiconformal.\n Clearly a quasiconformal conjugate of a conformal g
roup is a uniform quasiconformal group. A natural question is when the co
nverse holds.\n Tukia's theorem says that if a uniform quasiconformal grou
p of the n-sphere (for n at least 2) is big enough\, then the converse ho
lds. We present a generalization of Tukia's theorem to uniform quasiconfo
rmal groups of two classes of nilpotent Lie groups: Carnot groups and Carn
ot-by-Carnot groups. This has consequences for the quasiisometric rigidit
y of solvable Lie groups and finitely generated solvable groups. This tal
k is based on joint work with Tullia Dymarz and David Fisher.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raoni Ponciano (Universidade Federal do ABC (UFABC))
DTSTART:20241209T150000Z
DTEND:20241209T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/151
DESCRIPTION:Title: Sharp Sobolev and Adams-Trudinger-Moser embeddings for s
ymmetric functions without boundary conditions on hyperbolic spaces\nb
y Raoni Ponciano (Universidade Federal do ABC (UFABC)) as part of Geometri
c and functional inequalities and applications\n\n\nAbstract\nEmbedding th
eorems for symmetric functions without any boundary conditions have been s
tudied on flat Riemannian manifolds\, such as the Euclidean space. However
\, these theorems have only been established on hyperbolic spaces for func
tions with homogeneous Dirichlet conditions. In this presentation\, we foc
us on sharp Sobolev and Adams–Trudinger–Moser embeddings for radial fu
nctions in hyperbolic spaces\, considering both bounded and unbounded doma
ins. One of the main features of our approach is that we do not assume any
boundary conditions for symmetric functions on geodesic balls or the enti
re hyperbolic space. Our main results establish weighted Sobolev embedding
theorems and present Adams-Trudinger-Moser type of embedding theorems. In
particular\, a key result is a highly nontrivial comparison between norms
of the higher order covariant derivatives and higher order derivatives of
the radial functions. Higher order asymptotic behavior of radial function
s on hyperbolic spaces is established to prove our main theorems. This app
roach includes novel radial lemmas and decay properties of higher order ra
dial Sobolev functions defined in hyperbolic space.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugenia Malinnikova (Stanford University)
DTSTART:20250127T150000Z
DTEND:20250127T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/152
DESCRIPTION:Title: Inequalities for Laplace eigenfunction.\nby Eugenia
Malinnikova (Stanford University) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nFollowing the classical work of D
onnelly and Fefferman\, we consider eigenfunctions of the Laplace-Beltrami
operators on compact Riemannian manifolds and show that they behave as po
lynomials of bounded degree\, governed by the eigenvalue. In particular we
give a sharp version of the Bernstein inequality and a sharp version of t
he Remez inequality for the eigenfunctions. We will describe some applicat
ions of these inequalities if times permits.\n\n \n\nThe talk is based on
a joint work with Decio and Nazarov\, and a joint work with Logunov.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hongjie Dong (Brown University)
DTSTART:20250203T150000Z
DTEND:20250203T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/153
DESCRIPTION:Title: Recent results on Hopf type lemma\nby Hongjie Dong (
Brown University) as part of Geometric and functional inequalities and app
lications\n\n\nAbstract\nThe classical Hopf lemma asserts that if a positi
ve harmonic function in a sufficiently smooth domain vanishes at a boundar
y point\, then its inner normal derivative at that point is strictly posit
ive. This foundational result extends to elliptic equations in non-diverge
nce form with measurable coefficients\, provided the domain is $C^{1\,\\al
pha}$ or $C^{1\,Dini}$.\n\nIn this talk\, I will discuss some recent work
on related work for divergence form equations and double divergence form e
quations (adjoint equations for non-divergence form equations). Some appli
cations will also be mentioned.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Serena Dipierro (The University of Western Australia)
DTSTART:20250217T150000Z
DTEND:20250217T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/154
DESCRIPTION:Title: A strict maximum principle for nonlocal minimal surfaces
\nby Serena Dipierro (The University of Western Australia) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nSupp
ose that two nonlocal minimal surfaces are included one into the other and
touch at a point. Then\, they must coincide. But this is perhaps less obv
ious than what it seems at first glance.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Seiringer (Institute of Science and Technology Austria)
DTSTART:20250303T150000Z
DTEND:20250303T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/155
DESCRIPTION:Title: Lieb—Thirring Inequalities: a density-functional point
of view\nby Robert Seiringer (Institute of Science and Technology Aus
tria) as part of Geometric and functional inequalities and applications\n\
n\nAbstract\nThe Lieb—Thirring Inequalities give a lower bound on the su
m of the gradient norms of orthonormal functions in terms of an L^p-norm o
f the corresponding density (the sum of the squares of the functions). The
y improve upon Sobolev Inequalities by taking into account the orthogonali
ty of the functions. They were introduced in 1976 by Lieb and Thirring to
give a proof of the stability of matter in quantum mechanics\, and have si
nce proved very useful in many other applications. We present a new and si
mple proof of a refined version of the inequalities\, and explain their co
nnection to the local density approximation in density-functional theory.
(Joint work with Jan Philip Solovej)\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kiselev (Duke University)
DTSTART:20250224T150000Z
DTEND:20250224T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/157
DESCRIPTION:Title: Suppression of chemotactic blow up by active scalar\
nby Alexander Kiselev (Duke University) as part of Geometric and functiona
l inequalities and applications\n\n\nAbstract\nThere exist many regulariza
tion mechanisms in nonlinear PDE that help\nmake solutions more regular or
prevent formation\nof singularity: diffusion\, dispersion\, damping. A re
latively less\nunderstood regularization mechanism is transport.\nThere is
evidence that in the fundamental PDE of fluid mechanics such as\nEuler or
Navier-Stokes\, transport can play\na regularizing role. In this talk\, I
will discuss another instance where\nthis phenomenon appears: the Patlak-
Keler-Segel\nequation of chemotaxis. Chemotactic blow up in the context of
the\nPatlak-Keller-Segel equation is an extensively studied phenomenon.\n
In recent years\, it has been shown that the presence of a given fluid\nad
vection can arrest singularity\nformation given that the fluid flow posses
ses mixing or diffusion\nenhancing properties and its amplitude is suffici
ently strong.\nThis talk will focus on the case when the fluid advection i
s active: the\nPatlak-Keller-Segel equation coupled with fluid that obeys\
nDarcy's law for incompressible porous media flow via gravity.\nSurprising
ly\, in this context\, in contrast with the passive advection\,\nactive fl
uid is capable of suppressing chemotactic blow up at arbitrary\nsmall coup
ling strength: namely\, the system\nalways has globally regular solutions.
The talk is based on work joint\nwith Zhongtian Hu and Yao Yao.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giampiero Palatucci (Universita de Parma)
DTSTART:20250310T140000Z
DTEND:20250310T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/158
DESCRIPTION:Title: The De Giorgi-Nash-Moser theory for kinetic equations wi
th nonlocal diffusions\nby Giampiero Palatucci (Universita de Parma) a
s part of Geometric and functional inequalities and applications\n\n\nAbst
ract\nI will present some recent results in the spirit of the De Giorgi-Na
sh-Moser theory for a wide class of kinetic integral equations\, where the
diffusion term in velocity is an integro-differential operator having non
negative kernel of fractional order with merely measurable coefficients. I
will mainly focus on boundedness estimates and Harnack inequalities. The
talk is based on a series of papers by Anceschi\, Kassmann\, Piccinini\, W
eidner and myself.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:João Marcos do Ó (Universidade Federal da Paraíba)
DTSTART:20250331T140000Z
DTEND:20250331T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/159
DESCRIPTION:Title: Weighted Hardy-Sobolev type inequalities with remainder<
/a>\nby João Marcos do Ó (Universidade Federal da Paraíba) as part of G
eometric and functional inequalities and applications\n\n\nAbstract\nThis
talk focuses on indefinite quasilinear elliptic problems involving weighte
d terms on unbounded domains\, potentially with unbounded boundaries. We w
ill explore existence results using variational methods applied to weighte
d function spaces and present several Liouville-type results for this clas
s of problems.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Holmes Fay (University of Wyoming)
DTSTART:20250407T140000Z
DTEND:20250407T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/160
DESCRIPTION:Title: Weak-type inequalities for sparse operators via Bellman
functions\nby Irina Holmes Fay (University of Wyoming) as part of Geom
etric and functional inequalities and applications\n\n\nAbstract\nI’ll d
iscuss recent joint work with Guillermo Rey and Kristina Skreb\, where we
find the exact Bellman function governing a certain weak-type inequality f
or sparse operators. We work in the dyadic setting\, where sparse operator
s have become a standard tool in the last few years - largely due to the s
harp results one obtains from strong-type bounds. Several open problems in
volve weak-type bounds\, which are much more difficult to sharpen. We expl
ore this aspect through the Bellman function method.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Ionescu (Princeton University)
DTSTART:20250421T140000Z
DTEND:20250421T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/161
DESCRIPTION:Title: On the wave turbulence theory of 2D gravity waves\nb
y Alexander Ionescu (Princeton University) as part of Geometric and functi
onal inequalities and applications\n\n\nAbstract\nOur goal in joint work w
ith Yu Deng and Fabio Pusateri is to initiate \nthe rigorous investigation
of wave turbulence for water wave models. \nThis problem has received int
ense attention in recent years in the \ncontext of semilinear models\, suc
h as semilinear Schrodinger equations \nor multi-dimensional KdV-type equa
tions. However\, our situation is \ndifferent since water wave systems are
quasilinear and the solutions \ncannot be constructed by iteration of the
Duhamel formula due to \nunavoidable derivative loss.\n\nOur strategy con
sists of two main steps: (1) a deterministic energy \ninequality that prov
ides control of (possibly large) Sobolev norms of \nsolutions for long tim
es\, under the condition that a certain \n$L^\\infty$-type norm is small\,
and (2) a propagation of randomness \nargument to prove a probabilistic r
egularity result for long times\, in \na suitable scaling regime.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Huai-Dong Cao (Lehigh University)
DTSTART:20250505T140000Z
DTEND:20250505T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/162
DESCRIPTION:Title: Linear stability and first eigenvalue estimates\nby
Huai-Dong Cao (Lehigh University) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nIn this talk\, we will discuss es
timates of the first eigenvalue of the Laplace operator/Lichnerowicz Lapla
cian \nand linear stability of positive Einstein manifolds\, or compact sh
rinking Ricci solitons\, with respect to Perelman's $\\nu$-entropy.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Nardulli (Universidade Federal do ABC)
DTSTART:20250414T140000Z
DTEND:20250414T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/163
DESCRIPTION:Title: Interior regularity of area minimizing currents within a
$C^{2\,\\alpha}$-submanifold\nby Stefano Nardulli (Universidade Feder
al do ABC) as part of Geometric and functional inequalities and applicatio
ns\n\n\nAbstract\nGiven an area-minimizing integral m-current in Σ\, we p
rove that the Hausdorff dimension\nof the interior singular set of T canno
t exceed m−2\, provided that Σ is an embedded (m+n)-submanifold of Rm+n
of class C2\,α\, where α > 0. This result establishes the complete\ncou
nterpart\, in the arbitrary codimension setting\, of the interior regulari
ty theory for area-\nminimizing integral hypercurrents within a Riemannian
manifold of class C2\,α.\n\nThis is a joint work with Reinaldo Resende.\
n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marcello Lucia (College of Staten Island-CUNY)
DTSTART:20250428T140000Z
DTEND:20250428T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/164
DESCRIPTION:Title: A mountain pass Theorem with a lack of compactness\n
by Marcello Lucia (College of Staten Island-CUNY) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nIn a very interes
ting paper by Gonçalves-Uhlenbeck\, the authors found that the moduli spa
ce of minimal immersions of a given closed surface in hyperbolic 3-manifol
ds is given by the critical points of an energy functional. However\, this
one may fail to satisfy the standard compactness conditions\, which does
not allow to apply immediately the classical variational theory to discuss
existence/uniqueness of critical points. \n\nIn this talk I will discuss
this class of functionals in a more abstract framework\, stress what are t
he difficulties that need to be overcome\, and derive as a corollary exist
ence and uniqueness of a critical point for the Gonçalves-Uhlenbeck\nfunc
tional.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Van Hoang Nguyen (FPT University)
DTSTART:20250512T140000Z
DTEND:20250512T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/166
DESCRIPTION:Title: The stability estimate for sharp Caffarelli-Kohn-Nirenbe
rg inequality for curl-free vector fields.\nby Van Hoang Nguyen (FPT U
niversity) as part of Geometric and functional inequalities and applicatio
ns\n\n\nAbstract\nIn this talk\, we present a stablity version of the shar
p Caffarelli-Kohn-Nirenberg inequality for curl-free vector fields recentl
y established by Cazacu\, Flynn and Lam. We show that the discrepancy of b
oth sides of the inequality control the weighted $L_2$-norm from the curl-
free vector filed to the set of extremizers. Our approach is based on the
spherical decomposition and the sharp one dimensional inequalities. We als
o show that this approach yields the sharp Caffarelli-Kohn-Nirenberg inequ
ality for second order derivatives and its stablity version. (The talk is
based on joint work with Duong Anh Tuan)\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gunther Uhlmann (University of Washington)
DTSTART:20250908T140000Z
DTEND:20250908T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/168
DESCRIPTION:Title: The fractional Anisotropic Calderon Problem\nby Gunt
her Uhlmann (University of Washington) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nWe discuss some recent progr
ess on the anisotropic Calderón problem for the fractional Laplacian.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hong Wang (CIMS-New York University)
DTSTART:20250929T140000Z
DTEND:20250929T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/169
DESCRIPTION:Title: Furstenberg sets estimate in the plane\nby Hong Wang
(CIMS-New York University) as part of Geometric and functional inequaliti
es and applications\n\n\nAbstract\nA $(s\,t)$-Furstenberg set is a set $E$
in the plane with the following property: there exists a $t$-dim family o
f lines such that each line intersects $E$ in a $\\geq s$--dimensional set
. An unpublished conjecture of Furstenberg states that any $(s\,1)$-Furste
nberg set has dimension at least $(3s+1)/2$. The Furstenberg set problem
can be viewed as a natural generalization of Davies's result that a Kakeya
set in the plane (a set that contains a line segment in any direction) ha
s dimension 2.\n\nWe will survey a sequence of results by Orponen\, Shmerk
in\, and a joint work with Ren that lead to the solution of the Furstenbe
rg set conjecture in the plane: any $(s\,t)$-Furstenberg set has Hausdorff
dimension at least $\\min \\{s+t\, (3s+t)/2\, s+1\\}$.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Mazzucato (Penn State U.)
DTSTART:20250922T140000Z
DTEND:20250922T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/170
DESCRIPTION:Title: On the Euler equations with in-flow and out-flow boundar
y conditions\nby Anna Mazzucato (Penn State U.) as part of Geometric a
nd functional inequalities and applications\n\n\nAbstract\nI will discuss
recent results concerning the well-posedness and regularity for the incomp
ressible Euler equations when in-flow and out-flow boundary conditions a
re imposed on parts of the boundary\, motivated by applications to boundar
y layers. This is joint work with Gung-Min Gie (U. Louisville\, USA) and J
ames Kelliher (UC Riverside\, USA). I will also discuss energy dissipation
and enstrophy production in the zero-viscosity limit at outflow\, joint
work with (Jincheng Yang\, U Chicago and IAS)\, Vincent Martinez (CUNY\,
Hunter College)\, and Alexis Vasseur (UT Austin).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruoyu Wang (Yale University)
DTSTART:20250519T140000Z
DTEND:20250519T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/171
DESCRIPTION:Title: Carleman inequalities for fractional Laplacians\nby
Ruoyu Wang (Yale University) as part of Geometric and functional inequalit
ies and applications\n\n\nAbstract\nInside any fixed open subset of compac
t manifolds\, Laplace eigenfunctions cannot decay faster than exponentiall
y (in eigenvalue): this is called a Carleman inequality. We discuss a new
Carleman inequality for fractional Laplacian quasimodes\, and a few relate
d results\, with their application in obtaining new decay estimates for li
nearised gravity and capillary water waves.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk
DTSTART:20251006T130000Z
DTEND:20251006T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/172
DESCRIPTION:by No Talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yilin Wang (IHES)
DTSTART:20251020T140000Z
DTEND:20251020T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/174
DESCRIPTION:Title: Geometry of Riemann surfaces through the lens of probabi
lity\nby Yilin Wang (IHES) as part of Geometric and functional inequal
ities and applications\n\n\nAbstract\nThe goal of this talk is to showcase
how we can use stochastic processes to study the geometry of surfaces. Af
ter recalling basic facts about surfaces with constant curvature\, their l
ength spectrum\, and Brownian motion on them\, we use the Brownian loop me
asure to express the lengths of closed geodesics on a hyperbolic surface a
nd zeta-regularized determinant of the Laplace-Beltrami operator. This giv
es a tool to study the length spectra of a hyperbolic surface and we obtai
n a new identity between the length spectrum of a compact surface and that
of the same surface with an arbitrary number of additional cusps. This is
mainly based on a recent joint work with Yuhao Xue (IHES).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vlassis Mastrantonis (University of Maryland)
DTSTART:20250526T140000Z
DTEND:20250526T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/175
DESCRIPTION:Title: $L^p$-smoothing in the study of Mahler volumes\, Bergman
kernels\, and the isotropic constant\nby Vlassis Mastrantonis (Univer
sity of Maryland) as part of Geometric and functional inequalities and app
lications\n\n\nAbstract\nWe discuss three problems: Mahler’s conjectures
\, Bourgain’s slicing problem\, and Blocki’s conjecture on the sharp l
ower bound of Bergman kernels of tube domains. Starting from Nazarov’s b
ound on the Mahler volume\, and our subsequent extension to the non-symmet
ric case\, we show how Mahler’s and Blocki’s conjectures are two faces
of the same coin. By introducing an L^p smoothing of the support function
\, we define the L^p-polar body\, and interpret the Bergman kernels of tub
e domains as the L^1-Mahler volume. The classical Mahler volume correspond
s to the L^infty-case. This perspective allows us to use well-established
methods to obtain sharp lower bounds for the Bergman kernels of tube domai
ns in dimension two\, verifying Błocki’s conjecture in that dimension\,
as well as an upper bound in all dimensions. Time permitting\, we discuss
functional extensions and a functional L^p Santaló inequality\, followin
g the approach of Nakamura—Tsuji via the Fokker—Planck heat flow. We a
lso explore applications of L^p-polarity to Bourgain’s slicing problem\,
leading to an “easy” bound on the isotropic constant through complex
geometric methods involving Ricci curvature and Bergman Kahler metrics.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guido De Philippis (CIMS-New York University)
DTSTART:20251027T140000Z
DTEND:20251027T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/176
DESCRIPTION:Title: Min-max construction of anisotropic minimal hypersurface
s\nby Guido De Philippis (CIMS-New York University) as part of Geometr
ic and functional inequalities and applications\n\n\nAbstract\nWe use the
min-max construction to find closed hypersurfaces which are stationary wi
th respect to anisotropic elliptic integrands in any closed n-dimensiona
l manifold. These surfaces are regular outside a closed set of zero n-
3 dimension. The critical step is to obtain a uniform upper bound for dens
ity ratios in the anisotropic min-max construction. This confirms a conjec
ture posed by Allard. The talk is based on a joint work with A. De Rosa an
d Y. Li.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk
DTSTART:20251013T130000Z
DTEND:20251013T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/177
DESCRIPTION:by No Talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Brendle (Columbia University)
DTSTART:20251117T150000Z
DTEND:20251117T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/178
DESCRIPTION:Title: Systolic inequalities and the Horowitz-Myers conjecture<
/a>\nby Simon Brendle (Columbia University) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nLet $n$ be an integer w
ith $3 \\leq n \\leq 7$\, and let $g$ be a Riemannian metric on $B^2 \\tim
es T^{n-2}$ with scalar curvature at least $-n(n-1)$. We establish an ineq
uality relating the systole of the boundary to the infimum of the mean cur
vature on the boundary. As a consequence\, we obtain a new positive energy
theorem where equality holds for the Horowitz-Myers metrics. This is join
t work with Pei-Ken Hung.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natasa Sesum (Rutgers University)
DTSTART:20251201T150000Z
DTEND:20251201T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/179
DESCRIPTION:Title: Instability of peanut solution\nby Natasa Sesum (Rut
gers University) as part of Geometric and functional inequalities and appl
ications\n\n\nAbstract\nPeanut solutions are examples of degenerate neckpi
nches in both\, the Ricci flow and the mean curvature flow. We show that i
n every neighborhood of peanut solution there is an initial data developin
g a spherical singularity\, and at the same time there is an initial data
developing a nondegenerate neckpinch singularity. This shows the peanut t
ype solution is highly unstable. This is a joint work with Angenent and Da
skalopoulos.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicola Fusco (Università degli Studi di Napoli Federico II)
DTSTART:20251110T150000Z
DTEND:20251110T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/182
DESCRIPTION:Title: The isoperimetric inequality for the capillary energy ou
tside convex sets\nby Nicola Fusco (Università degli Studi di Napoli
Federico II) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nI will present an isoperimetric problem for capillary
surfaces with a general contact angle $ \\theta \\in (0\, \\pi) $\, lying
outside a convex set. We will see that the capillary energy of any set $E
$ contained in the complement of a convex set $C$ is strictly larger than
that of a spherical cap with the same volume and the same contact angle s
itting on a flat support\, unless $E$ is a spherical cap lying on a facet
of $C$. This result extends to the case of general contact angles a well-
known relative isoperimetric inequality\, corresponding to the case $ \\t
heta = \\pi/2$\, proved in 2007 by Choe-Ghomi-Ritoré. \nJoint paper with
Vesa Julin\, Massimiliano Morini and Aldo Pratelli.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eudes Barboza (Universidade Federal Rural do Pernambuco)
DTSTART:20250721T140000Z
DTEND:20250721T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/183
DESCRIPTION:Title: Existence results for some elliptic problems in $\\mathb
b{R}^N$ including variable exponents above the critical growth\nby Eud
es Barboza (Universidade Federal Rural do Pernambuco) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nIn this talk\
, we establish existence results for the following class of equations inv
olving variable exponents\n$$\n-\\Delta u +u=|u(x)|^{p(|x|)-1}u(x)+ \\lamb
da|u(x)|^{q(|x|)-1}u(x)\, \\quad x\\in\\mathbb{R}^{N}\,\n$$\nwhere $\\lamb
da\\geq0$\, $N\\geq 3$ and $p\,q:[0\,+\\infty)\\rightarrow(1\,+\\infty)$ a
re radial continuous functions which satisfy suitable conditions. For this
purpose\, it is sufficient to consider either subcriticality or criticali
ty within a small region near the origin. Surprisingly\, outside this regi
on\, the nonlinearity may oscillate between subcritical\, critical\, and s
upercritical growth in the Sobolev sense. Our approach enables the use of
the variational methods to tackle problems with variable exponents in $\
\mathbb{R}^N$ without imposing restrictions outside of a neighborhood of z
ero.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/183/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rodrigo Clemente (Federal Rural University of Pernambuco)
DTSTART:20250728T140000Z
DTEND:20250728T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/184
DESCRIPTION:Title: $p$-Harmonic Functions in the Upper Half-space\nby R
odrigo Clemente (Federal Rural University of Pernambuco) as part of Geomet
ric and functional inequalities and applications\n\n\nAbstract\nIn this ta
lk\, we will discuss the existence\, nonexistence\, and qualitative proper
ties of $p$-harmonic functions in the upper half-space that satisfy nonlin
ear boundary conditions\, for $1<\;p<\;N$. We will also present a symm
etry result for positive solutions\, obtained through the method of moving
planes.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/184/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Colding (MIT)
DTSTART:20251208T150000Z
DTEND:20251208T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/185
DESCRIPTION:Title: Deficit function and log Sobolev inequality\nby Tobi
as Colding (MIT) as part of Geometric and functional inequalities and appl
ications\n\n\nAbstract\nThere is a long history of parabolic monotonicity
formulas that developed independently from several different fields and a
much more recent elliptic theory. The elliptic theory can be localized and
there are additional monotone quantities. There is also a surprising link
: Taking a high-dimensional limit of the right elliptic monotonicity can g
ive a parabolic one as a limit. Poincare was the first to observe such a c
onnection. We introduce two deficit functions\, one elliptic and one parab
olic\, then show that the parabolic deficit is pointwise the limit of the
elliptic and\, that the elliptic satisfies an equation that converges to t
he equation for the parabolic. These pointwise quantities and their equati
ons recover the monotonicities and leads to an elliptic proof of the log S
obolev inequality as well as new concentration of measure phenomena. The
talk is based on joint work with Bill Minicozzi.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Piccione (University of São Paulo)
DTSTART:20260420T140000Z
DTEND:20260420T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/186
DESCRIPTION:Title: Cohomogeneity one minimal submanifolds\nby Paolo Pic
cione (University of São Paulo) as part of Geometric and functional inequ
alities and applications\n\n\nAbstract\nI will talk about minimal submanif
olds S of closed Riemannian M that are invariant by the isometric action o
f a compact connected Lie group G. Cohomogeneity one means that the quotie
nt space S/G has dimension one. By the symmetry reduction principle of Hsi
ang–Lawson]\, such submanifolds correspond to geodesics in the orbit spa
ce M/G\, endowed with a conformal metric which is singular on the boundary
\, I will discuss a result of existence of free boundary geodesics in this
singular space\, and present a related compactness result in the case of
smooth boundary. Finally\, I will discuss a bifurcation theoretic applicat
ion for minimal spheres in elongated ellipsoids. The talk is based on join
t works with R. Bettiol\, D. Corona and F. Giannoni.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Detang Zhou (Universidade Federal Fluminense)
DTSTART:20260119T150000Z
DTEND:20260119T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/187
DESCRIPTION:Title: Eigenvalue estimate on self shrinkers\nby Detang Zho
u (Universidade Federal Fluminense) as part of Geometric and functional in
equalities and applications\n\n\nAbstract\nIn this talk\, I will discuss t
he drifted Laplacian $\\Delta_f$ on a hypersurface $M$ in a Ricci shrinker
$(\\overline{M}\,g\,f)$. We proved that the spectrum of $\\Delta_f$ is d
iscrete for immersed hypersurfaces with bounded weighted mean curvature in
a Ricci shrinker. I will also discuss a lower bound for the first nonzero
eigenvalue of $\\Delta_f$ when the hypersurface is an embedded $f$-mini
mal one. This estimate contains the case of compact minimal hypersurfaces
in a positive Einstein manifold\, in particular Choi and Wang's estimate
for minimal hypersurfaces. The estimate also recovers the ones of Ding-Xin
and Brendle-Tsiamis on self-shrinkers.This is a joint work with Franciele
Conrado.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/187/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fausto Ferrari (Università di Bologna)
DTSTART:20260126T150000Z
DTEND:20260126T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/188
DESCRIPTION:Title: Recent regularity results in free boundary problems\
nby Fausto Ferrari (Università di Bologna) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nWe deal with regularity
results about one and two-phase free boundary problems. In particular\, w
e discuss the regularity of the free boundary in the viscosity setting. Mo
reover\, we introduce some recent results obtained in collaboration with C
laudia Lederman IMAS - CONICET and Departamento de Matematica\, Facultad
de Ciencias Exactas y Naturales\,Universidad de Buenos Aires\, Argentina\
, about flat free boundaries of viscosity solutions of two-phase probl
ems governed by the p(x)-Laplace operator. This research is part of a long
standing project\, where many people gave scientific contributions\, and
started with the pioneering papers of Luis Caffarelli.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/188/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Muratori (Politecnico di Milano)
DTSTART:20260202T150000Z
DTEND:20260202T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/189
DESCRIPTION:Title: A characterization of stochastic incompleteness via nonl
inear diffusion\nby Matteo Muratori (Politecnico di Milano) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nA R
iemannian manifold M is stochastically incomplete if the trajectories of t
he Brownian motion acting on it can diverge\, in finite time\, with non-ze
ro probability. Since the heat kernel of M is precisely the transition pro
bability density of the Brownian motion\, this is equivalent to the fact t
hat the heat semigroup loses mass. By exploiting the linearity of the heat
equation\, it is not difficult to see that such a property is also equiva
lent to the existence of multiple bounded solutions of parabolic Cauchy pr
oblems as well as to the existence of non-trivial bounded solutions of the
elliptic resolvent equation.\n\nThe main subject of this seminar is the e
xtension of this kind of characterizations to certain non-linear PDEs of d
iffusive type. Specifically\, I will describe some recent equivalence resu
lts between stochastic incompleteness and non-uniqueness properties of a g
eneral class of nonlinear diffusion equations\, known as filtration equati
ons\, and related semilinear elliptic equations\, obtained in collaboratio
n with G. Grillo\, K. Ishige\, and F. Punzo.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/189/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristina Trombetti (Università degli Studi di Napoli "Federico II
")
DTSTART:20260223T150000Z
DTEND:20260223T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/190
DESCRIPTION:Title: On a Class of Free Boundary Problems Related to Thermal
Insulation\nby Cristina Trombetti (Università degli Studi di Napoli "
Federico II") as part of Geometric and functional inequalities and applica
tions\n\n\nAbstract\nFree boundary problems in partial differential equati
ons (PDEs) constitute a class of mathematical models in which both the sol
ution and the domain where it is defined are unknown and must be determine
d simultaneously. Such problems naturally arise in a wide range of physica
l and engineering applications\, including fluid dynamics\, solid mechanic
s\, and heat conduction.\n\nIn this talk\, we focus on a class of free bou
ndary problems related to thermal insulation. In these models\, the free b
oundary may represent either an interface whose location is not known a pr
iori or the optimal configuration of an insulating material. The aim is to
characterize the resulting free boundaries and to analyze how their geome
try influences heat loss and energy efficiency.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/190/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dario Monticelli (Politecnico di Milano)
DTSTART:20260309T140000Z
DTEND:20260309T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/191
DESCRIPTION:Title: Rigidity and classification results for critical ellipti
c equations\nby Dario Monticelli (Politecnico di Milano) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nIn thi
s talk I will present some recent classification and rigidity results for
positive solutions to some classical nonlinear elliptic equations with cri
tical growth\, both in the Euclidean and in the Riemannian setting. If tim
e permits I will also briefly discuss extensions to degenerate/singular pr
oblems and to the subriemannian setting\, where similar rigidity and class
ification results occur. The results are joint works with G. Catino (Polit
ecnico di Milano)\, Y.Y. Li (Rutgers University)\, A. Roncoroni (Politecni
co di Milano) and X. Wang (Michigan State University).\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/191/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Pratelli (University of Pisa)
DTSTART:20260302T150000Z
DTEND:20260302T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/192
DESCRIPTION:Title: Small minimal clusters in manifolds\nby Aldo Pratell
i (University of Pisa) as part of Geometric and functional inequalities an
d applications\n\n\nAbstract\nIn this talk\, we will consider the isoperim
etric problem for $m$-clusters in a compact manifold. Since minimal cluste
rs in $\\mathbb{R}^N$ are known to be connected\, and a Riemannian manifol
d locally looks like $\\mathbb{R}^N$\, one might easily guess that minimal
clusters with sufficiently small total volume should be connected. This i
s well-known for the case of sets (so\, with $m=1$)\, and we will present
a recent proof of this fact for the general case. The situation becomes la
rgely different if one\, instead\, considers Finsler manifolds: in this ca
se we can prove that small minimal clusters can have up to m connected com
ponents\, and an explicit example shows that they do not have to be connec
ted. We will conclude by considering the intermediate case of the "fixed-n
orm" manifolds\, and by addressing the optimality issue for the possible n
umber of connected components. Most the results are contained in different
joint papers with D. Carazzato\, L. Felicetti\, S. Nardulli\, R. Ponciano
.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/192/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Olsen (University of St. Andrews)
DTSTART:20260209T150000Z
DTEND:20260209T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/193
DESCRIPTION:Title: The Baire Hierarchy\, multifractal decomposition sets an
d $\\Pi^0_{\\gamma}$-completeness\nby Lars Olsen (University of St. An
drews) as part of Geometric and functional inequalities and applications\n
\n\nAbstract\nThis talk will discuss the position of the so-called ``multi
fractal decomposition sets'' in the Baire Hierarchy. In particular\, we wi
ll prove that ``multifractal decomposition sets'' are the building blocks
from which all other $\\Pi^0_{\\gamma}$-sets can be constructed\; more pre
cisely\, ``multifractal decomposition sets'' are $\\Pi^0_{\\gamma}$-comple
te. As an application we find the position of the classical Eggleston-Besi
covitch set in the Baire Hierarchy.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/193/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Case (Penn State University)
DTSTART:20260323T140000Z
DTEND:20260323T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/194
DESCRIPTION:Title: The sharp $L^2$-Michael-Simon-Sobolev inequality via con
formal geometry\nby Jeffrey Case (Penn State University) as part of Ge
ometric and functional inequalities and applications\n\n\nAbstract\nWe use
methods from conformal geometry to show that the best constant in the $L^
2$-Michael-Simon-Sobolev inequality for an $n$-dimensional minimal submani
fold of Euclidean $(m+n)$-space is equal to the sharp constant in the usua
l $L^2$-Sobolev inequality on $\\mathbb{R}^n$ as computed by Aubin and Tal
enti. This removes the codimension restriction $m \\leq 2$ from a result o
f Brendle. This is joint work with Dawit Mengesha.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/194/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisabeth Werner (Case Western Reserve University)
DTSTART:20260216T150000Z
DTEND:20260216T160000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/195
DESCRIPTION:Title: The $L_p$-Floating Area\, Entropy\, and Isoperimetric In
equalities on the Sphere\nby Elisabeth Werner (Case Western Reserve Un
iversity) as part of Geometric and functional inequalities and application
s\n\n\nAbstract\nThe floating area was previously investigated as a natura
l extension of classical affine surface area to non-Euclidean convex bodie
s in spaces of constant positive curvature. We introduce the family of $L_
p$-floating areas for spherical convex bodies\, as an analog to $L_p$-aff
ine surface area measures \nfrom Euclidean geometry. We investigate a dual
ity formula\, monotonicity and isoperimetric inequalities for this new fam
ily of curvature measures on spherical convex bodies.\n \n\nIf time per
mits\, we introduce a new entropy functional for spherical convex bodies u
sing the $L_p$-floating area\, and a dual isoperimetric inequality is esta
blished.\n \n\nBased on joint works with Florian Besau.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/195/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christina Sormani (CUNY-Lehman College)
DTSTART:20260330T140000Z
DTEND:20260330T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/196
DESCRIPTION:Title: Geometric Stability of the Schoen-Yau Zero Mass Theorem:
A Survey\nby Christina Sormani (CUNY-Lehman College) as part of Geome
tric and functional inequalities and applications\n\n\nAbstract\nIn 1979\,
Schoen and Yau proved their 3D Positive Mass Theorem which is a compariso
n theorem and a rigidity theorem. The comparison theorem compares Euclid
ean space with a three dimensional asymptotically flat Riemannian manifold
\, M\, with nonnegative scalar curvature and concludes that the ADM mass o
f M is greater than or equal to the ADM mass of Euclidean Space (which is
zero). Their zero mass rigidity theorem states that if such a manifold\,
M\, has zero ADM mass then it is isometric to Euclidean space. Here
we review results and open questions on the geometric stability or almost
rigidity of their zero mass rigidity theorem: if such a manifold\, M\, h
as almost zero mass\, how close is the geometry of M to that of Euclidean
space? We will review examples of sequences of such manifolds with mass
approaching zero whose geometry is quite far from that of Euclidean space
and review theorems proving that (under additional hypotheses) regions in
these manifolds may have geometries that converge to regions in Euclidean
space. Although there has been much progress\, it is still an open ques
tion (even in dimension three): exactly which geometric notion of converge
nce and closeness works best to capture the geometric stability of this fa
mous rigidity theorem.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/196/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Sinestrari (Università di Roma "Tor Vergata")
DTSTART:20260413T140000Z
DTEND:20260413T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/197
DESCRIPTION:by Carlo Sinestrari (Università di Roma "Tor Vergata") as par
t of Geometric and functional inequalities and applications\n\nAbstract: T
BA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/197/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emanuel Milman (Israel Institute of Technology)
DTSTART:20260511T140000Z
DTEND:20260511T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/198
DESCRIPTION:by Emanuel Milman (Israel Institute of Technology) as part of
Geometric and functional inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/198/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruixiang Zhang (UC Berkely)
DTSTART:20260406T140000Z
DTEND:20260406T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/199
DESCRIPTION:Title: The Mizohata-Takeuchi Conjecture for convex hypersurfac
es\nby Ruixiang Zhang (UC Berkely) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nThe Mizohata-Takeuchi Conje
cture predicts an $L^2$ estimate of functions with Fourier support on a co
nvex hypersurface. It looks deceptively simple but remains a difficult pr
oblem to understand. I will talk about a recently found counterexample wit
h Cairo showing power blowups for this conjecture for many hypersurfaces i
n all dimensions. Our construction was inspired by intuitions from additiv
e combinatorics and lattice point counting for curves.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/199/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gigliola Staffilani (MIT)
DTSTART:20260504T140000Z
DTEND:20260504T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/201
DESCRIPTION:by Gigliola Staffilani (MIT) as part of Geometric and function
al inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/201/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liding Yao (Purdue University Fort Wayne)
DTSTART:20260518T140000Z
DTEND:20260518T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/202
DESCRIPTION:Title: The Newlander-Nirenberg Theorem below $C^{1/2}$\nby
Liding Yao (Purdue University Fort Wayne) as part of Geometric and functio
nal inequalities and applications\n\n\nAbstract\nThe celebrated Newlander-
Nirenberg theorem states that on a smooth manifold\, an almost complex str
ucture $J$ is a complex structure if and only if it is integrable\, namely
\, the Nijenhuis tensor $N_J$ vanishes. It was known from Hill and Taylor
that if $J$ has H\\"older regularity above $C^{1/2}$ then $N_J$ makes sens
e as a tensor with distributional coefficients. However $N_J$ is undefined
for generic $C^{1/2}$ tensor due to the failure of multiplication for $C^
{1/2}$ functions and $C^{-1/2}$ distributions.\n\nIn the talk\, we will ex
plore the integrability condition when $J$ has regularity below $C^{1/2}$.
We give a necessary and sufficient condition for $J$ being a complex stru
cture (at least) for $J\\in C^{1/3+}$ using Bony's paradifferential calcul
us.\n\nThis is an in progress work joint with Gennady Uraltsev.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/202/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiming Zhao (Syracuse University)
DTSTART:20260427T140000Z
DTEND:20260427T150000Z
DTSTAMP:20260404T111333Z
UID:GeomInequAndPDEs/203
DESCRIPTION:Title: SL(n)-invariant isoperimetric and Minkowski problems
\nby Yiming Zhao (Syracuse University) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nIn convex geometry\, where c
onvex bodies are the primary objects of study\, a central goal is to disco
ver geometric invariants and measures that can be used to recover or chara
cterize their shapes. Two intertwined lines of research pursue this object
ive. Isoperimetric inequalities involving geometric invariants\, including
the classical isoperimetric inequality and the celebrated Brunn–Minkows
ki inequality\, seek to identify special shapes as extremals. Minkowski pr
oblems\, a family of problems originating in the work of Minkowski\, aim t
o recover\, sometimes uniquely\, the shape of an arbitrary convex body by
solving measure equations that\, under additional but unnecessary smoothne
ss assumptions\, reduce to Monge–Ampère-type equations. In this talk\,
after giving some historical background\, I will discuss recent joint work
with Dongmeng Xi that continues this line of research through the study o
f integral affine surface area and radial mean bodies.\n
LOCATION:https://stable.researchseminars.org/talk/GeomInequAndPDEs/203/
END:VEVENT
END:VCALENDAR