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BEGIN:VEVENT
SUMMARY:William Beckner (University of Texas at Austin)
DTSTART;VALUE=DATE-TIME:20200720T130000Z
DTEND;VALUE=DATE-TIME:20200720T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/1
DESCRIPTION:Title: Symmetry in Fourier Analysis – Heisenberg to Stein-Weiss\nb
y William Beckner (University of Texas at Austin) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nEmbedded symmetry
within the Heisenberg group is used to couple geometric insight and analy
tic calculation to obtain a new sharp Stein-Weiss inequality with mixed ho
mogeneity on the line of duality. SL(2\,R) invariance and Riesz potentials
define a natural bridge for encoded information that connects distinct ge
ometric structures. Insight for Stein-Weiss integrals is gained from vorte
x dynamics\, embedding on hyperbolic space\, and conformal geometry. The i
ntrinsic character of the Heisenberg group makes it the natural playing fi
eld on which to explore the laws of symmetry and the interplay between ana
lysis and geometry on a manifold.\n\nZoom link: https://brown.zoom.us/j/91
683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Malchiodi (Scuola Normale Superiore)
DTSTART;VALUE=DATE-TIME:20200720T140000Z
DTEND;VALUE=DATE-TIME:20200720T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/2
DESCRIPTION:Title: On the Sobolev quotient in sub-Riemannian geometry\nby Andrea
Malchiodi (Scuola Normale Superiore) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nWe consider three-dimensional
CR manifolds\, which are modelled on the Heisenberg group.\nWe introduce
a natural concept of “mass” and prove its positivity under the conditi
on that\nthe scalar curvature is positive and in relation to their (holomo
rphic) embeddability properties.\nWe apply this result to the CR Yamabe pr
oblem\, and we discuss extremality of Sobolev-type\nquotients\, giving som
e counterexamples for “Rossi spheres”.\nThis is joint work with J.H.Ch
eng and P.Yang.\n\nZoom link: https://brown.zoom.us/j/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xavier Cabre (ICREA and UPC (Barcelona))
DTSTART;VALUE=DATE-TIME:20200727T130000Z
DTEND;VALUE=DATE-TIME:20200727T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/3
DESCRIPTION:Title: Stable solutions to semilinear elliptic equations are smooth up t
o dimension 9\nby Xavier Cabre (ICREA and UPC (Barcelona)) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nThe
regularity of stable solutions to semilinear elliptic PDEs has been studie
d since the 1970's. In dimensions 10 and higher\, there exist singular sta
ble energy solutions. In this talk I will describe a recent work in collab
oration with Figalli\, Ros-Oton\, and Serra\, where we prove that stable s
olutions are smooth up to the optimal dimension 9. This answers to an open
problem posed by Brezis in the mid-nineties concerning the regularity of
extremal solutions to Gelfand-type problems.\n\nZoom link: https://brown.z
oom.us/j/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Jerison (MIT)
DTSTART;VALUE=DATE-TIME:20201109T140000Z
DTEND;VALUE=DATE-TIME:20201109T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/4
DESCRIPTION:Title: Rescheduled to Spring Semester 2021\nby David Jerison (MIT) a
s part of Geometric and functional inequalities and applications\n\nAbstra
ct: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yiming Zhao (MIT)
DTSTART;VALUE=DATE-TIME:20200803T130000Z
DTEND;VALUE=DATE-TIME:20200803T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/5
DESCRIPTION:Title: Reconstruction of convex bodies via Gauss map\nby Yiming Zhao
(MIT) as part of Geometric and functional inequalities and applications\n
\n\nAbstract\nIn this talk\, we will discuss the Gauss image problem\, a p
roblem that reconstructs the shape of a convex body using partial data reg
arding its Gauss map. In the smooth category\, this problem reduces to a M
onge-Ampere type equation on the sphere. But\, we will use a variational a
rgument that works with generic convex bodies. This is joint work with Ká
roly Böröczky\, Erwin Lutwak\, Deane Yang\, and Gaoyong Zhang.\n\nZoom l
ink: https://brown.zoom.us/j/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juncheng Wei (University of British Columbia)
DTSTART;VALUE=DATE-TIME:20200817T140000Z
DTEND;VALUE=DATE-TIME:20200817T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/6
DESCRIPTION:Title: Rigidity Results for Allen-Cahn Equation\nby Juncheng Wei (Un
iversity of British Columbia) as part of Geometric and functional inequali
ties and applications\n\n\nAbstract\nI will discuss two recent rigidity re
sults 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-dimensional. Th
e 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/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fengbo Hang (New York University)
DTSTART;VALUE=DATE-TIME:20200810T140000Z
DTEND;VALUE=DATE-TIME:20200810T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/7
DESCRIPTION:Title: Concentration compactness principle in critical dimensions revisi
ted\nby Fengbo Hang (New York University) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nConcentration compact
ness principle for functions in $W^{1\,n}_0$ on a\nn-dimensional domain wa
s introduced by Lions in 1985 with the\nMoser-Trudinger inequality in mind
. We will discuss some further\nrefinements after Cerny-Cianchi-Hencl's im
provement in 2013. These\nrefinements unifiy the approach for n=2 and n>2
cases and work for higher\norder or fractional order Sobolev spaces as wel
l. They are motivated by\nand closely related to the recent derivation of
Aubin's Moser-Trudinger\ninequality for functions with vanishing higher or
der moments on the\nstandard 2-sphere.\n\nZoom link: https://brown.zoom.us
/j/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (CalTech)
DTSTART;VALUE=DATE-TIME:20200824T143000Z
DTEND;VALUE=DATE-TIME:20200824T153000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/8
DESCRIPTION:Title: REVERSE HARDY–LITTLEWOOD–SOBOLEV INEQUALITIES\nby Rupert
Frank (CalTech) as part of Geometric and functional inequalities and appli
cations\n\n\nAbstract\nWe are interested in a new family of reverse Hardy
–Littlewood–Sobolev inequalities which involve a power law kernel with
positive exponent and a Lebesgue exponent <1. We characterize the range o
f parameters for which the inequality holds and present results about the
existence of optimizers. A striking open question is the possibility of co
ncentration of a minimizing sequence.\n\nThis talk is based on joint work
with J. Carrillo\, M. Delgadino\, J. Dolbeault and F. Hoffmann.\n\nPlease
note the special time of this talk. \nFor Zoom link for each talk (future
links will not be posted here)\, please send an email to the organizers at
geometricinequalitiesandpdes@gmail.com\nZoom link: https://brown.zoom.us/
j/94525179475\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengfei Guan (McGill University)
DTSTART;VALUE=DATE-TIME:20200831T140000Z
DTEND;VALUE=DATE-TIME:20200831T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/9
DESCRIPTION:Title: A mean curvature type flow and isoperimetric problem in warped pr
oduct spaces\nby Pengfei Guan (McGill University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe will discu
ss a mean curvature type flow with the goal to solve isoperimetric problem
. The flow is induced from the variational properties associated to confor
mal Killing fields. Such flow was first introduced in space forms in a pre
vious joint work with Junfang Li\, where we provided a flow approach to th
e classical isoperimetric inequality in space form. Later\, jointly with J
unfang Li and Mu-Tao Wang\, we considered the similar flow in warped produ
ct spaces with general base. Under some natural conditions\, the flow pres
erves the volume of the bounded domain enclosed by a graphical hypersurfac
e\, and monotonically decreases the hypersurface area. Furthermore\, the r
egularity and convergence of the flow can be established\, thereby the iso
perimetric problem in warped product spaces can be solved. The flow serves
as an interesting way to achieve the optimal solution to the isoperimetri
c problem.\n\nZoom link: https://brown.zoom.us/j/99054390401\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Hebey (Université de Cergy-Pontoise)
DTSTART;VALUE=DATE-TIME:20201116T140000Z
DTEND;VALUE=DATE-TIME:20201116T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/10
DESCRIPTION:Title: Schrödinger-Proca constructions in the closed setting\nby E
mmanuel Hebey (Université de Cergy-Pontoise) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nWe discuss Schröding
er-Proca constructions in the context of closed manifolds leading \nto the
Bopp-Podolsky-Schrödinger-Proca and the Schrödinger-Poisson-Proca syste
ms.\nThe goal is to present an introduction to these equations (how we bui
ld 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-Po
dolsky parameter goes to zero.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yi Wang (Johns Hopkins University)
DTSTART;VALUE=DATE-TIME:20200907T130000Z
DTEND;VALUE=DATE-TIME:20200907T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/11
DESCRIPTION:Title: Rigidity of local minimizers of the $\\sigma_k$ functional\n
by Yi Wang (Johns Hopkins University) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nIn this talk\, I will present
a result on the rigidity of local minimizers of the functional $\\int \\s
igma_2+ \\oint H_2$ among all conformally flat metrics in the Euclidean (n
+ 1)-ball. We prove the metric is flat up to a conformal transformation i
n some (noncritical) dimensions. We also prove the analogous result in the
critical dimension n + 1 = 4. The main method is Frank-Lieb’s rearrange
ment-free argument. If minimizers exist\, this implies a fully nonlinear s
harp Sobolev trace inequality. I will also discuss a nonsharp Sobolev trac
e inequality. This is joint work with Jeffrey Case.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dongmeng Xi (NYU)
DTSTART;VALUE=DATE-TIME:20200803T140000Z
DTEND;VALUE=DATE-TIME:20200803T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/12
DESCRIPTION:Title: An isoperimetric type inequality via a modified Steiner symmetri
zation scheme\nby Dongmeng Xi (NYU) as part of Geometric and functiona
l inequalities and applications\n\n\nAbstract\nWe establish an affine isop
erimetric inequality using a symmetrization scheme that involves a total o
f 2n elaborately chosen Steiner symmetrizations at a time. The necessity o
f this scheme\, as opposed to the usual Steiner symmetrization\, will be d
emonstrated with an example. This is a joint work with Dr. Yiming Zhao.\n\
nZoom link: https://brown.zoom.us/j/91683612862\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phan Thành Nam (LMU Munich)
DTSTART;VALUE=DATE-TIME:20200921T130000Z
DTEND;VALUE=DATE-TIME:20200921T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/13
DESCRIPTION:Title: Lieb-Thirring inequality with optimal constant and gradient erro
r term\nby Phan Thành Nam (LMU Munich) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nIn 1975\, Lieb and Thir
ring conjectured that the kinetic energy of fermions is not smaller than i
ts Thomas-Fermi (semiclassical) approximation\, at least in three or highe
r dimensions. I will discuss a rigorous lower bound with the sharp semicla
ssical constant and a gradient error term which is normally of lower order
in applications. The proof is based on a microlocal analysis and a vari
ant of the Berezin-Li-Yau inequality. This approach can be extended to der
ive an improved Lieb-Thirring inequality for interacting systems\, where t
he Gagliardo-Nirenberg constant appears in the strong coupling limit.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Minicozzi (MIT)
DTSTART;VALUE=DATE-TIME:20201026T140000Z
DTEND;VALUE=DATE-TIME:20201026T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/14
DESCRIPTION:Title: Mean curvature flow in high codimension\nby William Minicozz
i (MIT) as part of Geometric and functional inequalities and applications\
n\n\nAbstract\nMean curvature flow (MCF) is a geometric heat equation wher
e a\nsubmanifold evolves to minimize its area. A central problem is to\nu
nderstand the singularities that form and what these imply for the\nflow.
I will talk about joint work with Toby Colding on higher\ncodimension MCF
\, where the flow becomes a complicated system of\nequations and much less
is known.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Saloff-Coste (Cornell University)
DTSTART;VALUE=DATE-TIME:20201005T140000Z
DTEND;VALUE=DATE-TIME:20201005T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/15
DESCRIPTION:Title: Heat kernel on manifolds with finitely many ends\nby Laurent
Saloff-Coste (Cornell University) as part of Geometric and functional ine
qualities 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 remains. In
this talk\, after giving an overview of what the problem is and what we k
now\, the main difficulty will be explained and recent progresses involvin
g joint work with Grigor'yan and Ishiwata will be explained. They provide
results concerning Poincaré inequality in large central balls on such man
ifold.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Dolbeault (Université Paris-Dauphine)
DTSTART;VALUE=DATE-TIME:20200914T130000Z
DTEND;VALUE=DATE-TIME:20200914T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/16
DESCRIPTION:Title: Stability in Gagliardo-Nirenberg inequalities\nby Jean Dolbe
ault (Université Paris-Dauphine) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nOptimal constants and optimal fun
ctions are known in some functional inequalities. The next question is the
stability issue: is the difference of the two terms controlling a distanc
e to the set of optimal functions ? A famous example is provided by Sobole
v's inequalities: in 1991\, G. Bianchi and H. Egnell proved that the diffe
rence of the two terms is bounded from below by a distance to the manifold
of the Aubin-Talenti functions. They argued by contradiction and gave a v
ery elegant although not constructive proof. Since then\, estimating the s
tability constant and giving a constructive proof has been a challenge. \n
\nThis lecture will focus mostly on subcritical inequalities\, for which e
xplicit constants can be provided. The main tool is based on entropy metho
ds and nonlinear flows. Proving stability amounts to establish\, under som
e constraints\, a version of the entropy - entropy production inequality w
ith an improved constant. In simple cases\, for instance on the sphere\, r
ather explicit results have been obtained by the « carré du champ » met
hod introduced by D. Bakry and M. Emery. In the Euclidean space\, results
based on constructive regularity estimates for the solutions of the nonlin
ear flow and corresponding to a joint research project with Matteo Bonfort
e\, Bruno Nazaret\, and Nikita Simonov will be presented.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungang Li (Brown University)
DTSTART;VALUE=DATE-TIME:20200928T130000Z
DTEND;VALUE=DATE-TIME:20200928T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/17
DESCRIPTION:Title: Higher order Brezis-Nirenberg problems on hyperbolic spaces\
nby Jungang Li (Brown University) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nThe Brezis-Nirenberg problem cons
iders elliptic equations whose nonlinearity is associated with critical So
bolev exponents. In this talk we will discuss a recent progress on higher
order Brezis-Nirenberg problem on hyperbolic spaces. The existence of solu
tions relates closely to the study of higher order sharp Hardy-Sobolev-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 method to
establish the symmetry of positive solutions. This is a joint work with Gu
ozhen Lu and Qiaohua Yang.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Flynn (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20201102T150000Z
DTEND;VALUE=DATE-TIME:20201102T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/18
DESCRIPTION:Title: Sharp Caffarelli-Kohn-Nirenberg Inequalities for Grushin Vector
Fields and Iwasawa Groups.\nby Joshua Flynn (University of Connecticut
) as part of Geometric and functional inequalities and applications\n\n\nA
bstract\nSharp Caffarelli-Kohn-Nirenberg inequalities are established for
the Grushin vector fields and for Iwasawa groups (i.e.\, the boundary grou
p of a real rank one noncompact symmetric space). For all but one paramete
r case\, this is done by introducing a generalized Kelvin transform which
is shown to be an isometry of certain weighted Sobolev spaces. For the exc
eptional parameter case\, the best constant is found for the Grushin vecto
r fields by introducing Grushin cylindrical coordinates and studying the t
ransformed Euler-Lagrange equation.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristian Cazacu (University of Bucharest)
DTSTART;VALUE=DATE-TIME:20201012T130000Z
DTEND;VALUE=DATE-TIME:20201012T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/19
DESCRIPTION:Title: Optimal constants in Hardy and Hardy-Rellich type inequalities<
/a>\nby Cristian Cazacu (University of Bucharest) as part of Geometric and
functional inequalities and applications\n\n\nAbstract\nIn this talk we d
iscuss Hardy and Hardy-Rellich type inequalities\, so important in establi
shing useful properties for differential operators with singular potential
s and their PDEs. We recall some well-known and recent results and present
some new extensions. We analyze singular potentials with one or various s
ingularities. The tools of our proofs are mainly based on the method of su
persolutions\, proper transformations and spherical harmonics decompositio
n. We also focus on the best constants and the existence/nonexistence of m
inimizers in the energy space. This presentation is partially supported b
y CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiuyi Zhu (Louisiana State University)
DTSTART;VALUE=DATE-TIME:20201019T140000Z
DTEND;VALUE=DATE-TIME:20201019T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/20
DESCRIPTION:Title: The bounds of nodal sets of eigenfunctions\nby Jiuyi Zhu (Lo
uisiana State University) as part of Geometric and functional inequalities
and applications\n\n\nAbstract\nMotivated by Yau's conjecture\, the study
of the measure of nodal sets (Zero level sets) for eigenfunctions is inte
resting. We investigate the measure of nodal sets for Steklov\, Dirichlet
and Neumann eigenfunctions in the domain and on the boundary of the domai
n. For Dirichlet or Neumann eigenfunctions in the analytic domains\, we s
how some sharp upper bounds of nodal sets which touch the boundary. We w
ill also discuss some upper bounds of nodal sets for eigenfunctions of ge
neral eigenvalue problems. Furthermore\, some sharp doubling inequalities
and vanishing order are obtained. Part of the talk is based on joint wor
k with Fanghua Lin.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Almut Burchard (University of Toronto)
DTSTART;VALUE=DATE-TIME:20201214T140000Z
DTEND;VALUE=DATE-TIME:20201214T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/21
DESCRIPTION:Title: Rearrangement inequalities on spaces of bounded mean oscillation
\nby Almut Burchard (University of Toronto) as part of Geometric and f
unctional inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giovanna Citti (University of Bologna)
DTSTART;VALUE=DATE-TIME:20201207T140000Z
DTEND;VALUE=DATE-TIME:20201207T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/22
DESCRIPTION:Title: Degree preserving variational formulas for submanifolds\nby
Giovanna Citti (University of Bologna) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nI present a joint work with
M. Ritoré and G. Giovannardi related to an area functional for \nsubmanif
olds of fixed degree immersed in a graded manifold. The expression of this
area functional \nstrictly depends on the degree of the manifold\, so tha
t\, while computing the first variation\, \nwe need to keep fixed its degr
ee. We will show that there are isolated surfaces\, \nfor which this type
of degree preserving variations do not exist: they can be considered \nhi
gher dimensional extension of the subriemannian abnormal geodesics.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annalisa Baldi (University of Bologna)
DTSTART;VALUE=DATE-TIME:20201130T140000Z
DTEND;VALUE=DATE-TIME:20201130T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/23
DESCRIPTION:Title: Poincaré and Sobolev inequalities for differential forms in Euc
lidean spaces and Heisenberg groups\nby Annalisa Baldi (University of
Bologna) as part of Geometric and functional inequalities and applications
\n\n\nAbstract\nIn this talk I present some recent results obtained in col
laboration with B. Franchi and P. Pansu about Poincaré and Sobolev inequa
lities for differential forms in Heisenberg groups (some results are new a
lso for Euclidean spaces). For L^p\, p>1\, the estimates are consequence o
f singular integral estimates. In the limiting case L^1\, the singular i
ntegral estimates are replaced with inequalities which go back to Bourgain
-Brezis and Lanzani-Stein in Euclidean spaces\, and to Chanillo-Van Schaft
ingen and Baldi-Franchi-Pansu in Heisenberg groups. Also the case p=Q (Q i
s the homogeneous dimension of the Heisenberg group ) is considered.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Betsy Stovall (University of Wisconsin-Madison)
DTSTART;VALUE=DATE-TIME:20201214T150000Z
DTEND;VALUE=DATE-TIME:20201214T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/24
DESCRIPTION:Title: Fourier restriction to degenerate hypersurfaces\nby Betsy St
ovall (University of Wisconsin-Madison) as part of Geometric and functiona
l inequalities and applications\n\n\nAbstract\nIn this talk\, we will desc
ribe various open questions and recent progress on the Fourier restriction
problem associated to hypersurfaces with varying or vanishing curvature.\
n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew Gursky (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20210118T140000Z
DTEND;VALUE=DATE-TIME:20210118T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/25
DESCRIPTION:Title: Extremal Eigenvalues of the conformal laplacian\nby Matthew
Gursky (University of Notre Dame) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nI will report on joint work with
Samuel Perez-Ayala in which we consider the problem of extremizing eigenva
lues of the conformal laplacian in a fixed conformal class. This generali
zes the problem of extremizing the eigenvalues of the laplacian on a compa
ct surface. I will explain the connection of this problem to the existenc
e of harmonic maps\, and to nodal solutions of the Yamabe problem (first n
oticed by Ammann-Humbert).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhen-Qing Chen (University of Washington)
DTSTART;VALUE=DATE-TIME:20210125T150000Z
DTEND;VALUE=DATE-TIME:20210125T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/26
DESCRIPTION:Title: Stability of Elliptic Harnack Inequality\nby Zhen-Qing Chen
(University of Washington) as part of Geometric and functional inequalitie
s and applications\n\n\nAbstract\nHarnack inequality\, if it holds\, is a
useful tool in analysis and probability theory.\nIn this talk\, I will dis
cuss scale invariant elliptic Harnack inequality for general diffusions\,
or equivalently\, for general differential operators on metric measure spa
ces\, and show that it is stable under form-comparable perturbations for s
trongly local Dirichlet forms on complete locally compact separable metri
c spaces that satisfy metric doubling property. \nBased on Joint work with
Martin Barlow and Mathav Murugan.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gilles Carron (University of Nantes)
DTSTART;VALUE=DATE-TIME:20210111T140000Z
DTEND;VALUE=DATE-TIME:20210111T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/27
DESCRIPTION:Title: Euclidean heat kernel rigidity\nby Gilles Carron (University
of Nantes) as part of Geometric and functional inequalities and applicati
ons\n\n\nAbstract\nThis is joint work with David Tewodrose (Bruxelles). I
will explain that a metric measure space with Euclidean heat kernel are E
uclidean. An almost rigidity result comes then for free\, and this can be
used to give another proof of Colding's almost rigidity for complete mani
fold with non negative Ricci curvature and almost Euclidean growth.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yehuda Pinchover (Technion -Israel Institute of Technology)
DTSTART;VALUE=DATE-TIME:20210215T140000Z
DTEND;VALUE=DATE-TIME:20210215T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/28
DESCRIPTION:Title: On families of optimal Hardy-weights for linear second-order ell
iptic operators.\nby Yehuda Pinchover (Technion -Israel Institute of T
echnology) as part of Geometric and functional inequalities and applicatio
ns\n\n\nAbstract\nWe construct families of optimal Hardy-weights for a sub
critical linear second-order elliptic operator using a one-dimensional red
uction. More precisely\, we first characterize all optimal Hardy-weights
with respect to one-dimensional subcritical Sturm-Liouville operators on $
(a\,b)$\, $\\infty \\leq a < b \\leq \\infty$\, and then apply this re
sult to obtain families of optimal Hardy inequalities for general linear s
econd-order elliptic operators in higher dimensions. This is a joint work
with Idan Versano.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ling Xiao (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20201109T140000Z
DTEND;VALUE=DATE-TIME:20201109T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/29
DESCRIPTION:Title: Entire spacelike constant $\\sigma_{n-1}$ curvature in Minkowski
space\nby Ling Xiao (University of Connecticut) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe prove that\
, in the Minkowski space\, if a spacelike\, (n − 1)-convex hypersurface
M with constant $\\sigma_{n−1}$ curvature has bounded principal curvatur
es\, then M is convex. Moreover\, if M is not strictly convex\, after 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 hypersurface
M with constant $\\sigma_{n−1}$ curvature and bounded principal curvatu
res. This is a joint work with Changyu Ren and Zhizhang Wang.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martin Dindos (The University of Edinburgh)
DTSTART;VALUE=DATE-TIME:20210208T140000Z
DTEND;VALUE=DATE-TIME:20210208T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/30
DESCRIPTION:Title: On p-ellipticity and connections to solvability of elliptic comp
lex valued PDEs\nby Martin Dindos (The University of Edinburgh) as par
t of Geometric and functional inequalities and applications\n\n\nAbstract\
nThe notion of an elliptic partial differential equation (PDE)\ngoes back
at least to 1908\, when it appeared in a paper J. Hadamard. In\nthis talk
we present a recently discovered structural condition\, called\n$p$-ellip
ticity\, which generalizes classical ellipticity. It was\nco-discovered i
ndependently by Carbonaro and Dragicevic on one hand\, and\nPipher and mys
elf on the other\, and plays a fundamental role in many\nseemingly mutuall
y 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) c
onvexity of power functions (Bellman functions)\n(ii) dimension-free bilin
ear embeddings\,\n(iii) $L^p$-contractivity and boundedness of semigroups
$(P_t^A)_{t>0}$\nassociated with elliptic operators\,\n(iv) holomorphic fu
nctional calculus\,\n(v) multilinear analysis\,\n(vi) regularity theory of
elliptic PDE with complex coefficients.\n\nDuring the talk\, I will descr
ibe my contribution to this development\, in\nparticular to (vi).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Saikat Mazumdar (Indian Institute of Technology Bombay)
DTSTART;VALUE=DATE-TIME:20210222T140000Z
DTEND;VALUE=DATE-TIME:20210222T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/31
DESCRIPTION:Title: EXISTENCE RESULTS FOR THE HIGHER-ORDER $Q$-CURVATURE EQUATION\nby Saikat Mazumdar (Indian Institute of Technology Bombay) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nIn t
his talk\, we will obtain some existence results for the $Q$-curvature equ
ation\nof arbitrary $2k$-th order\, where $k \\geq 1$ is an integer\, on a
compact Riemannian\nmanifold of dimension $n \\geq 2k + 1$. This amounts
to solving a nonlinear elliptic\nPDE involving the powers of Laplacian cal
led the GJMS operator. The difficulty\nin determining the explicit form of
this GJMS operator together with a lack of\nmaximum principle complicates
the issues of existence.\nThis is a joint work with Jérôme Vétois (McG
ill University).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Del Pino (University of Bath)
DTSTART;VALUE=DATE-TIME:20210301T140000Z
DTEND;VALUE=DATE-TIME:20210301T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/33
DESCRIPTION:Title: Dynamics of concentrated vorticities in 2d and 3d Euler flows\nby Manuel Del Pino (University of Bath) as part of Geometric and functi
onal inequalities and applications\n\n\nAbstract\nA classical problem that
traces back to Helmholtz and Kirchoff is the understanding \nof the dynam
ics of solutions to the 2d and 3d Euler equations of an inviscid incompres
sible \nfluid\, when the vorticity of the solution is initially concentrat
ed near isolated points in 2d or \nvortex lines in 3d. We discuss some rec
ent result on existence and asymptotic behaviour of \nthese solutions. We
describe\, with precise asymptotics\, interacting vortices and travelling
helices. We rigorously establish the law of of motion of of "leapfroggin
g vortex rings"\, originally conjectured by Helmholtz in 1858. This is jo
int work with Juan Davila\, Monica Musso and Juncheng Wei.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enrique Zuazua (Friedrich-Alexander-Universität)
DTSTART;VALUE=DATE-TIME:20210308T140000Z
DTEND;VALUE=DATE-TIME:20210308T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/34
DESCRIPTION:Title: UNILATERAL BOUNDS FOR NONLINEAR SEMIGROUPS AND TIME-INVERSION\nby Enrique Zuazua (Friedrich-Alexander-Universität) as part of Geometr
ic and functional inequalities and applications\n\n\nAbstract\nSome classi
cal nonlinear semigroups arising in mechanics induce unilateral bounds on
solutions. \nHamilton--Jacobi equations and 1-d scalar conservation laws
are classical examples of such nonlinear effects: solutions spontaneously
develop one-sided Lipschitz or semi-concavity conditions.\n\nWhen this occ
urs the range of the semigroup is unilaterally bounded by a threshold.\n\n
On the other hand\, in practical applications\, one is led to consider the
problem of time-inversion\, so to identify the initial sources that have
led to the observed dynamics at the final time.\n\nIn this lecture we shal
l discuss this problem answering to the following two questions: On one ha
nd\, to identify the range of the semigroup and\, given a target\, to char
acterize and reconstruct the ensemble of initial data leading to it.\n\nIl
lustrative numerical simulations will be presented\, and a complete geome
tric interpretation will also be provided.\n\nWe shall also present a numb
er of open problems arising in this area and the possible link with reinfo
rcement learning.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brian Street (University of Wisconsin-Madison)
DTSTART;VALUE=DATE-TIME:20210329T130000Z
DTEND;VALUE=DATE-TIME:20210329T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/35
DESCRIPTION:Title: Maximal Hypoellipticity\nby Brian Street (University of Wisc
onsin-Madison) as part of Geometric and functional inequalities and applic
ations\n\n\nAbstract\nIn 1974\, Folland and Stein introduced a generalizat
ion of ellipticity known as maximal hypoellipticity. This talk will be an
introduction to this concept and some of the ways it generalizes elliptic
ity.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Kenig (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210426T140000Z
DTEND;VALUE=DATE-TIME:20210426T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/36
DESCRIPTION:Title: Wave maps into the sphere\nby Carlos Kenig (University of Ch
icago) as part of Geometric and functional inequalities and applications\n
\n\nAbstract\nWe will introduce wave maps\, an important geometric flow\,
and\ndiscuss\, for the case when the target is the sphere\, the asymptotic
\nbehavior near the ground state (without symmetry) and recent results in\
nthe general case (under co-rotational symmetry) in joint work with\nDuyc
kaerts\, Martel and Merle.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Man Wah Wong (York University)
DTSTART;VALUE=DATE-TIME:20210315T140000Z
DTEND;VALUE=DATE-TIME:20210315T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/37
DESCRIPTION:Title: Spectral Theory and Number Theory of the Twisted Bi-Laplacian\nby Man Wah Wong (York University) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nWe begin with the sub-Laplacia
n on the Heisenberg group and then the twisted Laplacian by taking its inv
erse Fourier transform with respect to the center of the group. The eigenv
alues and the eigenfunctions of the twisted Laplacian are computed explici
tly. Then we turn our attention to the product of the twisted Laplacian an
d its transpose\, thus obtaining a fourth order partial differential opera
tor dubbed the twisted bi-Laplacian. The connections between the spectral
analysis of the twisted bi-Laplacian and Dirichlet divisors\, the Riemann
zeta function and the Dixmier trace are explained.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yanyan Li (Rutgers University)
DTSTART;VALUE=DATE-TIME:20210419T130000Z
DTEND;VALUE=DATE-TIME:20210419T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/38
DESCRIPTION:Title: Regular solutions of the stationary Navier-Stokes equations on h
igh dimensional Euclidean space\nby Yanyan Li (Rutgers University) as
part of Geometric and functional inequalities and applications\n\n\nAbstra
ct\nWe study the existence of regular solutions of the incompressible stat
ionary Navier-Stokes equations in n-dimensional Euclidean space with a giv
en bounded external force of compact support. In dimensions $n\\le 5$\, th
e existence of such solutions was known. In this paper\, we extend it to d
imensions $n\\le 15$. This is a joint work with Zhuolun Yang.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spring Recess (No Talk)
DTSTART;VALUE=DATE-TIME:20210412T130000Z
DTEND;VALUE=DATE-TIME:20210412T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/39
DESCRIPTION:by Spring Recess (No Talk) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenxiong Chen (Yeshiva University)
DTSTART;VALUE=DATE-TIME:20210322T140000Z
DTEND;VALUE=DATE-TIME:20210322T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/40
DESCRIPTION:Title: Asymptotic radial symmetry\, monotonicity\, non-existence for so
lutions to fractional parabolic equations\nby Wenxiong Chen (Yeshiva U
niversity) as part of Geometric and functional inequalities and applicatio
ns\n\n\nAbstract\nIn this talk\, we will consider nonlinear parabolic frac
tional equations\n\nWe develop a systematical approach in applying an asym
ptotic method\nof moving planes to investigate qualitative properties of p
ositive solutions for\nfractional parabolic equations. To this end\, we de
rive a series of needed key\ningredients such as narrow region principles\
, and various asymptotic maximum and strong maximum principles for antisym
metric functions in both bounded and unbounded domains. Then we illustrate
how these new methods can be employed to obtain asymptotic radial symmetr
y 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 so
lutions will eventually approach to radially symmetric functions. We will
also consider the entire positive solutions on a half space\, in\nthe whol
e space\, and with indefinite nonlinearity. Monotonicity and nonexistence
of solutions are obtained. This is joint work with P. Wang\, Y. Niu\, Y. H
u and L. Wu.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yunfeng Zhang (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20210201T140000Z
DTEND;VALUE=DATE-TIME:20210201T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/41
DESCRIPTION:Title: Schr\\"odinger equations on compact globally symmetric spaces\nby Yunfeng Zhang (University of Connecticut) as part of Geometric and f
unctional inequalities and applications\n\n\nAbstract\nLet $M$ be a compac
t manifold of dimension $d$. Scale-invariant Strichartz estimates of the f
orm\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 repo
rt that the above estimate holds true on an arbitrary compact globally sym
metric space $M$ equipped with the canonical Killing metric\, for all $p\\
geq 2+8/r$\, where $r$ denotes the rank of $M$. As an immediate applicatio
n\, we provide local well-posedness results for nonlinear Schr\\"odinger e
quations of polynomial nonlinearities of degree $\\beta\\geq 4$ on any com
pact globally symmetric space of large enough rank\, in all subcritical sp
aces.\n\nWe also discuss bilinear Strichartz estimates on compact globally
symmetric spaces\, and critical and subcritical local well-posedness resu
lts for the cubic nonlinearity.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungang Li (Brown University)
DTSTART;VALUE=DATE-TIME:20210503T140000Z
DTEND;VALUE=DATE-TIME:20210503T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/42
DESCRIPTION:Title: Sharp critical and subcritical Moser-Trudinger inequalities on c
omplete and noncompact Riemannian manifolds\nby Jungang Li (Brown Univ
ersity) as part of Geometric and functional inequalities and applications\
n\n\nAbstract\nMoser-Trudinger inequality is the borderline case of the So
bolev inequality and has many applications in differential geometry. In th
is talk\, I will report a recent progress of critical and subcritical Mose
r-Trudinger inequalities on complete noncompact Riemannian manifolds. Clas
sical results depend heavily on the validity of some rearrangement inequal
ities\, which are unavailable on general manifolds. To overcome this diffi
culty\, we applied a rearrangement-free approach to obtain those inequalit
ies on manifolds\, together with their sharp constants. This is a joint wo
rk with Guozhen Lu.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Struwe (ETH Zürich)
DTSTART;VALUE=DATE-TIME:20210510T140000Z
DTEND;VALUE=DATE-TIME:20210510T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
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\nA
bstract\nFinding non-constant harmonic 3-spheres for a closed target manif
old N\nis a prototype of a super-critical variational problem. In fact\, t
he\ndirect method fails\, as the infimum of Dirichlet energy in any homoto
py\nclass of maps from the 3-sphere to any closed N is zero\; moreover\, t
he\nharmonic map heat flow may blow up in finite time\, and even the ident
ity\nmap from the 3-sphere to itself is not stable under this flow.\n\nTo
overcome these difficulties\, we propose the normalized harmonic map\nheat
flow as a new tool\, and we show that for this flow the identity map\nfro
m the 3-sphere to itself now\, indeed\, is stable\; moreover\, the flow\nc
onverges to a harmonic 3-sphere also when we perturb the target\ngeometry.
While our results are strongest in the perturbative setting\,\nwe also ou
tline a possible global theory.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No talk
DTSTART;VALUE=DATE-TIME:20210517T140000Z
DTEND;VALUE=DATE-TIME:20210517T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/44
DESCRIPTION:by No talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk (Memorial Day)
DTSTART;VALUE=DATE-TIME:20210531T140000Z
DTEND;VALUE=DATE-TIME:20210531T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/45
DESCRIPTION:by No Talk (Memorial Day) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaojun Huang (Rutgers University)
DTSTART;VALUE=DATE-TIME:20210524T130000Z
DTEND;VALUE=DATE-TIME:20210524T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/46
DESCRIPTION:Title: Revisit to a non-degeneracy property for extremal mappings\n
by Xiaojun Huang (Rutgers University) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nI will discuss a generalizati
on of my previous result on the localization of extremal maps near a stron
gly pseudo-convex point.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sun-Yung Alice Chang (Princeton University)
DTSTART;VALUE=DATE-TIME:20210607T140000Z
DTEND;VALUE=DATE-TIME:20210607T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/47
DESCRIPTION:Title: On bi-Lipschitz equivalence of a class of non-conformally flat s
pheres\nby Sun-Yung Alice Chang (Princeton University) as part of Geom
etric and functional inequalities and applications\n\n\nAbstract\nThis is
a report of some recent joint work with Eden Prywes and Paul Yang. The mai
n\nresult is a bi-Lipschitz equivalence of a class of metrics on 4-shpere
under curvature constraints. The proof involves two steps: first a constru
ction of quasiconformal maps between\ntwo conformally related metrics in a
positive Yamabe class\, followed by the step of applying\nthe Ricci flow
to establish the bi-Lipschitz equivalence from such a conformal class to t
he\nstandard conformal class on 4-sphere.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svitlana Mayboroda (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20210614T140000Z
DTEND;VALUE=DATE-TIME:20210614T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/48
DESCRIPTION:Title: Green Function vs. Geometry\nby Svitlana Mayboroda (Universi
ty of Minnesota) as part of Geometric and functional inequalities and appl
ications\n\n\nAbstract\nIn this talk we will discuss connections between t
he geometric and PDE properties of sets. The emphasis is on quantifiable\,
global results which yield true equivalence between the geometric and PDE
notions in very rough scenarios\, including domains and equations with si
ngularities and structural complexity. The main result establishes that in
all dimensions $d < n$\, a $d$-dimensional set in $\\mathbb{R}^n$ is regu
lar (rectifiable) if and only if the Green function for elliptic operators
is well approximated by affine functions (distance to the hyperplanes). T
o the best of our knowledge\, this is the first free boundary result of th
is type for lower dimensional sets and the first free boundary result in t
he classical case $d=n-1$ without restrictions on the coefficients of the
equation.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susanna Terracini (Universitá di Torino)
DTSTART;VALUE=DATE-TIME:20210628T130000Z
DTEND;VALUE=DATE-TIME:20210628T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/49
DESCRIPTION:Title: Free boundaries in segregation problems\nby Susanna Terracin
i (Universitá di Torino) as part of Geometric and functional inequalities
and applications\n\n\nAbstract\nWe first consider classes of variational
problems for densities that repel each other at distance. Examples are giv
en by the minimizers of Dirichlet functional or the Rayleigh quotient\n\\[
\n D({\\bf u}) = \\sum_{i=1}^k \\int_{\\Omega} |\\nabla u_i|^2 \\quad \\t
ext{or} \\quad R({\\bf u}) = \\sum_{i=1}^k \\frac{\\int_{\\Omega} |\\nabl
a u_i|^2}{\\int_{\\Omega} u_i^2}\n\\]\nover the class of $H^1(\\Omega\,\\R
^k)$ functions attaining some boundary conditions on $\\partial \\Omega$\,
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\nAs seco
nd class of problems\, we consider energy minimizers of Dirichlet energies
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 constraint\n\\[
\n u_i(x)\\cdot u_j(x)=0\, \\qquad \\forall x\\in \\Omega\\\;\, \\forall i
\\neq j.\n\\]\n\nFor these problems\, we investigate the optimal regulari
ty of the solutions\, prove a free-boundary extremality condition\, and de
rive some preliminary results characterising the emerging free boundary.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roger Moser (University of Bath)
DTSTART;VALUE=DATE-TIME:20210621T130000Z
DTEND;VALUE=DATE-TIME:20210621T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/50
DESCRIPTION:Title: The infinity-elastica problem\nby Roger Moser (University of
Bath) as part of Geometric and functional inequalities and applications\n
\n\nAbstract\nThe Euler elastica problem seeks to minimise the $L^2$-norm
of\nthe curvature of curves under certain boundary conditions. If we\nrepl
ace the $L^2$-norm with the $L^\\infty$-norm\, then we obtain a\nvariation
al problem with quite different properties. Nevertheless\, even\nthough th
e underlying functional is not differentiable\, it turns out\nthat the sol
utions of the problem can still be described by\ndifferential equations. A
n analysis of these equations then gives a\nclassification of the solution
s.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jingzhi Tie (University of Georgia)
DTSTART;VALUE=DATE-TIME:20210405T140000Z
DTEND;VALUE=DATE-TIME:20210405T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/51
DESCRIPTION:Title: CR analogue of Yau’s Conjecture on pseudo harmonic functions o
f polynomial growth.\nby Jingzhi Tie (University of Georgia) as part o
f Geometric and functional inequalities and applications\n\n\nAbstract\nCh
eng and Yau derived the well-known gradient estimate for positive harmonic
functions and obtained the classical Liouville theorem\, which states tha
t any bounded harmonic function is constant in complete noncompact Riemann
ian manifolds with nonnegative Ricci curvature. I will talk about the CR a
nalogue of Yau’s conjecture. We need to derive the CR volume doubling pr
operty\, CR\\ Sobolev inequality\, and mean value inequality. Then we can
apply them to prove the CR analogue of Yau's conjecture on the space consi
sting of all pseudoharmonic functions of polynomial growth of degree at mo
st $d$ 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://researchseminars.org/talk/GeomInequAndPDEs/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lorenzo D'Ambrosio (Universita di Bari)
DTSTART;VALUE=DATE-TIME:20210705T130000Z
DTEND;VALUE=DATE-TIME:20210705T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/52
DESCRIPTION:Title: Liouville theorems for semilinear biharmonic equations and inequ
alities\nby Lorenzo D'Ambrosio (Universita di Bari) as part of Geometr
ic and functional inequalities and applications\n\n\nAbstract\nWe study no
nexistence results for a coercive semilinear biharmonic equation on the wh
ole $R^N$. The analysis is made for general solutions without any assumpti
on on their sign nor on their behaviour at infinity. A relevant role is pl
ayed by some extensions of the Hardy-Rellich inequalities for general func
tions (not necessarily compactly supported).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Silvestre (University of Chicago)
DTSTART;VALUE=DATE-TIME:20210913T140000Z
DTEND;VALUE=DATE-TIME:20210913T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/53
DESCRIPTION:Title: Regularity estimates for the Boltzmann equation without cutoff\nby Luis Silvestre (University of Chicago) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nWe study the regulari
zation effect of the inhomogeneous Boltzmann equation without cutoff. We o
btain a priori estimates for all derivatives of the solution depending onl
y on bounds of its hydrodynamic quantities: mass density\, energy density
and entropy density. As a consequence\, a classical solution to the equati
on 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 mode
rately soft and hard potentials. We use methods that originated in the stu
dy of nonlocal elliptic and parabolic equations: a weak Harnack inequality
in the style of De Giorgi\, and a Schauder-type estimate.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yoshikazu Giga (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20210927T130000Z
DTEND;VALUE=DATE-TIME:20210927T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/54
DESCRIPTION:Title: On the Helmholtz decomposition of BMO spaces of vector fields\nby Yoshikazu Giga (University of Tokyo) as part of Geometric and functi
onal inequalities and applications\n\n\nAbstract\nThe Helmholtz decomposit
ion of vector fields is a fundamental tool for analysis of vector fields e
specially to analyze the Navier-Stokes equations in a domain. It gives a u
nique decomposition of a (tangential) vector field defined in a domain of
an Euclidean space (or a riemannian maniford) into a sum of a gradient fie
ld and a solenoidal field with supplemental condition like a boundary cond
ition.It is well-known that such decomposition gives an orthogonal decomp
osition of the space of L^2 vector fields in an arbitrary domain and known
as the Weyl decomposition. It is also well-studied that in various domain
s including the half space\, smooth bounded and exterior domain\, it 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 impossible
because otherwise it would imply the boundedness of the Riesz type operato
r in L^∞ (or L^1) which is absurd.\n In this talk\, we extend the Hemlh
oltz decomposition in a space of vector fields with bounded mean oscillati
ons (BMO) when the domain of vector field is a smooth bounded domain in an
Euclidean space. There are several possible definitions of\na BMO space o
f vector fields. However\, to have a topological direct sum decomposition\
, it turns out that components of normal and tangential to the boundary s
hould be handled separately.\n This decomposition problem is equivalent t
o solve the Poisson equation with the divergence of the original vector fi
eld v as a data with the Neumann data with the normal trace of v. The desi
red gradient field is the gradient of the solution of this Poisson equatio
n. To solve this problem we construct a kind of volume potential so that t
he problem is reduced to the Neumann problem for the Laplace equation. Unf
ortunately\, taking usual Newton potential causes a problem to estimate ne
cessary norm so we construct another volume potential based on normal coor
dinate.We need a trace theorem to control L^∞ norm of the normal trace.
This is of independent interest. Finally\,we solve the Neumann problem wit
h L^∞ data in a necessary space. The Helmholtz decomposition for BMO v
ector 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://researchseminars.org/talk/GeomInequAndPDEs/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jill Pipher (Brown University)
DTSTART;VALUE=DATE-TIME:20210920T140000Z
DTEND;VALUE=DATE-TIME:20210920T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/55
DESCRIPTION:Title: Boundary value problems for p-elliptic operators\nby Jill Pi
pher (Brown University) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nWe give some background about the regularit
y of solutions to real and complex elliptic operators\, motivating a new a
lgebraic condition (p-ellipticity). We introduce this condition in order t
o solve new boundary value problems for operators with complex coefficient
s. Results with M. Dindos\, and with M. Dindos and J. Li\, are discussed.\
n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sundaram Thangavelu (INDIAN INSTITUTE OF SCIENCE)
DTSTART;VALUE=DATE-TIME:20211004T130000Z
DTEND;VALUE=DATE-TIME:20211004T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/56
DESCRIPTION:Title: On the extension problem for the sublaplacian on the Heisenberg
group\nby Sundaram Thangavelu (INDIAN INSTITUTE OF SCIENCE) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nIn
this talk we plan to describe some results on the extension problem associ
ated to the sublaplacian $ \\mathcal{L} $ on the Heisenberg group $ \\H^n
.$ The Dirichlet to Neumann map induced by this problem leads to conformal
ly invariant fractional powers of $ \\mathcal{L}.$ We use the results to p
rove a version of Hardy's inequality for such fractional powers. These res
ults are based on my joint work with Luz Roncal.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gabriele Grillo (Politecnico di Milano)
DTSTART;VALUE=DATE-TIME:20211011T130000Z
DTEND;VALUE=DATE-TIME:20211011T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/57
DESCRIPTION:Title: Nonlinear characterizations of stochastic completeness\nby G
abriele Grillo (Politecnico di Milano) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nA manifold is said to be sto
chastically complete if the free heat semigroup preserves probability. It
is well-known that this property is equivalent to nonexistence of nonnegat
ive\, bounded solutions to certain (linear) elliptic problems\, and to uni
queness 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 fa
ct\, and the known analytic-geometric characterizations of stochastic comp
leteness\, allow to give new explicit criteria for existence/nonexistence
of solutions to certain nonlinear elliptic equations on manifolds\, and fo
r uniqueness/nonuniqueness of solutions to certain nonlinear diffusions on
manifolds.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (University of Washington)
DTSTART;VALUE=DATE-TIME:20211025T140000Z
DTEND;VALUE=DATE-TIME:20211025T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/58
DESCRIPTION:Title: Mean-Value Inequalities for Convex Domains\nby Stefan Steine
rberger (University of Washington) as part of Geometric and functional ine
qualities and applications\n\n\nAbstract\nThe Mean Value Theorem implies t
hat the average value of a subharmonic\nfunction in a disk can be bounded
from above by the average value on the boundary. \nWhat happens if we rep
lace the disk by another domain? Maybe surprisingly\, the problem \nhas a
relatively clean answer -- we discuss a whole range of mean value inequal
ities for \nconvex domains in IR^n. The extremal domain remains a mystery
for most of them. \nThe techniques are an amusing mixture of classical po
tential theory\, complex analysis\,\na little bit of elliptic PDEs and\, s
urprisingly\, the theory of solids from the 1850s.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juan Manfredi (University of Pittsburgh)
DTSTART;VALUE=DATE-TIME:20211129T140000Z
DTEND;VALUE=DATE-TIME:20211129T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/62
DESCRIPTION:Title: NATURAL $p$-MEANS FOR THE $p$-LAPLACIAN IN EUCLIDEAN SPACE AND
THE HEISENBERG GROUP\nby Juan Manfredi (University of Pittsburgh) as p
art of Geometric and functional inequalities and applications\n\n\nAbstrac
t\nWe consider semi-discrete approximations to $p$-harmonic functions base
d 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 consider
adapted semi-discrete approximations in the Heisenberg group $\\mathbb{H}$
and prove uniform convergence in bounded $C^{1\,1}$-domains.\n\nThis talk
is based in joint work with András Domokos and Diego Ricciotti (Sacramen
to)\nand Bianca Stroffolini (Naples)\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guofang Wang (University of Freiburg)
DTSTART;VALUE=DATE-TIME:20211101T140000Z
DTEND;VALUE=DATE-TIME:20211101T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/63
DESCRIPTION:Title: Geometric inequalities in the hyperbolic space and their applica
tions.\nby Guofang Wang (University of Freiburg) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe will talk a
bout Alexandrov-Fenchel type inequalities in the hyperbolic space and thei
r applications in a higher order mass of asymptotically hyperbolic manifol
ds. The talk is based on a series of work joint with Yuxin Ge\, Jie Wu and
Chao Xia\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Yang (Princeton University)
DTSTART;VALUE=DATE-TIME:20220207T150000Z
DTEND;VALUE=DATE-TIME:20220207T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/64
DESCRIPTION:Title: Sturm comparison for Jacobi vector fields and applications\n
by Paul Yang (Princeton University) as part of Geometric and functional in
equalities and applications\n\n\nAbstract\nFor CR manifolds of real dimens
ion three\, we study the Jacobi field equation. Under the condition that t
he torsion be parallel\, we obtain comparison results against a family of
homogeneous CR structures. As application\, we describe the singularities
of contact forms on the the homogeneous structures with finite total Q-pri
me curvature. This is ongoing joint work with Sagun Chanillo.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Chen (University of California at Berkeley)
DTSTART;VALUE=DATE-TIME:20211108T150000Z
DTEND;VALUE=DATE-TIME:20211108T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/65
DESCRIPTION:Title: Integral curvature pinching and sphere theorems via the Ricci fl
ow\nby Eric Chen (University of California at Berkeley) as part of Geo
metric and functional inequalities and applications\n\n\nAbstract\nI will
discuss how uniform Sobolev inequalities obtained from the monotonicity of
Perelman's W-functional can be used to prove curvature pinching theorems
on Riemannian manifolds. These are based on scale-invariant integral norm
s and generalize some earlier pointwise and supercritical integral pinchin
g statements. This is joint work with Guofang Wei and Rugang Ye.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Qing Han (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20211206T150000Z
DTEND;VALUE=DATE-TIME:20211206T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/67
DESCRIPTION:Title: A Concise Boundary Regularity for the Loewner-Nirenberg Problem<
/a>\nby Qing Han (University of Notre Dame) as part of Geometric and funct
ional inequalities and applications\n\n\nAbstract\nLoewner and Nirenberg d
iscussed complete metrics conformal to the Euclidean metric and with a con
stant scalar curvature in bounded domains in the Euclidean space. The conf
ormal factors blow up on boundary. The asymptotic behaviors of the conform
al factors near boundary are known in smooth and sufficiently smooth domai
ns. In this talk\, we introduce the logarithm of the distance to boundary
as an additional independent self-variable and establish a concise boundar
y regularity.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jyotshana Prajapat (University of Mumbai)
DTSTART;VALUE=DATE-TIME:20211122T140000Z
DTEND;VALUE=DATE-TIME:20211122T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/68
DESCRIPTION:Title: Geodetically convex sets in Heisenberg group $H^n$\nby Jyots
hana Prajapat (University of Mumbai) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nA classification of geodetical
ly convex subsets of Heisenberg group of homogeneous dimension 4 was prov
ed by Monti-Rickly. We extend their result to a higher dimension Heisenb
erg group. This is ongoing work with my PhD student Anoop Varghese.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McCann (University of Toronto)
DTSTART;VALUE=DATE-TIME:20211220T150000Z
DTEND;VALUE=DATE-TIME:20211220T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/69
DESCRIPTION:Title: Inscribed radius bounds for lower Ricci bounded metric measure s
paces with mean convex boundary\nby Robert McCann (University of Toron
to) as part of Geometric and functional inequalities and applications\n\n\
nAbstract\nConsider an essentially nonbranching metric measure space with
the measure contraction property of Ohta and Sturm. We prove a sharp upper
bound on the inscribed radius of any subset whose boundary has a suitably
signed lower bound on its generalized mean curvature. This provides a non
smooth analog of results dating back to Kasue (1983) and subsequent author
s. We prove a stability statement concerning such bounds and --- in the Ri
emannian curvature-dimension (RCD) setting --- characterize the cases of e
quality. This represents joint work with Annegret Burtscher\, Christian Ke
tterer and Eric Woolgar.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitali Kapovitch (University of Toronto)
DTSTART;VALUE=DATE-TIME:20220131T140000Z
DTEND;VALUE=DATE-TIME:20220131T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/70
DESCRIPTION:Title: Mixed curvature almost flat manifolds\nby Vitali Kapovitch (
University of Toronto) as part of Geometric and functional inequalities an
d applications\n\n\nAbstract\nA celebrated theorem of Gromov says that giv
en $n>1$ there is an $\\epsilon(n)>0$ such that if a closed Riemannian man
ifold $M^n$ satisfies $-\\epsilon < sec_M < \\epsilon\, diam(M) < 1$ then
$M$ is diffeomorphic to an infranilmanifold. I will show that the lower se
ctional curvature bound in Gromov’s theorem can be weakened to the lower
Bakry-Emery Ricci curvature bound. I will also discuss the relation of th
is result to the study of manifolds with Ricci curvature bounded below.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Vega (Basque Center for Applied Mathematics)
DTSTART;VALUE=DATE-TIME:20220124T150000Z
DTEND;VALUE=DATE-TIME:20220124T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/71
DESCRIPTION:Title: New Conservation Laws and Energy Cascade for 1d Cubic NLS\nb
y Luis Vega (Basque Center for Applied Mathematics) as part of Geometric a
nd functional inequalities and applications\n\n\nAbstract\nI’ll present
some recent results concerning the IVP of 1d cubic NLS at the critical le
vel of regularity. I’ll also exhibit a cascade of energy for the 1D Schr
ödinger map which is related to NLS through the so called Hasimoto transf
ormation. For higher regularity these two equations are completely integra
ble systems and therefore no cascade of energy is possible.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Mondino (University of Oxford)
DTSTART;VALUE=DATE-TIME:20220509T130000Z
DTEND;VALUE=DATE-TIME:20220509T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/72
DESCRIPTION:Title: Optimal transport and quantitative geometric inequalities\nb
y Andrea Mondino (University of Oxford) as part of Geometric and functiona
l inequalities and applications\n\n\nAbstract\nThe goal of the talk is to
discuss a quantitative version of the Levy-Gromov isoperimetric inequality
(joint with Cavalletti and Maggi) as well as a quantitative form of Obata
's rigidity theorem (joint with Cavalletti and Semola). Given a closed Rie
mannian manifold with strictly positive Ricci tensor\, one estimates the m
easure of the symmetric difference of a set with a metric ball with the de
ficit in the Levy-Gromov inequality. The results are obtained via a quanti
tative analysis based on the localisation method via L1-optimal transport.
For simplicity of presentation\, the talk will present the results in cas
e of smooth Riemannian manifolds with Ricci Curvature bounded below\; more
over it will not require previous knowledge of optimal transport theory.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linhan Li (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20211213T150000Z
DTEND;VALUE=DATE-TIME:20211213T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/73
DESCRIPTION:Title: Comparison between the Green function and smooth distances\n
by Linhan Li (University of Minnesota) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nIn the upper half-space\, th
e distance function to the boundary is a positive solution to Laplace's eq
uation that vanishes on the boundary\, which can be interpreted as the Gre
en function with pole at infinity for the Laplacian. We are interested in
understanding the exact relations between the behavior of the Green functi
on\, the structure of the underlying operator\, and the geometry of the do
main. In joint work with G. David and S. Mayboroda\, we obtain a precise a
nd quantitative control of the proximity of the Green function and the dis
tance function on the upper half-space by the oscillation of the coefficie
nts 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 dista
nce function. More recently\, together with J. Feneuil and S. Mayboroda\,
we obtain analogous results for domains with uniformly rectifiable boundar
ies.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaodan Zhou (Okinawa Institute of Science and Technology)
DTSTART;VALUE=DATE-TIME:20220221T140000Z
DTEND;VALUE=DATE-TIME:20220221T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/74
DESCRIPTION:Title: Quasiconvex envelope in the Heisenberg group\nby Xiaodan Zho
u (Okinawa Institute of Science and Technology) as part of Geometric and f
unctional inequalities and applications\n\n\nAbstract\nVarious notions of
convexity of sets and functions in the Heisenberg group have been studied
in the past two decades. In this talk\, we focus on the horizontally quasi
convex ($h$-quasiconvex) functions in the Heisenberg group. Inspired by th
e first-order characterization and construction of quasiconvex envelope by
Barron\, Goebel and Jensen in the Euclidean space\, we obtain a PDE appro
ach to construct the $h$-quasiconvex envelope for a given function $f$ in
the Heisenberg group. In particular\, we show the uniqueness and existence
of viscosity solutions to a non-local Hamilton-Jacobi equation and iterat
e the equation to obtain the $h$-quasiconvex envelope. Relations between $
h$-convex hull of a set and the $h$-quasiconvex envelopes are also investi
gated. This is joint work with Antoni Kijowski (OIST) and Qing Liu (Fukuok
a University/OIST).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jian Song (Rutgers University)
DTSTART;VALUE=DATE-TIME:20220214T150000Z
DTEND;VALUE=DATE-TIME:20220214T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/75
DESCRIPTION:Title: Positivity conditions for complex Hessian equations\nby Jian
Song (Rutgers University) as part of Geometric and functional inequalitie
s and applications\n\n\nAbstract\nIn this talk\, we will discuss the relat
ion between complex Hessian equation and positivity of algebraic numerical
conditions. In particular\, we will prove a Naki-Moishezon criterion for
Donaldson's J-equation.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Enno Lenzmann (University of Basel)
DTSTART;VALUE=DATE-TIME:20220228T140000Z
DTEND;VALUE=DATE-TIME:20220228T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
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 Geometric
and functional inequalities and applications\n\n\nAbstract\nIn this talk\,
I will review recent results on symmetry and symmetry-breaking for optimi
zing solutions of a general class of nonlinear elliptic PDEs. On one 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 Polya-Szeg
ö principle fail short). On the other hand\, I will discuss recent result
s on symmetry-breaking for optimizers by using Fourier methods and the Ste
in-Tomas inequality. This talk is based on joint work with Tobias Weth and
Jeremy Sok.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carolyn Gordon (Dartmouth College)
DTSTART;VALUE=DATE-TIME:20220328T130000Z
DTEND;VALUE=DATE-TIME:20220328T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/77
DESCRIPTION:Title: Inverse spectral problems on compact Riemannian orbifolds\nb
y Carolyn Gordon (Dartmouth College) as part of Geometric and functional i
nequalities and applications\n\n\nAbstract\nOrbifolds are a generalization
of manifolds in which various types of singularities may occur. After r
eviewing the notion of Riemannian orbifolds and their Hodge Laplacians\, w
e will address the question: Does the spectrum of the Hodge Laplacian 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)\, althoug
h many partial results are known. We will show that the spectra of the Hod
ge Laplacians on functions and 1-forms together suffice to distinguish man
ifolds from orbifolds with sufficiently large singular set. In particular
\, these spectra always distinguish low-dimensional orbifolds (dimension a
t most 3) with singularities from smooth manifolds. We also obtain weaker
affirmative results for the spectrum on 1-forms alone and show via counte
rexamples that some of these results are sharp.\n\n(This is based on recen
t joint work with Katie Gittins\, Magda Khalile\, Ingrid Membrillo Solis\,
Mary Sandoval\, and Elizabeth Stanhope and work in progress with the same
co-authors along with Juan Pablo Rossetti.) \n\nTime permitting\, we will
also make a few remarks concerning the Steklov spectrum on Riemannian orb
ifolds with boundary. The Steklov spectrum is the spectrum of the Dirich
let-to-Neumann operator\, which maps Dirichlet boundary values of harmonic
functions to their Neumann boundary values.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jérôme Vétois (McGill University)
DTSTART;VALUE=DATE-TIME:20220307T140000Z
DTEND;VALUE=DATE-TIME:20220307T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/78
DESCRIPTION:Title: Stability and instability results for sign-changing solutions to
second-order critical elliptic equations\nby Jérôme Vétois (McGill
University) as part of Geometric and functional inequalities and applicat
ions\n\n\nAbstract\nIn this talk\, we will consider a question of stabilit
y (i.e. compactness of solutions to perturbed equations) for sign-changing
solutions to second-order critical elliptic equations on a closed Riemann
ian manifold. I will present a stability result obtained in the case of di
mensions greater than or equal to 7. I will then discuss the optimality of
this result by constructing counterexamples in every dimension. This is a
joint work with Bruno Premoselli (Université Libre de Bruxelles\, Belgiu
m).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Loss (Georgia Institute of Technology)
DTSTART;VALUE=DATE-TIME:20220404T130000Z
DTEND;VALUE=DATE-TIME:20220404T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/79
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Michael Loss
(Georgia Institute of Technology) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nI present some results concerning
the size of magnetic fields that support zero modes for the three dimensi
onal Dirac equation and related problems for spinor equations. The critica
l quantity\, is the $3/2$ norm of the magnetic field $B$. The point is tha
t the spinor structure enters the analysis in a crucial way. This is joint
work with Rupert Frank at LMU Munich.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fritz Gesztesy (Baylor University)
DTSTART;VALUE=DATE-TIME:20220516T140000Z
DTEND;VALUE=DATE-TIME:20220516T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/80
DESCRIPTION:Title: Continuity properties of the spectral shift function for massles
s Dirac operators and an application to the Witten index\nby Fritz Ges
ztesy (Baylor University) as part of Geometric and functional inequalities
and applications\n\n\nAbstract\nWe report on recent results regarding the
limiting absorption principle for multi-dimensional\, massless Dirac-type
operators (implying absence of singularly continuous spectrum) and contin
uity properties of the associated spectral shift function.\n\nWe will moti
vate our interest in this circle of ideas by briefly describing the connec
tion to the notion of the Witten index for a certain class of non-Fredholm
operators.\n\nThis is based on various joint work with A. Carey\, J. Kaad
\, G. Levitina\, R. Nichols\, D. Potapov\, F. Sukochev\, and D. Zanin.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eric Carlen (Rutgers University)
DTSTART;VALUE=DATE-TIME:20220411T130000Z
DTEND;VALUE=DATE-TIME:20220411T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/83
DESCRIPTION:Title: Some trace inequalities related to quantum entropy\nby Eric
Carlen (Rutgers University) as part of Geometric and functional inequaliti
es and applications\n\n\nAbstract\nMany inequalities for trace functional
are formulated as concavity/convexity theorems. These generally have an eq
uivalent monotonicity version asserting monotonicity of the functional und
er some class of completely positive maps. The monotonicty formulation has
advantages: (1) Often this has a direct physical interpretation. (2) Ofte
n a direct proof of the monotonicity version is simpler than a direct proo
f of the concavity/convexity version\, and the later is always recovered u
sing a simple partial trace argument. (3) Often the monotonicty theorem ho
lds for a broader class of maps\, not\, necessarily completely positive\,
and is thus a strictly stronger result. We discus significant examples\,
some coming from recent joint work with Alexander Mueller-Hermes.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiaping Wang (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20220418T140000Z
DTEND;VALUE=DATE-TIME:20220418T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/84
DESCRIPTION:Title: Spectrum of complete manifolds\nby Jiaping Wang (University
of Minnesota) as part of Geometric and functional inequalities and applica
tions\n\n\nAbstract\nSpectrum of Laplacian is an important set of geometri
c invariants. The talk\, largely based on\njoint work with Peter Li and Ov
idiu Munteanu\, concerns its structure and size on complete manifolds unde
r various curvature conditions. The focus is on sharp estimates of the bot
tom spectrum in terms of either Ricci or scalar curvature lower bound.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Galia Dafni (Concordia University)
DTSTART;VALUE=DATE-TIME:20220523T140000Z
DTEND;VALUE=DATE-TIME:20220523T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/85
DESCRIPTION:Title: Locally uniform domains and extension of nonhomogeneous BMO spac
es\nby Galia Dafni (Concordia University) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nIn joint work with Al
maz Butaev (Cincinnati)\, we study local versions of uniform domains\, whi
ch can be identified with the epsilon-delta domains used by Jones to exten
d Sobolev spaces. We show that a domain is locally uniform if and only if
it is an extension domain for the nonhomogeneous (also known as "local") s
pace of functions of bounded mean oscillation introduced by Goldberg\, and
denoted by bmo. We also prove analogous results for functions of vanishi
ng mean oscillation.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alina Stancu (Concordia University)
DTSTART;VALUE=DATE-TIME:20220425T130000Z
DTEND;VALUE=DATE-TIME:20220425T140000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/86
DESCRIPTION:Title: On the fundamental gap of convex sets in hyperbolic space\nb
y Alina Stancu (Concordia University) as part of Geometric and functional
inequalities and applications\n\n\nAbstract\nThe difference between the fi
rst two eigenvalues of the Dirichlet Laplacian on convex sets of R^n and\,
respectively S^n\, satisfies the same strictly positive lower bound depen
ding on the diameter of the domain. In work with collaborators\, we have f
ound that the gap of the hyperbolic space on convex sets behaves strikingl
y different even if a stronger notion of convexity is employed. This is ve
ry interesting as many other features of first two eigenvalues behave in t
he same way on all three spaces of constant sectional curvature.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pei-Yong Wang (Wayne State University)
DTSTART;VALUE=DATE-TIME:20220502T140000Z
DTEND;VALUE=DATE-TIME:20220502T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/88
DESCRIPTION:Title: A Bifurcation Phenomenon Of The Perturbed Two-Phase Transition P
roblem\nby Pei-Yong Wang (Wayne State University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nThis talk pre
sents a joint work with F. Charro\, A. Haj Ali\, M. Raihen\, and\nM. Torre
s 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 data decr
eases through\na threshold value. For boundary values below the threshold\
, there are at least\nthree solutions\, namely\, the harmonic solution whi
ch is treated as a trivial solution\nin the absence of a free boundary\, a
nontrivial minimizer of the functional under\nconsideration\, and a third
solution of the mountain-pass type. We classify these\nsolutions accordin
g to the stability through evolution. The evolution with initial\ndata nea
r a stable solution\, such as the trivial harmonic solution or a minimizer
of\nthe functional\, converges to the stable solution. On the other hand\
, the evolution\ndeviates away from a non-minimal solution of the free bou
ndary problem.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bianca Stroffolini (Universit`a degli Studi di NAPOLI ”Federico
II”)
DTSTART;VALUE=DATE-TIME:20220919T140000Z
DTEND;VALUE=DATE-TIME:20220919T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/89
DESCRIPTION:Title: Taylor formula and regularity properties for degenerate Kolmogor
ov equations with Dini continuous coefficients\nby Bianca Stroffolini
(Universit`a degli Studi di NAPOLI ”Federico II”) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nWe study the
regularity properties of the second order linear operator in $\\mathbb{R}^
{N+1}$:\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\, with $A$ s
ymmetric 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 solution $u
$ to the equation $\\mathscr{L}u = f$ are Dini continuous functions as wel
l. 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 $\\m
athscr{L}u=f$ that we establish under minimal regularity assumptions on $u
$.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mimi Dai (University of Illinois at Chicago)
DTSTART;VALUE=DATE-TIME:20221003T140000Z
DTEND;VALUE=DATE-TIME:20221003T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/91
DESCRIPTION:Title: Navier-Stokes equation: determining wavenumber\, Kolmogorov’s
dissipation number\, and Kraichnan’s number\nby Mimi Dai (University
of Illinois at Chicago) as part of Geometric and functional inequalities
and applications\n\n\nAbstract\nWe show the existence of determining waven
umber for the Naiver-Stokes equation in both 3D and 2D. Estimates on the d
etermining wavenumber are established in term of the phenomenological Kolm
ogorov’s dissipation number (3D) and Kraichnan’s number (2D). The resu
lts rigorously justify the criticality of Kolmogorov’s dissipation numbe
r and Kraichnan’s number.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias König (Goethe-Universität Frankfurt)
DTSTART;VALUE=DATE-TIME:20221010T140000Z
DTEND;VALUE=DATE-TIME:20221010T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/92
DESCRIPTION:Title: Multibubble blow-up analysis for the Brezis-Nirenberg problem in
three dimensions\nby Tobias König (Goethe-Universität Frankfurt) as
part of Geometric and functional inequalities and applications\n\n\nAbstr
act\nIn this talk\, I will present a recent result about blow-up asymptoti
cs in the three-dimensional Brezis-Nirenberg problem. More precisely\, fo
r a smooth bounded domain $\\Omega \\subset \\R^3$ and smooth functions $a
$ and $V$\, consider a sequence of positive solutions $u_\\epsilon$ to $-\
\Delta u_\\epsilon + (a+\\epsilon V) u_\\epsilon = u_\\epsilon^5$ on $\\Om
ega$ with zero Dirichlet boundary conditions\, which blows up as $\\epsilo
n \\to 0$. We derive the sharp blow-up rate and characterize the location
of concentration points in the general case of multiple blow-up\, thereby
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 fort
hcoming 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://researchseminars.org/talk/GeomInequAndPDEs/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jianxiong Wang (University of Connecticut)
DTSTART;VALUE=DATE-TIME:20221017T140000Z
DTEND;VALUE=DATE-TIME:20221017T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/93
DESCRIPTION:Title: Symmetry of solutions to higher and fractional order semilinear
equations on hyperbolic spaces\nby Jianxiong Wang (University of Conne
cticut) as part of Geometric and functional inequalities and applications\
n\n\nAbstract\nWe show that nontrivial solutions to higher and fractional
order equations with certain nonlinearity are radially symmetric and nonin
creasing on geodesic balls in the hyperbolic space $\\mathbb{H}^n$ as well
as on the entire space $\\mathbb{H}^n$ . Applying Helgason-Fourier analys
is techniques on $\\mathbb{H}^n$ \, we developed a moving plane approach f
or integral equations on $\\mathbb{H}^n$. We also established the symmetry
to solutions of certain equations with singular terms on Euclidean spaces
. Moreover\, we obtained symmetry to solutions of some semilinear equation
s involving fractional order derivatives.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joshua Flynn (Centre de Recherches Mathématiques)
DTSTART;VALUE=DATE-TIME:20221024T140000Z
DTEND;VALUE=DATE-TIME:20221024T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/94
DESCRIPTION:Title: Sharp Uncertainty Principles for Physical Vector Fields and Seco
nd Order Derivatives\nby Joshua Flynn (Centre de Recherches Mathémati
ques) as part of Geometric and functional inequalities and applications\n\
n\nAbstract\nThe Heisenberg uncertainty principle is a fundamental result
in quantum mechanics. Related inequalities are the hydrogen and Hardy unce
rtainty principles and all three belong to the family of geometric inequal
ities 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 and vect
or-valued cases. Presented works were done jointly with G. Lu\, N. Lam and
C. Cazacu.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandru Kristaly (Babes-Bolyai University)
DTSTART;VALUE=DATE-TIME:20221031T140000Z
DTEND;VALUE=DATE-TIME:20221031T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/95
DESCRIPTION:Title: Lord Rayleigh’s conjecture for clamped plates in curved spaces
\nby Alexandru Kristaly (Babes-Bolyai University) as part of Geometric
and functional inequalities and applications\n\n\nAbstract\nThis talk is
focused on the vibrating clamped plate problem\, initially formulated by L
ord Rayleigh in 1877\, and solved by M. Ashbaugh & R. Benguria (Duke Math.
J.\, 1995) and N. Nadirashvili (ARMA\, 1995) in 2 and 3 dimensional eucli
dean spaces. We consider the same problem on both negatively and positivel
y curved spaces\, and provide various answers depending on the curvature\,
dimension and the width/size of the clamped plate.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xiaolong Han (California State University)
DTSTART;VALUE=DATE-TIME:20221107T150000Z
DTEND;VALUE=DATE-TIME:20221107T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/96
DESCRIPTION:Title: Fractal uncertainty principle for discrete Fourier transform and
random Cantor sets\nby Xiaolong Han (California State University) as
part of Geometric and functional inequalities and applications\n\n\nAbstra
ct\nThe Fourier uncertainty principle describes a fundamental phenomenon t
hat a function and its Fourier transform cannot simultaneously localize. D
yatlov and his collaborators (Zahl\, Bourgain\, Jin\, Nonnenmacher) recent
ly introduced a concept of Fractal Uncertainty Principle (FUP). It is a ma
thematical formulation concerning the limit of localization of a function
and its Fourier transform on sets with certain fractal structure. \n\nThe
FUP has quickly become an emerging topic in Fourier analysis and also has
important applications to other fields such as wave decay in obstacle scat
tering. In this talk\, we consider the discrete Fourier transform and 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 b
efore\, when the Cantor sets are constructed by a random procedure. This i
s a joint work with Suresh Eswarathasan.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Reiter (University of Technology at Chemnitz)
DTSTART;VALUE=DATE-TIME:20221114T150000Z
DTEND;VALUE=DATE-TIME:20221114T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/97
DESCRIPTION:Title: Elasticity models with self-contact\nby Philipp Reiter (Univ
ersity of Technology at Chemnitz) as part of Geometric and functional ineq
ualities and applications\n\n\nAbstract\nMaintaining the topology of objec
ts undergoing deformations is a crucial\naspect of elasticity models. In t
his talk we consider two different\nsettings in which impermeability is im
plemented via regularization by a\nsuitable nonlocal functional.\n\nThe be
havior of long slender objects may be characterized by the classic\nKirchh
off model of elastic rods. Phenomena like supercoiling which play an\nesse
ntial role in molecular biology can only be observed if\nself-penetrations
are precluded. This can be achieved by adding a\nself-repulsive functiona
l such as the tangent-point energy. We discuss the\ndiscretization of this
approach and present some numerical simulations.\n\nIn case of elastic so
lids whose shape is described by the image of a\nreference domain under a
deformation map\, self-interpenetrations can be\nruled out by claiming glo
bal invertibility. Given a suitable stored energy\ndensity\, the latter is
ensured by the Ciarlet–Nečas condition which\,\nhowever\, is difficult
to handle numerically in an efficient way. This\nmotivates approximating
the latter by adding a self-repulsive functional\nwhich formally correspon
ds to a suitable Sobolev–Slobodeckiĭ seminorm of\nthe inverse deformati
on.\n\nThis is joint work with Sören Bartels (Freiburg) and Stefan Kröme
r\n(Prague).\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Falocchi (Politecnico di Milano)
DTSTART;VALUE=DATE-TIME:20221121T150000Z
DTEND;VALUE=DATE-TIME:20221121T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/100
DESCRIPTION:Title: Some results on the 3D Stokes eigenvalue problem under Navier b
oundary conditions\nby Alessio Falocchi (Politecnico di Milano) as par
t of Geometric and functional inequalities and applications\n\n\nAbstract\
nWe study the Stokes eigenvalue problem under Navier boundary conditions i
n $C^{1\,1}$-domains $\\Omega\\subset \\mathbb{R}^3$. Differently from the
Dirichlet boundary conditions\, zero may be the least eigenvalue. We full
y characterize the domains where this happens\, showing the related validi
ty/failure of a suitable Poincar\\'{e}-type inequality.\n\nAs application
we prove regularity results for the solution of the evolution Navier-Stok
es equations under Navier boundary conditions in a class of merely {\\em L
ipschitz domains} of physical interest\, that we call {\\em sectors}.\n\nT
his is a joint work with Filippo Gazzola\, Politecnico di Milano.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Schlein (University of Zurich)
DTSTART;VALUE=DATE-TIME:20230306T150000Z
DTEND;VALUE=DATE-TIME:20230306T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/101
DESCRIPTION:Title: Gross-Pitaevskii and Bogoliubov theory for trapped Bose-Einstei
n condensates.\nby Benjamin Schlein (University of Zurich) as part of
Geometric and functional inequalities and applications\n\n\nAbstract\nWe c
onsider a quantum system consisting of N bosons (particles described by a
permutation symmetric wave function) trapped in a volume of order one and
interacting through a short range potential\, with scattering length of t
he order 1/N (this is known as the Gross-Pitaevskii regime). First\, we wi
ll show how non-linear Gross-Pitaevskii theory describes\, to leading orde
r\, the ground state energy of the gas and the time-evolution resulting fr
om a change of the external fields. In the second part of the talk\, I wil
l then explain how Bogoliubov theory predicts the next order corrections.\
n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk (Spring Break)
DTSTART;VALUE=DATE-TIME:20230313T130000Z
DTEND;VALUE=DATE-TIME:20230313T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/103
DESCRIPTION:by No Talk (Spring Break) as part of Geometric and functional
inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre-Damien Thizy (University of Lyon 1 (Claude Bernard))
DTSTART;VALUE=DATE-TIME:20230410T140000Z
DTEND;VALUE=DATE-TIME:20230410T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/104
DESCRIPTION:Title: Large blow-up sets for Q-curvature equations.\nby Pierre-Da
mien Thizy (University of Lyon 1 (Claude Bernard)) as part of Geometric an
d functional inequalities and applications\n\n\nAbstract\nOn a bounded dom
ain 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 equations
$(-\\Delta)^m u= Q e^{2m u}$ without boundary conditions may blow-up not
only at points\, but also on the zero set of some nonpositive nontrivial p
olyharmonic function. This is in striking contrast with the two dimensiona
l case ($m=1$). During this talk\, starting from a work in progress with A
li Hyder and Luca Martinazzi\, we will discuss the construction of such so
lutions which involves (possible generalizations of) the Walsh-Lebesgue th
eorem and some issues about elliptic problems with measure data.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deping Ye (Memorial University of Newfoundland)
DTSTART;VALUE=DATE-TIME:20230123T150000Z
DTEND;VALUE=DATE-TIME:20230123T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/105
DESCRIPTION:Title: The $L_p$ surface area measure and related Minkowski problem fo
r log-concave functions\nby Deping Ye (Memorial University of Newfound
land) as part of Geometric and functional inequalities and applications\n\
n\nAbstract\nThe Minkowski type problems for convex bodies are fundamental
\nin convex geometry\, and have found many important connections and\nappl
ications in analysis\, partial differential equations\, etc. It is\nwell-k
nown that the log-concave functions behave rather similar to\nconvex bodie
s in many aspects\, for example the famous Prékopa–Leindler\ninequality
to the (dimension free) Brunn-Minkowski inequality.\n\nIn this talk\, I w
ill present an $L_p$ theory for the log-concave\nfunctions\, which is anal
ogous to the $L_p$ Brunn-Minkowski theory of\nconvex bodies. In particular
\, I will explain how to define the $L_p$ sum\nof log-concave functions\,
present a variational formula related to the\n$L_p$ addition\, and talk ab
out the corresponding $L_p$ Minkowski\nproblems as well as their solutions
.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Walter Strauss (Brown University)
DTSTART;VALUE=DATE-TIME:20230227T150000Z
DTEND;VALUE=DATE-TIME:20230227T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/106
DESCRIPTION:Title: Instability of Water Waves (even small ones)\nby Walter Str
auss (Brown University) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nAfter a gentle introduction on water waves\
, I will present an exposition of joint work with Huy Quang Nguyen. We pro
ve rigorously that the classical (small-amplitude irrotational steady peri
odic) water waves are unstable with respect to long-wave perturbations. T
hat is\, the perturbations grow exponentially in time. This instability w
as first observed heuristically more than half a century ago by Benjamin a
nd Feir. However\, a rigorous proof was never found except in the case of
finite depth. We provide a completely different and self-contained proof o
f both the finite and infinite depth cases that retains the physical varia
bles. The proof reduces to an analysis of the spectrum of an explicit ope
rator. The growth is obtained by means of a Liapunov-Schmidt reduction th
at more or less reduces the analysis to four dimensions.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Leonard Gross (Cornell University)
DTSTART;VALUE=DATE-TIME:20230911T140000Z
DTEND;VALUE=DATE-TIME:20230911T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/107
DESCRIPTION:by Leonard Gross (Cornell University) as part of Geometric and
functional inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Monica Visan (University of California at Los Angeles)
DTSTART;VALUE=DATE-TIME:20230220T150000Z
DTEND;VALUE=DATE-TIME:20230220T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/108
DESCRIPTION:Title: The derivative nonlinear Schrodinger equation\nby Monica Vi
san (University of California at Los Angeles) as part of Geometric and fun
ctional inequalities and applications\n\n\nAbstract\nI will discuss the de
rivative nonlinear Schrodinger equation\, how some inherent instabilities
have hindered the study of this equation\, and how we were able to demonst
rate global well-posedness in the natural scale-invariant space. This is j
oint work with Ben Harrop-Griffiths\, Rowan Killip\, and Maria Ntekoume.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rowan Killip (University of California at Los Angles)
DTSTART;VALUE=DATE-TIME:20230320T140000Z
DTEND;VALUE=DATE-TIME:20230320T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/109
DESCRIPTION:Title: From Optics to the Deift Conjecture\nby Rowan Killip (Unive
rsity of California at Los Angles) as part of Geometric and functional ine
qualities and applications\n\n\nAbstract\nAfter providing a mathematical b
ackground for some curious\noptical experiments in the 19th century\, we w
ill then describe how\nthese ideas inform our understanding of the Deift c
onjecture for the\nKorteweg--de Vries equation. Specifically\, they allow
us to show that the\nevolution of almost-periodic initial data need not r
emain almost\nperiodic. This is joint work with Andreia Chapouto and Moni
ca Visan.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Silvia Cingolani (Università degli Studi di Bari Aldo Moro)
DTSTART;VALUE=DATE-TIME:20230508T140000Z
DTEND;VALUE=DATE-TIME:20230508T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/111
DESCRIPTION:by Silvia Cingolani (Università degli Studi di Bari Aldo Moro
) as part of Geometric and functional inequalities and applications\n\nAbs
tract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:No Talk
DTSTART;VALUE=DATE-TIME:20230327T130000Z
DTEND;VALUE=DATE-TIME:20230327T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/112
DESCRIPTION:by No Talk as part of Geometric and functional inequalities an
d applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenchuan Tian (UC Santa Barbara)
DTSTART;VALUE=DATE-TIME:20230130T150000Z
DTEND;VALUE=DATE-TIME:20230130T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/113
DESCRIPTION:Title: On a family of integral operators on the ball\nby Wenchuan
Tian (UC Santa Barbara) as part of Geometric and functional inequalities a
nd applications\n\n\nAbstract\nIn this work\, we prove an extension inequa
lity in the hyperbolic space. The inequality involves the hyperbolic harm
onic extension of a function on the boundary and the Fefferman-Graham comp
actification of the hyperbolic metric. We offer an interpretation of the e
xtension inequality as a conformally invariant generalization of Carleman'
s inequality to higher dimensions. \nIn addition to that\, we classify all
the solutions to the Euler-Lagrange equation of the extension inequality.
The proof uses the moving sphere method and relies on the properties of t
he Fefferman-Graham compactification of the hyperbolic metric.\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Roysdon (Brown University)
DTSTART;VALUE=DATE-TIME:20230213T150000Z
DTEND;VALUE=DATE-TIME:20230213T160000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/114
DESCRIPTION:Title: Intersection Functions\nby Michael Roysdon (Brown Universit
y) as part of Geometric and functional inequalities and applications\n\n\n
Abstract\nThe classical Busemann-Petty Problem from the 1950s asked the fo
llowing tomographic question:\n\nAssuming you have two origin-symmetric co
nvex bodies $K$ and $L$ in the $n$-dimensional Euclidean space satisfying
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 foll
ow that $|K| \\leq |L|$? The answer is affirmative for $n \\leq 4$ and ne
gative whenever $n >5$. However\, if $K$ belongs to a certain class of co
nvex bodies\, the intersection bodies\, then the answer to the Busemann-Pe
tty problem is affirmative in all dimension. Several extensions of this r
esult have been shown in the case of measures on convex bodies\, and isomo
rphic results of the same type have been established. Moreover\, the isom
orphic Busemann-Petty problem is actually equivalent to the isomorphic sli
cing problem of Bourgain (1986)\, which remains open to this day. \n\nIn
this talk\, we will introduce the notion of an intersection function\, pro
vide a Fourier analytic characterization for such functions\, and show som
e versions of the Busemann-Petty problem in this setting. In particular\,
we will show that if you have a pair of continuous\, even\, integrable fu
nctions $f\,g \\colon \\R^n \\to \\R_+$ 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 function. \
n\nThis is based on a joint work with Alexander Koldobsky and Artem Zvavit
ch\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luca Capogna (Smith College)
DTSTART;VALUE=DATE-TIME:20230417T140000Z
DTEND;VALUE=DATE-TIME:20230417T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/115
DESCRIPTION:by Luca Capogna (Smith College) as part of Geometric and funct
ional inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jie Qing (UC Santa Cruz)
DTSTART;VALUE=DATE-TIME:20230501T140000Z
DTEND;VALUE=DATE-TIME:20230501T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/116
DESCRIPTION:by Jie Qing (UC Santa Cruz) as part of Geometric and functiona
l inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lu Wang (Yale University)
DTSTART;VALUE=DATE-TIME:20231009T140000Z
DTEND;VALUE=DATE-TIME:20231009T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/117
DESCRIPTION:by Lu Wang (Yale University) as part of Geometric and function
al inequalities and applications\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:María del Mar González (Universidad Autónoma de Madrid)
DTSTART;VALUE=DATE-TIME:20230424T140000Z
DTEND;VALUE=DATE-TIME:20230424T150000Z
DTSTAMP;VALUE=DATE-TIME:20230331T094228Z
UID:GeomInequAndPDEs/118
DESCRIPTION:by María del Mar González (Universidad Autónoma de Madr
id) as part of Geometric and functional inequalities and applications\n\nA
bstract: TBA\n
LOCATION:https://researchseminars.org/talk/GeomInequAndPDEs/118/
END:VEVENT
END:VCALENDAR