Geometric
Analysis Seminar
Organizers: Pengfei Guan and
Jérôme Vétois
WINTER 2024
Tuesday,
February 6, 14:35-15:35,
Burnside Hall 1205
Fang Hong (McGill
University)
Title: On normal derivative of Dirichlet
eigenfunctions
Abstract: Normal derivative gives a lot of useful
information of Dirichlet eigenfunctions and eigenvalues on Riemannian
manifolds. We will introduce Hadamard's formula for eigenvalues and
local relationship between normal derivative and curvature of boundary.
Then we will review Ozawa's and Tao's results, giving more detailed
estimates to normal derivatives.
Tuesday,
February 20, 14:35-15:35,
Burnside Hall 1205
Cale Rankin (University
of Toronto)
Title: A geometric approach to the
Ma–Trudinger–Wang estimates
Abstract: For a long time the regularity of optimal
transport
with general costs was a fundamental open problem. Whilst this question
had been answered in particular cases by Delanoë, Caffarelli,
and
Urbas (separately) via apriori estimates for
Monge–Ampère
equations, it remained unknown how to deal with general costs. The
extension of these estimates and the regularity of the optimal
transport maps was proved in a groundbreaking work by Ma, Trudinger,
and Wang who introduced a mysterious fourth order condition on the
cost. Kim and McCann realised this condition amounts to a curvature
condition in a particular pseudo-Riemannian geometry and, with Warren,
realised after a conformal rescaling the graph of optimal transport
maps are maximal surfaces in this geometry. In this talk, which
describes joint work with Brendle, Léger, and McCann we show
how
the original MTW estimates may be realized via estimates for maximal
surfaces in pseudo-Riemannian geometry.
Tuesday,
March 12, 14:35-15:35,
Burnside Hall 1205
Pengfei Guan (McGill
University)
Title: Anisotropic Gauss curvature flows and Lp
Minkowski problem
Abstract: The classical Minkowski problem is a
problem of area
measures. Lutwak introduced Lp version of the Minkowski problem. A
basic question is under what conditions one may produce a weak
solution. The problem can be reframed as a variational problem for the
associated entropies with volume constraint. We use Andrews anisotropic
Gauss curvature flows provide good path to achieve the goal. The talk
is based on a recent joint work with Karoly Boroczky.
Tuesday,
March 19, 14:35-15:35,
Burnside Hall 1205
Hyun Chul Jang (Caltech)
Title: Instability of minimal entropy rigidity for
product spaces of rank-one symmetric spaces
Abstract: The volume entropy is a geometric
invariant defined as
the volume growth of geodesic balls in the universal cover equipped
with the pull-back metric. The minimal entropy and the corresponding
rigidity for some special cases have been established in the
literature. In this talk, I will review the minimal entropy rigidity
results and discuss a couple of new results concerning product spaces
of rank-one symmetric spaces. Firstly, I will present the uniqueness of
the spherical Plateau solution, which is closely related to the minimal
entropy rigidity. Secondly, I will demonstrate the geometric
instability of the minimal entropy rigidity for product spaces with a
counterexample. This observation differs from the recent results of
Song for locally symmetric spaces.
Tuesday,
March 26, 14:35-15:35,
Burnside Hall 1205
Joshua Flynn (McGill
University)
Title: Conformal Boundary Operators, Extension
Theorems and Trace Inequalities.
Abstract: The Caffarelli-Silvestre extension theorem
allows one to obtain the fractional Laplacian of fractional order in
(0,1) via a Dirichlet-to-Neumann map which maps solutions to a weighted
Dirichlet problem on Euclidean halfspace to functions on its boundary.
Dirichlet's principle gives a resulting sharp Sobolev trace inequality.
Given that the fractional Laplacian may be defined for higher
fractional orders and analogous fractional operators are naturally
defined on conformal or CR infinities of certain geometries, it is
natural to extend Caffarelli-Silvestre's results to the higher order
and for general geometries. In this talk, we discuss relevant recent
results obtained in collaboration with G. Lu and Q. Yang.
Tuesday,
April 2, 14:35-15:35,
Burnside Hall 1205
Min Chen (McGill
University)
Title:Inhomogeneous Gauss Curvature flow.
Abstract:We consider the flow of convex
hypersurfaces in Euclidean space R^{n+1} under the in-homogeneous speed
functions of Gauss curvature. We establish the existence and
convergence of the flow to a limit which is the round sphere (after
rescaling) under appropriate conditions of the speed functions for
$\alpha>\frac{1}{n+1}$. This generalized the celebrated results
on Gauss curvature flow by Andrews-Guan-Ni and
Brendle-Choi-Daskalopoulos. This is joint work with Prof.Pengfei Guan
and Jiuzhou Huang.
Friday,
April 5, 13:30-14:30,
Burnside Hall 1104,
joint with the CRM-Montreal-Quebec
Analysis
Seminar
Yi Wang (Johns
Hopkins University)
Title: Yamabe flow of asymptotically flat metrics
Abstract: In this talk, we will discuss the behavior
of the
Yamabe flow on an asymptotically flat (AF) manifold. We will first show
the long-time existence of the Yamabe flow starting from an AF manifold
and discuss the uniform estimates on manifolds with positive Yamabe
constant. This would allow us to prove global weighted convergence
along the Yamabe flow on such manifolds. We will also talk about the
case when the Yamabe constant is nonpositive. This is joint work with
Eric Chen and Gilles Carron.
Tuesday,
April 9, 14:35-15:35,
Burnside Hall 1205
Sébastien Picard (University
of British Columbia)
Title: G2 flows and parabolic complex Monge-Ampere
equations
Abstract: We will discuss how parabolic complex
Monge-Ampere
equations arise in the context of G2 geometry. Estimates for these
general Monge-Ampere flows can be used to provide examples of the G2
flow which exist for all time and converge to a torsion-free G2
structure on a compact manifold. This talk will survey joint works with
X.-W. Zhang and C. Suan.
Tuesday,
April 23, 14:35-15:35,
Burnside Hall 1205
Xiangwen Zhang (University
of California, Irvine)
Title:
A geometric flow in symplectic geometry
Abstract: Geometric flows have been proven to be
powerful tools
in the study of many important problems arising from both geometry and
theoretical physics. In this talk, we will discuss the progress on the
so-called Type IIA flows, introduced in a joint work with Fei, Phong
and Picard, aiming to study the Type IIA equations from the flux
compactifications of superstrings.
Tuesday,
April 30, 14:35-15:30,
Burnside Hall 1205
Samuel Zeitler (McGill
University)
Title: A Sharp Higher Order Sobolev Inequality
Abstract: We establish a sharp Sobolev inequality of
higher order on closed Riemannian manifolds. We first introduce some
background behind the problem, which dates back to the first order
asymptotically sharp Sobolev inequality of Aubin and the concentration
compactness principle of P.L. Lions. We then discuss the main ideas of
the proof, which involves an asymptotic analysis of a blowing up
sequence of solutions to elliptic semilinear PDEs of critical growth at
the lowest energy level.
Tuesday,
April 30, 15:35-16:30,
Burnside Hall 1205
Siyuan Lu (McMaster
University)
Title:
Interior C^2 estimate for Hessian quotient equations
Abstract: In this talk, I will first review the
history of
interior C^2 estimate for fully nonlinear equations. As a matter of
fact, only very few equations were shown to have such properties. Even
the Monge-Ampere equations for dimension three and higher do not have
interior C^2 estimate due to the famous example by Pogorelov. In the
second part, I will discuss my recent work on interior C^2 estimate for
Hessian quotient equations. Such equations have deep connections with
Monge-Ampere equations, Hessian equations as well as special Lagrangian
equations. I will then discuss the main ideas behind the proof. The new
method we adopted to prove interior C^2 estimate has independent
interest and can be used in other settings.
Tuesday,
May 7,
14:35-15:35,
Burnside Hall 1205
Mariel Sáez Trumper (Pontificia
Universidad Católica de Chile)
Title: Uniqueness of Semigraphical Translators of
Mean Curvature Flow
Abstract: In this talk we prove the uniqueness of
pitchfork and
helicoid translators of the mean curvature flow in $\mathbb{R}^3$. This
solves a conjecture by Hoffman, White and Martin.
The proof is based on an arc-counting argument motivated by
Morse-Rad\'o theory for translators and a rotational maximum principle.
This is joint work with F. Martin and R. Tsiamis.
Tuesday,
May 14,
14:35-15:35,
Burnside Hall 1205
Junfang Li (The
University of Alabama at Birmingham)
Title: Riemannian geometric properties and Weyl
structures
Abstract: In this talk, we discuss a tie between
Riemannian and
semi-Riemannian geometry. More specifically, we will relate several
classical Riemannian geometric properties, such as integral Reilly
formula, local Bochner technique, and its applications including
inequalities, geometric bounds, rigidity results etc. to the well-known
Weyl conformal and projective structures from general relativity.
Tuesday,
May 21,
10:00-10:55,
Burnside Hall 1205
Meijun Zhu (The
University of Oklaoma)
Title: On the extension operators - sharp
inequalities and the applications
Abstract: Define the general extension operator
$E_{alpha,beta}f(x^2,x_n)=\int_{\partial R^n_+}x_n^\beta
f(y)/(|x'-y|^2+x_n^2)^((n-\alpha)/2))dy$
In this talk, I shall recall how we arrive at this general form in the
study of the
extension operators, what kind of sharp estimates we have obtained so
far in the
last ten years. I shall also report some recent results on the
application of these
estimates, in particular in the study of Carleman type inequalities in
Hardy space.
Tuesday,
May 21,
11:00-11:55,
Burnside Hall 1205
Jyotshana Prajapat (University
of Mumbai)
Title: Convexity and symmetry results on Heisenberg
group
Abstract: I will talk about my recent results
related to
geodetic convex sets in the Heisenberg group. I will also illustrate a
proof of symmetry of solution to the CR-Yamabe problem on the
Heisenberg group and non existence result for the subcritical values,
without any partial symmetry assumptions on the solution.
Tuesday,
May 28,
14:35-15:30,
Burnside Hall 1205
Yevgeny Liokumovich (University
of Toronto)
Title: Asymptotics of the volume spectrum
Abstract: Morse theory on the space of generalized
k-dimensional
submanifolds on a Riemannian manifold M gives rise to a sequence of
min-max widths w_k(p) that are sometimes called the "volume spectrum"
of M. It was conjectured by Gromov that (like eigenvalues of the
Laplacian) the asymptotic behavior of w_k(p) satisfies a Weyl law. I
will describe the proof of this conjecture for some values of k, and
discuss its connection to minimal submanifolds and stationary geodesic
networks.
The talk will be based on joint works with Marques and Neves, Larry
Guth and Bruno Staffa.
Tuesday,
May 28, 15:35-16:30,
Burnside Hall 1205
Florica Cîrstea (The
University
of Sydney)
Title: Existence and boundedness of solutions to
singular anisotropic elliptic equations
Abstract: In this talk, we present new results on
the existence
and uniform boundedness of solutions for a general class of Dirichlet
anisotropic elliptic problems of the form
$$-\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f
\quad \mbox{in } \Omega, \qquad u=0 \quad \mbox{on }\partial \Omega,$$
where $\Omega$ is a bounded domain in $\mathbb R^N$ $(N\geq 2)$, $
\Delta_{\overrightarrow{p}}u=\sum_{j=1}^N \partial_j (|\partial_j
u|^{p_j-2}\partial_j u)$ and $\Phi_0(u,\nabla
u)=\left(\mathfrak{a}_0+\sum_{j=1}^N \mathfrak{a}_j |\partial_j
u|^{p_j}\right)|u|^{m-2}u$, with $\mathfrak{a}_0>0$,
$m,p_j>1$,
$\mathfrak{a}_j\geq 0$ for $1\leq j\leq N$ and $N/p=\sum_{k=1}^N
(1/p_k)>1$. We assume that $f \in L^r(\Omega)$ with
$r>N/p$. The
feature of this study is the inclusion of a possibly singular
gradient-dependent term $\Psi(u,\nabla u)=\sum_{j=1}^N
|u|^{\theta_j-2}u\, |\partial_j u|^{q_j}$, where $\theta_j>0$
and
$0\leq q_j.
FALL 2023
Tuesday,
October 3, 14:30-15:30,
Burnside Hall 1104
Marcin Sroka (McGill
University)
Title: Second order estimate for Monge-Ampere type
equations on Riemannian manifolds with additional structure
Abstract: We will
discuss the challenges and
differences in obtaining second order estimates for some second order,
elliptic equations (of Monge-Ampere type) arising naturally on
Riemannian manifolds endowed with additional structure (like complex or
quaternionic one). This type of equations emerged for example in the
course of proving Calabi's volume
prescribing
conjecture or its analogue in hypercomplex case
Alesker-Verbitsky conjecture.
Tuesday,
October 17, 14:35-15:35,
Burnside Hall 1104
Dylan Cant (McGill
University)
Title:
Floer theory in convex-at-infinity symplectic manifolds
Abstract: "Floer
theory" is a sort of
elliptic Morse homology constructed by counting solutions to a certain
inhomogeneous perturbation of the J-holomorphic curve equation. The
nonlinear PDE is now called "Floer's equation." As discovered by Gromov
and Floer, if one works in a symplectic manifold which "tames" the
almost complex structure J and uses inhomogeneous perturbations coming
from certain Hamiltonian vector fields one has a priori W1,2 estimates
(for suitable choice of Riemannian metric). When the symplectic
manifold is non-compact, the variety of invariants which can be
constructed using Floer theory is still an open field of research. A
well-studied class of noncompact manifolds are those which are
symplectically "convex" at infinity, in the sense of Eliashberg-Gromov.
I will present some recent work which proves a maximum principle for
solutions to Floer's equation in convex-at-infinity symplectic
manifolds.
Tuesday,
October 24, 14:35-15:35,
Burnside Hall 1104
Min Chen (McGill
University)
Title:Alexandrov-Fenchel type inequalities for
hypersurfaces in the sphere
Abstract: The
Alexandrov-Fenchel inequalities
in the Euclidean space are inequalities involving quermassintegrals of
different orders and are classical topics in convex geometry and
differential geometry. Brendle-Guan-Li proposed a conjecture on the
corresponding inequalities for quermassintegrals in the sphere. In this
talk, we introduce a new progress to this Conjecture.
Tuesday,
October 31, 14:35-15:35,
Burnside Hall 1104
Amir Moradifam(University
of California Riverside)
Title: The Sphere Covering Inequality and Its
Applications
Abstract: We show
that the total area of two
distinct Gaussian curvature 1 surfaces with the same conformal factor
on the boundary, which are also conformal to the Euclidean unit disk,
must be at least 4Ï€. In other words, the areas of
these
surfaces must cover the whole unit sphere after a proper rearrangement.
We refer to this lower bound of total areas as the Sphere Covering
Inequality. This inequality and it’s
generalizations
are applied to a number of open problems related to Moser-Trudinger
type inequalities, mean field equations and Onsager vortices, etc, and
yield optimal results. In particular we confirm the best constant of a
Moser-Truidinger type inequality conjectured by A. Chang and P. Yang in
1987. This is a joint work Changfeng Gui.
Tuesday,
November 14, 14:35-15:35,
Burnside Hall 1104
Jacob Reznikov (McGill
University)
Title: Isoperimetric inequality using curvature
flows of conformal vector fields
Abstract:
We describe a flow approach to the Isoperimetric inequality
first
derived by Guan-Li in 2013 and then improved by Guan-Li-Wang in 2018,
and its generalization by Li-Pan in 2023. We describe the key algebraic
properties of a special type of conformal vector field which allows us
to define a curvature flow on embedded hypersurfaces. We then derive
the evolution equations of geometric properties along this flow and
finish by proving its convergence.
Tuesday,
November 28, 14:35-15:35,
Burnside Hall 1104
Edward Chernysh (McGill
University)
Title: A Struwe-Type Decomposition for
Caffarelli-Kohn-Nirenberg Equations
Abstract:
In this talk, we establish a Struwe-type decomposition result
for a
class of critical $p$-Laplace equations of the Cafarelli-Kohn-Nirenberg
type, in smoothly bounded domains for $n \ge 3$. In doing so, we
highlight crucial differences between the weighted setting and the
pioneering work of Michael Struwe in the unweighted model $p=2$ case.
Tuesday,
December 5, 14:35-15:35,
Burnside Hall 1104
Bartosz Syroka (McGill
University)
Title: A harmonic flow of geometric structures
Abstract:
We will consider minimizers of a Dirichlet-type energy of
tensor
fields on a Riemannian manifold, left invariant by some Lie group H
inside SO(n). To this end, we will explain how to deform these
H-structures, define their torsion and some of its basic properties.
The Euler-Lagrange equations for the Dirichlet-type energy functional
can then be phrased in terms of a deformation of the structure by its
torsion tensor. We will show some of the analytic results for the
corresponding gradient flow of the energy, if time allows. Throughout
the talk, we will reference the unitary group U(m) as a guiding
example.
WINTER 2023
Wednesday,
February 1,
15:15-16:15,
Burnside Hall 1234 and Zoom
Hugues Auvray (Université
Paris-Sud)
Title: Bergman kernels on punctured Riemann surfaces
Abstract: In joint
works with X. Ma (Paris 7)
and G. Marinescu (Cologne) we obtain refined asymptotics for Bergman
kernels computed from singular data. We work on the complement of a
finite number of points, seen as punctures, on a compact Riemann
surface, that we endow with a metric extending Poincaré's
cusp
metric around the puncture points. We moreover fix a holomorphic line
bundle polarizing such a metric. I'll explain how an advanced
description of the model (on the punctured unit disc) and weighted
analusis techniques in a singular context allow to describe the Bergman
kernels associated to such Riemann surfaces, up to the singularities.
Wednesday,
February 8,
15:00-16:00,
Burnside Hall 1234 and Zoom
Marcin Sroka (McGill
University)
Title: Gradient estimates for complex PDEs
Abstract: We will
discuss the recent paper of
Guo, Phong and Tong on the gradient estimate for the complex
Monge-Ampere equation. In general we will outline troubles with
obtaining this bound for more general PDEs on complex manifolds.
Wednesday,
February 15,
15:00-16:00,
Burnside Hall 1234 and Zoom
Fang Hong (McGill
University)
Title: Sharp Minkowski inequalities in Hadamard
manifolds and their applications
Abstract: Extension
of Minkowski inequality
to hyperbolic space H^3 and Finding the sharp inequality have been a
long standing problem. We will discuss the recent paper by M. Ghomi and
J. Spruck, in which they generalized Minkowski inequality to
Cartan-Hadamard manifolds via harmonic mean curvature flow. We will
further discuss sharper inequality we get based on their results.
Wednesday,
February 22,
15:00-16:00,
Burnside Hall 1234 and Zoom
Pengfei Guan (McGill
University)
Title: Remarks on the gradient estimate for real and
complex Hessian type equations
Abstract: This is a
follow-up of Marcin Sroka's talk. We will discuss various methods to
obtain the global and interior the gradient estimates for the real
Hessian type equations. We then switch the attention to global estimate
of complex Hessian equations. There are still several open problems in
this case, we will discuss why is so difficulty in complex. We will
provide a proof of gradient estimate for geometric solutions of a class
of complex Hessian equations on Hermitian manifolds.
Wednesday,
March 8,
15:00-16:00,
Burnside Hall 1234 and Zoom
Jacob Reznikov (McGill
University)
Title: Entropy and singularities in mean curvature
flow.
Abstract: We will
discuss the basics of mean curvature flow and construct some examples
of its singularities. We will then discuss entropy and
Huisken’s monotonicity formula and some
of its basic properties. We will further discuss a recent paper by
Chodosh, Choi, Mantoulidis and Schulze on low-entropy initial data and
further classification we get based on their results.
Wednesday,
March 22,
15:00-16:00,
Burnside Hall 1234 and Zoom
Min Chen (McGill
University)
Title: Foliations by stable spheres with constant
Gauss curvature in an asymptotically flat Riemannian manifold.
Abstract: We will
discuss using a heat flow method to deform a coordinate sphere into a
constant Gauss curvature surface. With the positivity of the mass, we
then prove that the constructed surfaces form a stable constant Gauss
curvature foliation. It defines a natural coordinate system near
infinity. Finally, we will briefly introduce a method to obtain the
uniqueness of the constant mean curvature foliation for ends with
positive mass.
Wednesday,
March 29
15:00-16:00,
Burnside Hall 1234 and Zoom
Samuel Zeitler (McGill
University)
Title: Sharp Sobolev inequalities of arbitrary
order.
Abstract: Best
constants for Sobolev inequalities on closed Riemannian manifolds have
been the target of investigation for decades. We will introduce the
first order case and discuss a natural extension of this problem to
higher order embeddings. In particular we present the value of the best
first constant and give examples of classes of manifolds where this
constant is achieved.
Wednesday,
April 5,
15:00-16:00,
Burnside Hall 1234 and Zoom
Carlo Scarpa (UQAM)
Title: Kahler forms and B-fields
Abstract: Motivated
by some constructions in Mirror Symmetry, we will consider the problem
of finding a canonical representative of a complexified Kahler class on
a compact complex manifold. In 2020, Schlitzer and Stoppa proposed a
geometric PDE, whose solutions conjecturally give the required
canonical representative of the class. I will explain a variational
framework in which to consider their equation, focussing on an
associated system of geodesic equations for
Kähler potentials. In particular, I will explain
how to prove uniqueness of solutions of the PDE in the toric setting.
Wednesday,
April 19,
15:00-16:00,
Burnside Hall 1234 and Zoom
Jérôme Vétois
(McGill
University)
Title: On the entire solutions to the critical
p-Laplace equation
Abstract: We will
discuss the problem of classifying the entire, positive solutions to
the critical p-Laplace equation in the Euclidean space. In the
case where p=2
(i.e. for the classical
Laplace operator), this
problem was solved by Caffarelli,
Gidas and Spruck in 1989 (see also the work of Obata in 1971 for
finite-energy solutions). I will review some recent work extending this
result to
the case of the p-Laplace operator for large values of p.
Thursday
June 22,
15:00-16:00,
Burnside Hall 920
YanYan Li (Rutgers
University)
Title: Some recent results on conformally invariant
equations
Abstract:
We will present some recent work on conformally invariant nonlinear
elliptic equations.
This includes results on Liouville type theorems, derivative estimates,
isolated singularities, existence and nonexistence of solutions.
FALL 2022
Wednesday,
October 5,
15:00-16:00,
Burnside Hall 1234 and Zoom
Bartosz Syroka (McGill
University)
Title: Balanced metrics on 6-manifolds of
cohomogeneity one
Abstract:
We will introduce the notions of a cohomogeneity one group action on a
6-manifold, and of balanced, non-Kahler SU(3)
structures invariant with respect to the action. We will then state and
describe an existence result contingent on the decomposition of the Lie
algebra of the group. Such structures form a part of the Strominger
system playing a role in string theory, inspiring recent work on the
interplay of Lie group symmetries and the associated set of PDEs.
Wednesday,
October 19,
15:00-16:00,
Burnside Hall 1234 and Zoom
Jack Borthwick (Université
de Bourgogne-Franche Comté)
Title: Projective differential geometry and
asymptotic analysis in General Relativity
Abstract: Every
Lorentzian manifold $(M,g)$
has a natural projective structure induced by its Levi-Civita
connection. In some cases, $M$ can be embedded into a manifold with
boundary $\overline{M}$, in
which the projective structure extends to the boundary: $(M,g)$ is then
said to be projectively
compact. In this talk, we will discuss applications of the projective
structure to the asymptotic
analysis of partial differential equations, in particular a generalised
Proca equation, on projectively
compact Lorentzian manifolds.
Wednesday,
October 26,
15:00-16:00,
Burnside Hall 1234 and Zoom
Huangchen Zhou (McGill
University)
Title: From isometric embedding to a sum of squares
theorem
Abstract: Motivated
by an isometric embedding
problem in the graph setting, we'll discuss a sum of squares theorem
for Holder function. Given a non-negative C^{2,2\alpha} function f over
R, can we find a function g in C^{1,\alpha} such that f=g^2? We've
found a necessary and sufficient condition for this problem, which is
related to the non-zero strict local minimum points of the function f.
Wednesday,
November 2,
15:00-16:00,
Burnside Hall 1234 and Zoom
Joshua Flynn (McGill
University)
Title: Sharp Hardy-Sobolev-Maz'ya inequalities for
noncompact rank one symmetric spaces
Abstract:
The Hardy-Sobolev-Maz'ya inequality combines the Hardy and Sobolev
inequalities into a single inequality on the halfspace.
Using conformal equivalence, this inequality is equivalent to the
Poincare-Sobolev inequality on the real hyperbolic space.
Using the Helgason-Fourier analysis, higher order versions of these
inequalities were established by G. Lu and Q. Yang.
We introduce these results and indicate how they may be extended to the
other noncompact rank one symmetric spaces.
Discussed works are that of J. Li, G. Lu, Q. Yang and myself.
Wednesday,
November 9,
15:00-16:00,
Burnside Hall 1234 and Zoom
Zhizhang Wang (Fudan
University)
Title: The prescribed curvature problem in Minkowski
space
Abstract: In this
talk, we will first discuss the existence of hypersurfaces with
constant hessian curvature in Minkowski space.
Using the similar method, we can obtain some existence theorems for the
prescribed hessian curvature equations. We may
further consider the hessian curvature flow in Minkowski space.
Wednesday,
November 16,
15:00-16:00,
Burnside Hall 1234 and Zoom
Marcin Sroka (McGill
University)
Title: On certain, geometrically motivated,
Monge-Ampere type equation
Abstract: I will
discuss the equation
Alesker and Verbitsky introduced on hyperKahler with torsion manifolds
as a device for proving “quaternionic version” of
the
Calabi conjecture. The equation is called quaternionic Monge-Ampere
equation and has many common features with its real and complex
counterparts. Its solvability has applications to obtaining Calabi-Yau
type theorems for different classes of hermitian and hyperhermitian
metrics.
Wednesday,
November 23,
15:00-16:00,
Burnside Hall 1234 and Zoom
Ramya Dutta (TIFR
Bangalore)
Title: Apriori decay estimates for
Hardy-Sobolev-Maz'ya equations and application to a Brezis-Nirenberg
problem
Abstract:
In this talk we will discuss some qualitative properties and sharp
decay estimates of solutions to the Euler-Lagrange equation
corresponding to Hardy-Sobolev-Maz'ya inequality with cylindrical
weight. Using these sharp asymptotics we will establish a
Brezis-Nirenberg type existence result for class of $C^1$ sublinear
perturbations of the p-Hardy-Sobolev equation with cylindrical weight
in a bounded domain in dimensions $n > p^2$ and an appropriate
notion of positivity for these perturbations.
Wednesday,
November 30,
15:00-16:00,
Burnside Hall 1234 and Zoom
Bruno Premoselli (Université
Libre de Bruxelles)
Title: (Non)-Compactness for sign-changing solutions
of the Yamabe equation at the lowest energy level
Abstract: The goal
of this talk is to describe the behavior of least energy sign-changing
solutions of the Yamabe equation. Sign-changing solutions of the
celebrated Yamabe equation naturally appear as extremals for the
minimization problem of eigenvalues of the conformal Laplacian in a
fixed conformal class. We will review their link with the original
geometric problem and will describe in this talk their behavior at the
lowest energy level. Our main focus will in particular be a compactness
result in small dimensions or in the locally conformally flat case. The
results in this talk have been obtained in collaboration with J.
Vétois.
Wednesday,
December 7,
15:00-16:00,
Burnside Hall 1234 and Zoom
Clara Aldana (Universidad
del Norte, Barranquilla)
Title: Isospectral and quasi-isospectral Schrodinger
operators
Abstract: On this
talk I will first briefly talk about the isospectral problem in
geometry and about isospectrality of Strum-Liouville operators on a
finite interval in the simplified form of a Schrodinger operator. I
will mention very interesting known results about isospectral
potentials. I will introduce quasi-isospectrality as a generalization
of isospectrality. I will mention the history of the problem and how to
construct quasi-isospectral potentials. I will present what we know so
far about them. The work presented here is still on-going joint work
with Camilo Perez.
WINTER 2022
Wednesday,
February 9, 9:30-10:30, Zoom
Meeting
Pengfei Guan (McGill
University)
Title: Diameter estimates for solutions of
$L^p$-Minkowski problem with weak data
Abstract: $L^p$-Minkowski
problem corresponds
to the following Monge-Ampere equation on $S^n$:
$\det(u_{ij}(x)+u(x)\delta_{ij})=u^p(x)f(x)$. We establish estimate for
solution $u$ with the prescribed data $f$ only in certain integrable
space. In the talk, we will review literature and discuss some open
problems.
Wednesday,
February 16, 9:30-10:30, Zoom
Meeting
Gantumur Tsogtgerel (McGill
University)
Title: Perron’s solution and the Weyl
projection method
Abstract: In this talks we
will focus on the
relationship between the Perron-Wiener method, and the variational
method based on Sobolev spaces for constructing solutions of the
classical Dirichlet problem. After talking about some facets of the
history of the Dirichlet problem, we will discuss an elementary method
to deal with the aforementioned issue
Wednesday,
February 23, 9:30-10:30, Zoom
Meeting
Jérôme
Vétois (McGill
University)
Title: Stability and instability results for
sign-changing solutions to second-order critical elliptic equations
Abstract: In this talk, we
will consider a question of stability (i.e. compactness of solutions to perturbed equations) for
sign-changing solutions to second-order
critical elliptic equations on a closed Riemannian
manifold.
I will present a stability result obtained in the case of dimensions
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, Belgium).
Wednesday,
March 9, 9:30-10:30, Zoom
Meeting
Jiuzhou Huang (McGill
University)
Title: A warped product metric, Hilbert Einstein
functional and Weyl’s problem
Abstract: In this talk, we
use a warped
product metric introduced by Izmestiev and the Einstein Hilbert
functional to study Weyl’s embedding problem. The warped
metric
can be used to give a new proof of the closeness of the problem, and
the Hilbert-Einstein functional is related to the variational property
and stability of the embedding.
Wednesday,
March 16, 9:30-10:30, Zoom
Meeting
Sisi Shen (Columbia
University)
Title: A Chern-Calabi flow on Hermitian Manifolds
Abstract: We discuss the
existence problem of
constant Chern scalar curvature metrics on a compact complex manifold
and introduce a Hermitian analogue of the Calabi flow on compact
complex manifolds with vanishing first Bott-Chern class.
Wednesday,
March 23, 9:30-10:30, Zoom
Meeting
Bartosz Siroka (McGill
University)
Title: A cohomogeneity one approach to
Kahler-Einstein metrics
Abstract: We will discuss
the work of Andrew
Dancer and McKenzie Wang on the classification of solutions to the
Kahler-Einstein equations using cohomogeneity one group actions,
reducing the equations to a system of nonlinear ODEs. We will first
cover the cohomogeneity one setup, then the explicit local solutions to
the Einstein equations. Lastly, we will look at extending the local
solutions to the global case.
Wednesday,
March 30, 9:30-10:30, Zoom
Meeting
Min Chen (McGill
University)
Title: Alexandrov-Fenchel type inequalities in the
sphere
Abstract: The
Alexandrov-Fenchel inequalities
for quermassintegrals in the Euclidean spaces are classical topics in
differential geometry and convex geometry. The corresponding problem in
non-convex hypersurfaces in space forms has gained much interest
recently but remains largely unsettled. The application of curvature
flows to prove the geometric inequalities is nowadays classical. In
this talk, I will introduce a recent work about the Alexandrov-Fenchel
inequalities in the sphere by employing suitable curvature flows.
Wednesday,
April 6, 9:30-10:30, Zoom
Meeting
Valentino Tosatti (McGill
University)
Title: The Chern-Ricci flow
Abstract: The Chern-Ricci
flow is a flow of Hermitian metrics by their Chern-Ricci form, which
generalizes the Kähler-Ricci flow to the setting of
non-Kähler metrics on complex manifolds,
introduced by Weinkove and myself around 10 years ago. I will give an
overview of known results for
this flow, including a detailed discussion of the case of compact
complex surfaces, and describe some
open problems.
Wednesday,
April 20, 9:30-10:30, Zoom
Meeting
Huangchen Zhou (McGill
University)
Title: Carleman type estimates and uniqueness of
Cauchy problem
Abstract: Carleman estimate
is a weighted
estimate in proving the uniqueness of Cauchy problem. I will review
some results and discuss an explicit example. In this example, we will
see 1. How to get the estimate, 2. How to prove the uniqueness with
this estimate.
Wednesday,
April 27, 9:30-10:30, Zoom
Meeting
Vladmir Sicca (McGill
University)
Title: A prescribed scalar and boundary mean
curvature problem on asymptotically Euclidean manifolds with boundary
Abstract: (Joint work with
Gantumur
Tsogtgerel) We consider the problem of finding a metric in a given
conformal class with prescribed non-positive scalar curvature and
non-positive boundary mean curvature on an asymptotically Euclidean
manifold with inner boundary. We obtain a necessary and sufficient
condition in terms of a conformal invariant of the zero sets of the
target curvatures for the existence of solutions to the problem and use
this result to establish the Yamabe classification of metrics in those
manifolds with respect to the solvability of the prescribed curvature
problem.
Wednesday,
May 4, 9:30-10:30, Zoom
Meeting
Nick McCleerey (University
of Michigan)
Title: Lelong Numbers of m-Subharmonic Functions
Along Submanifolds
Abstract: We study the
possible
singularities of an m-subharmonic function φ along a complex
submanifold V of a compact Kähler manifold, finding a maximal
rate
of growth for φ which depends only on m and k, the codimension
of
V. When k < m, we show that φ has at worst log poles
along V,
and that the strength of these poles is moveover constant along V. This
can be thought of as an analogue of Siu's theorem. This is joint work
with Jianchun Chu.
Wednesday,
May 11, 9:30-10:30, Zoom
Meeting
Carlos Valero (McGill
University)
Title: Stability in the Inverse Steklov Problem for
Warped Products
Abstract: We review some
results from a
recent paper of Daudé, Kamran and Nicoleau in which it is
shown
that approximate knowledge of the Steklov spectrum, that is the
spectrum of the Dirichlet-to-Neumann map, determines a warped product
metric in a neighbourhood of the boundary; some stability estimates on
the metric are also proved. We briefly touch on a possible extension to
the spinor Laplacian.
FALL 2021
Wednesday,
October 20,
12:30-13:30, Zoom
Meeting
Bartosz Syroka (McGill
University)
Title: An overview of the Hull-Strominger system
Abstract: We will present
the basics of
conformally balanced Calabi-Yau manifolds, with a view towards the
Hull-Strominger system of PDEs coming from heterotic string theory. We
will discuss some of the known solutions, including the first
non-Kähler cases such as Goldstein-Prokushkin fibrations.
Finally,
we will look at the approach to the system via the Anomaly flow.
Wednesday,
October 27,
12:30-13:30, Zoom
Meeting
Bartosz Syroka (McGill
University)
Title: The Hull-Strominger system on Riemann surface
fibrations
Abstract: We present the
construction of
generalized Calabi-Gray manifolds, which are Riemann surface fibrations
with hyperkähler fibres. The Anomaly flow reduces to a single
scalar equation for a smooth function on the base surface. We will
consider its properties of long time existence and convergence under
the assumptions of large initial data.
Wednesday, November
3,
12:30-13:30, Zoom
Meeting
Pengfei Guan (McGill
University)
Title: Gauss curvature type flows: an introduction
Abstract: The talk is an
introduction to some
recent results on Gauss curvature type flows in $\mathbb R^{n+1}$. The
main focus is the flow by powers of Gauss curvature
$X_t=-K^{\alpha}\nu, \alpha>0$ in ambient space $\mathbb
R^{n+1}$,
where $\nu$ the outer normal and $K$ the Gauss curvature of the
evolving convex hypersurfaces. After a brief review of curve shorting
flow, we discuss the work of Guan-Ni, Andrews-Guan-Ni on entropies
associated to the flows. The crucial entropy point estimate enable us
to control the lower bound of the support function of normalized flow
and deduce the convergence of the flow to solitons. Finally, we present
the main arguments of the proof of beautiful uniqueness theorem of
solitons for $\alpha>\frac{1}{n+2}$ by
Brendle-Choi-Daskalopoulos.
Wednesday,
November 10,
12:30-13:30, Zoom
Meeting
Min Chen (McGill
University)
Title: Flow by powers of the Gauss curvature in
space forms
Abstract: In this talk, we
consider flow by
powers of the Gauss curvature in space forms
$\mathbb{N}^{n+1}(\kappa)$. Our approach to this flow (in space forms)
is to deduce it to a flow in the Euclidean space by proper projections.
The key in our proof is an almost monotonicity formula for associated
entropies considered in Guan-Ni and Andrews-Guan-Ni. We could obtain a
new monotone quantity
$\mathcal{E}_{\alpha}(\hat{\Omega}_t)+C(n,\alpha,\tilde{X}_0)e^{-\frac{2(n+1)}{2n+1}t}$
along the normalized flow by modifying the monotone quantity used in
Euclidean space. This allows us to extend the known results in
Euclidean space to space forms completely.
Wednesday, November
17,
12:30-13:30, Zoom
Meeting
Jiuzhou Huang (McGill
University)
Title: Flow by powers of the Gauss curvature in
space forms (continued)
Abstract: This is a
continuation of
Chen’s talk last week about our paper for flows by powers of
the
Gauss curvature in space forms. I this talk, I will go into more
details of the proof. The focus will be on the difference between the
space forms and the Euclidean space.
Wednesday, November
24,
12:30-13:30, Zoom
Meeting
Carlos Valero (McGill
University)
Title: A Calderon problem for U(1)-connections
coupled to spinors
Abstract: We introduce the
Dirichlet-to-Neumann (DN) map for the Dirac Laplacian coupled to a
U(1)-connection A on a spin manifold with boundary, and show that it is
a pseudodifferential operator of order 1 whose symbol determines the
metric and connection to infinite order at the boundary. We go on to
show that A can be recovered up to gauge equivalence from the DN map in
the real analytic category if it satisfies a Yang-Mills-Dirac equation
in the interior.
Wednesday, December
8,
12:30-13:30, Zoom
Meeting
Valentino Tosatti (McGill
University)
Title: Immortal solutions of the
Kähler-Ricci flow
Abstract: I will discuss
what is known,
not known, and conjectured about solutions of the Kähler-Ricci
flow on compact Kähler manifolds which exist for all positive
time.
WINTER 2021
Wednesday,
February 24,
13:30-14:30, Zoom
Meeting
Bartosz Syroka (McGill
University)
Title: Quasi-local mass in spacetimes
Abstract: We will present
some basic ideas of
general relativity, and the problem of calculating mass and energy
quantities associated to regions of spacetime. To this end, we will
explain the equations of curvature for submanifolds of Lorentzian
manifolds, and conformal Killing-Yano tensors which allow us access to
Minkowski curvature formulas in the spacetimes which admit them.
Wednesday,
March 10,
13:30-14:30, Zoom
Meeting
Gábor
Székelyhidi (University of Notre Dame)
Title: Uniqueness of certain cylindrical tangent
cones
Abstract: Leon Simon showed
that if an area minimizing hypersurface
admits a cylindrical tangent cone of the form C x R, then this tangent
cone is unique for a large class of minimal cones C. One of the
hypotheses in this result is that C x R is integrable and this
excludes the case when C is the Simons cone over S^3 x S^3. The main
result in this talk is that the uniqueness of the tangent cone holds
in this case too. The new difficulty in this non-integrable situation
is to develop a version of the Lojasiewicz-Simon inequality that can
be used in the setting of tangent cones with non-isolated
singularities.
Wednesday,
March 17,
13:30-14:30, Zoom
Meeting
Sébastien Picard (University
of British Columbia)
Title: Metrics Through Non-Kahler Transitions
Abstract: It was proposed by
Clemens, Friedman
and Reid to connect Calabi-Yau threefolds of different topologies by an
operation known as a conifold transition. However, this process may
produce a non-Kahler complex manifold with trivial canonical bundle. We
will consider conifold transitions from the point of view of
differential geometry and discuss passing special metrics through a
non-Kahler transition. This is joint work with T.C. Collins and S.-T.
Yau.
Wednesday,
March 24,
13:30-14:30, Zoom
Meeting
Jianchun Chu (Northwestern
University)
Title: The k-Ricci curvature in Kahler geometry
Abstract: In 2018, Lei Ni
introduced the
definition of k-Ricci curvature, which can be regarded as a natural
generalization of holomorphic sectional curvature and Ricci curvature.
There are some connections between these curvatures and the properties
of the underlying manifold. In this talk, I will show that a compact
Kahler manifold with quasi-negative k-Ricci curvature is projective.
This is a joint work with Man-Chun Lee and Luen-Fai Tam.
Wednesday,
March 31,
13:30-14:30, Zoom
Meeting
Marc-Andrew Lavigne (McGill
University)
Title: Concentration Compactness Principle for
Variable Exponent Spaces
Abstract: In 1985, Lions
published his paper
on the concentration compactness principle, which became very useful
when proving existence of solutions for PDE with critical growth (with
respect to Sobolev embeddings). In the fields of electro-rheological
fluids and image processing, the need for variable exponents in PDEs
gave rise to many new questions. Several results were obtained for
nonlinear elliptic equations when the growth is subcritical. As for the
critical case, a concentration compactness principle for variable
exponents proves again very useful. In this talk, I will prove this
principle by combining the proofs of Fu (2009) and of Bonder and Silva
(2010) into a shorter one.
Wednesday,
April 7,
13:30-14:30, Zoom
Meeting
Valentino Tosatti (McGill
University)
Title: Higher order estimates for collapsing complex
Monge-Ampère equations
Abstract: I will consider a
family of
complex Monge-Ampère equations on a compact Calabi-Yau
manifold
which has a fibration structure, with fiber size that is shrinking to
zero, whose solutions of these equations give Ricci-flat
Kähler
metrics on the total space. Each equation is uniquely solvable by
classical work of Yau, but understanding their asymptotic behavior as
the fiber size shrinks is a very challenging problem. I will discuss
very recent work with Hans-Joachim Hein where we prove a priori Ck
estimates for all k, away from the singular fibers of the fibration.
This is a consequence of an aymptotic expansion for the solution, which
relies on a new analytic method where each additional term of the
expansion arises as the obstruction to proving a uniform bound on one
additional derivative of the remainder.
Wednesday,
April 14,
13:30-14:30, Zoom
Meeting
Jérôme
Vétois (McGill University)
Title: Sign-changing blow-up for the Moser-Trudinger
equation
Abstract: In this talk, we
will discuss a
question of stability for the energy levels of the Moser-Trudinger
functional on a smooth, bounded domain of R^2. This functional involves
an exponential non-linearity, which is critical with respect to Sobolev
embeddings. We will present an existence result of sign-changing
blowing-up solutions, which stands in sharp contrast to the
quantization result for positive solutions, recently obtained by
Olivier Druet and Pierre-Damien Thizy. This is a joint work with Luca
Martinazzi (University of Padua) and Pierre-Damien Thizy (University of
Lyon).
Wednesday,
May 5,
13:30-14:30, Zoom
Meeting, joint
with the Analysis seminar
Frédéric
Naud (Sorbonne Université)
Title: The spectral gap of random hyperbolic surfaces
Abstract: We will start with
a survey on (some
very recent) results on the low spectrum of "random" compact hyperbolic
surfaces, for various models including discrete and continuous
Teichmuller spaces. We will then give some ideas of the proofs by
emphasizing the analogy with radom graphs. We will also dicuss non
compact situations where similar results on resonances can be obtained.
Joint works with Michael Magee and Doron Puder.
Wednesday,
May 19,
13:30-14:30, Zoom
Meeting, joint
with the Analysis seminar
Laurent Moonens (Université
Paris-Saclay)
Title: Solving the divergence equation with measure
data in non-regular domains
Abstract: In this talk, we
shall present a
recent joint work with E. Russ (Grenoble) concerning the equation
$\mathrm{div}\,v=\mu$ in a (rather general) open domain $\Omega$, where
$\mu$ is a (signed) Radon measure in $\Omega$ satisfying
$\mu(\Omega)=0$. We show in particular that, under mild assumptions on
the geometry of $\Omega$ (and some assumptions on $\mu$), one can
provide a constructive way to build solutions $v$ in a weighted
$L^\infty$ space enjoying weak Neumann-type boundary conditions.
FALL 2020
Wednesday,
October 7,
13:30-14:30, Zoom
Meeting
Moritz Reintjes (University
of Konstanz)
Title: Uhlenbeck compactness and optimal regularity
in Lorentzian geometry
Abstract: We resolve two
problems of Mathematical Physics. First,
we prove that any L^\infty connection \Gamma on the tangent bundle of
an arbitrary
differentiable manifold with L^\infty Riemann curvature can be smoothed
by
coordinate transformation to optimal regularity \Gamma\in W^{1,p} (one
derivative smoother than the curvature), any p<\infty. This
implies in particular that Lorentzian metrics of shock wave solutions
of the Einstein-Euler
equations are non-singular-geodesic curves, locally inertial
coordinates
and the resulting Newtonian limit all exist in a classical sense. This
result is based on a system of nonlinear elliptic partial differential
equations, the Regularity Transformation equations, and an existence
theory
for them at the level of L^\infty connections. Secondly, we prove that
this
existence theory suffices to extend Uhlenbeck compactness from the
case
of connections on vector bundles over Riemannian manifolds, to the case
of connections on the tangent bundle of arbitrary manifolds, including
Lorentzian manifolds of General Relativity.
Wednesday, October
21,
13:30-14:30, Zoom
Meeting
Fengrui
Yang (McGill
University)
Title: Prescribed curvature measure problem in
hyperbolic space
Abstract: The problem of the
prescribed curvature measure is one of the important problems in
differential geometry and nonlinear partial differential equations. In
this talk, we are going to talk about our recent result about
prescribed curvature measure problem in hyperbolic space. We obtained
the existence of star-shaped k-convex bodies with prescribed (n-k)-th
curvature measures (k<n) by establishing crucial C^2 regularity
estimates for solutions to the corresponding fully nonlinear PDE in the
hyperbolic space.
Wednesday, October
28,
13:30-14:30, Zoom
Meeting
Jiawei
Liu (McGill
University)
Title: The Kähler-Ricci flows with cusp
singularity
Abstract: In this talk, I
will talk about the
existence, uniqueness and convergence of the Kähler-Ricci flow
with cusp singularity on a compact Kähler manifold M which
carries
a smooth hypersurface D such that the twisted canonical bundle K_M+D is
ample. We deduce this flow by limiting a sequence of conical
Kähler-Ricci flows as the cone angles tend to zero.
Wednesday, November 11,
13:30-14:30, Zoom
Meeting
Valentino
Tosatti (McGill
University)
Title: Regularity of envelopes on Kähler
manifolds
Abstract: I will give an
introduction to the topic of envelopes of quasi-plurisubharmonic
functions (also known as extremal functions) on compact Kähler
manifolds. I will discuss the optimal C1,1
regularity of such envelopes, by approximating the envelope by a family
of complex Monge-Ampère equations and using an a priori C1,1
estimate developed by Chu, Weinkove and myself. I will also mention
some related open questions.
Wednesday, November 18,
13:30-14:30, Zoom
Meeting
Bruno Premoselli
(ULB,
Brussels)
Title: Towers of bubbles for Yamabe-type equations
in dimensions larger than 7
Abstract: In this talk we
consider
perturbations of Yamabe-type equations on closed Riemannian manifolds.
In dimensions larger than 7 and on locally conformally flat manifolds
we construct blowing-up solutions that behave like towers of bubbles
concentrating at a critical point of the mass function. Our result does
not assume any symmetry on the underlying manifold. We perform our
construction by combining finite-dimensional reduction methods with a
linear blow-up analysis. Our approach works both in the positive and
sign-changing case. As an application we prove the existence, on a
generic bounded open set of R^n, of blowing-up solutions of the
Brézis-Nirenberg equation that behave like towers of bubbles
of
alternating signs.
Wednesday, November
25,
13:30-14:30, Zoom
Meeting
Vladmir
Sicca (McGill
University)
Title: A prescribed scalar and boundary mean
curvature problem on compact manifolds with boundary
Abstract: In this talk I
will present our
recent result in the problem of finding a metric in a given conformal
class with prescribed nonpositive scalar curvature and nonpositive
boundary mean curvature, on a compact manifold with boundary. We
established a necessary and sufficient condition in terms of a
conformal invariant that measures the zero set of the target
curvatures, which we call the relative Yamabe invariant of the set.
Wednesday, December 2,
13:30-14:30, Zoom
Meeting
Jiuzhou
Huang (McGill
University)
Title: Approximation of convex surfaces by Ricci
flow
Abstract: Ricci flow has
been an important
tool in geometric analysis since Hamilton’s seminal paper in
1982. In this talk, we are going to discuss an approximation of general
convex surfaces by Ricci flow and mention some of its possible
applications.
WINTER 2020
Friday, January 17,
13:30-14:30,
Burnside Hall 1104,
joint
with the Analysis seminar
Henrik Matthiesen (University
of Chicago)
Title:
Handle attachment and the normalised first eigenvalue
Abstract: I will discuss
asymptotic lower
bounds of the first eigenvalue for two constructions of attaching
degenerating handles to a given closed Riemannian surface. One of these
constructions is relatively simple but often fails to strictly increase
the first eigenvalue normalized by area. Motivated by this negative
result, we then give a much more involved construction that always
strictly increases the first eigenvalue normalized by area.
As a consequence we obtain the existence of a metric that maximizes the
first eigenvalue among all unit area metrics on a given closed surface.
This is based on joint work with Anna Siffert.
Wednesday, January 22,
13:30-14:30,
Burnside Hall 920
Dmitry
Jakobson (McGill University)
Title:
Zero and negative eigenvalues of conformally covariant operators, and
nodal sets in conformal geometry
Abstract: We first review
some old results about conformal invariants that arise from nodal sets
and negative eigenvalues of conformally covariant operators, as well as
applications to curvature prescription problems. Next, we discuss
related results on manifolds with boundary. We relate Dirichlet and
Neumann eigenvalues for conformally covariant boundary value problems.
If time permits, we shall discuss related results for boundary
operators of arbitrary order, as well as for weighted graphs. This is
joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M.
Levitin, M. Karpukhin, G. Cox and Y. Sire.
Wednesday, January 29,
13:30-14:30,
Burnside Hall 920
Jérôme
Vétois (McGill
University)
Title:
Blowing-up solutions for critical elliptic equations in low dimensions:
the impact of the mass and the scalar curvature
Abstract: In this talk, we
will consider the
question of existence of positive blowing-up solutions to a class of
elliptic equations with critical Sobolev growth on a closed Riemannian
manifold. A result of Olivier Druet provides necessary conditions for
the existence of such solutions. We will present new results showing
the optimality of Druet's conditions. We will see that the scalar
curvature of the manifold plays a crucial role in this question.
Furthermore, we will give special attention to the case of dimensions 4
and 5, where a mass term arises and plays an important role in the
analysis. This is a joint work with Frédéric
Robert
(Université de Lorraine).
Wednesday, February 5,
13:30-14:30,
Burnside Hall 920
Pengfei
Guan (McGill University)
Title:
Locally constrained flows, isoperimetric type inequalities, and open
problems
Abstract: We discuss a new
type of
hypersurface flows with constraints. This type of flows enjoy certain
monotonicity properties which make them a natural PDE tool to prove
various isoperimetric geometric inequalities. Yet, there are several
challenging problems for the longtime existence and regularity of these
flows. We will discuss their background and open problems arising from
these new flows. The talk is aiming for graduate students and young
researchers.
Wednesday,
February 19, 13:30-14:30,
Burnside Hall 920
Siyuan Lu
(McMaster University)
Title:
Monge-Ampere equation with bounded periodic data
Abstract: We consider the
Monge-Ampere
equation det(D^2u) = f in R^n, where f is a positive bounded periodic
function. We prove that u must be the sum of a quadratic polynomial and
a periodic function. For f =1, this is the classic result by Jorgens,
Calabi and Pogorelov. For f \in C^\alpha, this was proved by Caffarelli
and Li. This is a joint work with Y.Y. Li.
Wednesday, February 26,
13:30-14:30,
Burnside Hall 920
Niky
Kamran (McGill
University)
Title:
Solving the Einstein equations holographically
Abstract: We will discuss
the problem of
solving the Einstein equations with boundary data at infinity close to
the conformal infinity of anti-de-Sitter space and present some recent
well-posedness results obtained in collaboration with Alberto Enciso
(ICMAT). We will also list some open problems. The talk will have a
significant introductory component and should be of interest to
graduate students and post-docs.
Wednesday, March 11,
13:30-14:30,
Burnside Hall 920
Bartosz
Syroka (McGill University)
Title:
Minkowski formulas and quasi-local mass
Abstract: The classical
Minkowski formulas for
curvature were extended to spacetimes in a paper of Wang, Wang, and
Zhang, making use of the presence of conformal Killing-Yano tensors. We
will look at the use of Minkowski formulas in spacetime to analyze a
quasi-local mass definition given by Wang and Yau. We will then
consider some applications to rigidity theorems if time allows.
Wednesday, April 29,
13:30-14:30, Zoom Meeting,
joint
with the Analysis seminar
Julian Scheuer (University
of Freiburg)
Title:
Concavity of solutions to elliptic equations on the sphere
Abstract: An important
question in PDE is when
a solution to an elliptic equation is concave. This has been of
interest with respect to the spectrum of linear equations as well as in
nonlinear problems. An old technique going back to works of Korevaar,
Kennington and Kawohl is to study a certain two-point function on a
Euclidean domain to prove a so-called concavity maximum principle with
the help of a first and second derivative test. To our knowledge, so
far this technique has never been transferred to other ambient spaces,
as the nonlinearity of a general ambient space introduces geometric
terms into the classical calculation, which in general do not carry a
sign.
In this talk we have a look at this situation on the unit sphere. We
prove a concavity maximum principle for a broad class of degenerate
elliptic equations via a careful analysis of the spherical Jacobi
fields and their derivatives. In turn we obtain concavity of solutions
to this class of equations. This is joint work with Mat Langford,
University of Tennessee Knoxville.
Wednesday, May 6,
13:30-14:30, Zoom
Meeting
Jiawei
Liu (McGill
University)
Title:
Stability of the Conical Kähler-Ricci flows on Fano manifolds
Abstract: I will talk about
the stability of
the conical Kähler-Ricci flows on Fano manifolds. That is, if
there exists a conical Kähler-Einstein metric with cone angle
2πβ along the divisor, then for any β'
sufficiently close
to β, the corresponding conical Kähler-Ricci flow
converges
to a conical Kähler-Einstein metric with cone angle
2πβ'
along the divisor. As corollaries, we give parabolic proofs of
Donaldson's openness theorem and his existence conjecture for the
conical Kähler-Einstein metrics with positive Ricci
curvatures.
Wednesday, May
13,
13:30-14:30, Zoom
Meeting
Sébastien
Picard (Harvard University)
Title:
Non-Kahler Calabi-Yau manifolds and nonlinear PDE
Abstract: We will discuss a
certain class of
manifolds introduced by string theorists C. Hull and A. Strominger.
These spaces are non-Kahler Calabi-Yau threefolds. We propose to study
this geometry by using the Anomaly flow, which is a nonlinear flow of
non-Kahler metrics. This talk will contain joint work with T. Fei, D.H.
Phong, and X.-W. Zhang.
Wednesday, May
20,
13:30-14:30, Zoom
Meeting
Frédéric
Robert (Université de Lorraine)
Title:
Impact of localization of the Hardy potential on the stability of
Pohozaev obstructions
Abstract: The Pohozaev
obstruction yields a
sufficient condition (say (C)) for the absence of positive solutions to
critical nonlinear elliptic equations on domains of the flat space.
On the round sphere, this condition is essentially the Kazdan-Warner
obstruction.
Condition (C) is not stable under reasonable perturbations of the
potential. However, for the classical Brezis-Nirenberg problem, the
absence of solutions is preserved in small dimension only, see
Druet-Laurain. In this talk, I will address the same issue when a
Hardy-type potential is added. When the Hardy potential is centered in
the interior of the domain, there is also a small dimensions
phenomenon. Surprisingly, when the Hardy potential is centered on the
boundary of the domain, the obstruction is satble, whatever the
dimension.
Wednesday, May
27,
13:30-14:30, Zoom
Meeting
Edward
Chernysh (McGill
University)
Title:
A Compactness Theorem for Weighted Critical p-Laplace Equations
Abstract: We investigate the
compactness of
Palais-Smale sequences for a class of critical p-Laplace equations with
weights. More precisely, we establish a Struwe-type decomposition
result for Palais-Smale sequences extending a recent result of
Mercuri-Willem to weighted equations. In sharp contrast to the model
case of the critical p-Laplace equation, all bubbling must occur at the
origin.
Wednesday, June 3,
13:30-14:30, Zoom Meeting,
joint
with the Analysis seminar
Sagun Chanillo
(Rutgers University)
Title:Bourgain-Brezis inequalities, applications and
Borderline
Sobolev inequalities on Riemannian Symmetric spaces of non-compact type.
Abstract: Bourgain and
Brezis discovered a
remarkable inequality which is borderline for the Sobolev inequality in
Eulcidean spaces. In this talk we obtain these inequalities on
nilpotent Lie groups and on Riemannian symmetric spaces of non-compact
type. We obtain applications to Navier Stokes eqn in 2D and to
Strichartz inequalities for wave and Schrodinger equations and to the
Maxwell equations for Electromagnetism. These results were obtained
jointly with Jean Van Schaftingen and Po-lam Yung.
FALL 2019
Wednesday,
August 7,
13:30-14:30,
Burnside Hall 1104
Karoly
Boroczky
(Central
European
University, Budapest)
Title:
The logarithmic Minkowski problem and some relatives
Abstract: Some exciting
recent developments are surveyed concerning the celebrated logarithmic
Brunn-Minkowski conjecture and the related conjectured
uniqueness of the solution of the even logarithmic Minkowski
problem.
Friday,
August 9,
13:30-14:30,
Burnside Hall 1104
, joint
with the Analysis seminar
Ali
Aleyasin (Stony
Brook University)
Title:
Singular and degenerate Monge-Ampère equations
Abstract: It is well known
that several
non-linear elliptic partial differential equations have applications in
various fields of geometry and analysis, including but not limited to
the Calabi problem, the Weyl, and Minkowski problems, and optimal
transport. An important class of such non-linear equations are the real
and complex Monge-Ampère equations. Although the case of
strictly elliptic equations with smooth source term has been rather
well-understood, the behaviour of solutions in the vicinity of possible
singularities or degeneracies of the source term is far from being
understood. This corresponds to the vanishing or blowing up of the
prescribed curvature in the Weyl problem. In this talk, I will present
an application of a differential geometric approach to the study of
certain singularities and degeneracies of elliptic complex
Monge-Ampère equations. This approach will allow new
estimates
for solutions to be derived. I shall also outline how the idea works in
the case of several other important geometric partial differential
equations.
Wednesday, August
21,
13:30-14:30,
Burnside Hall 1104
Guofang
Wang (University
of Freiburg)
Title:
Alexandrov-Fenchel type inequalities and applications
Abstract: In the first part of
this talk we review our previous work on Alexandrov-Fenchel
inequalities and weighted Alexandrov-Fenchel inequalities in the
hyperbolic space and applications on a higher order hyperbolic, which
leads to a Penrose type inequality. In the second part, we prove the
remaining open case of weighted Alexandrov-Fenchel inequalities and
introduce an extrinsic mass. For this mass, we also obtain a Penrose
type inequality.
Wednesday, August
21,
14:40-15:40,
Burnside Hall 1104
Julian
Scheuer (University
of Freiburg)
Title:
New estimates for the Willmore flow and applications
Abstract: The Minkowski
inequality for a mean-convex closed surface in three-dimensional
Euclidean space provides a lower bound on the total mean curvature in
terms of surface area, which is optimal on round spheres. Until today,
it is only known to be valid under additional assumptions, for example
starshapedness or outward minimality. It remains open, if these
additional assumptions can be dropped.
In this talk we prove a so-called "almost-Minkowski-inequality" within
the class of closed surfaces with L^2-small traceless second
fundamental form. No further curvature assumptions are made. This
result is based on new asymptotic estimates for the Willmore flow with
small energy, which were achieved in a recent joint work with Ernst
Kuwert.
Further direct applications will be discussed as well.
Wednesday, October 23,
13:30-14:30,
Burnside Hall 920
Thierry
Daudé (Université de
Cergy-Pontoise)
Title:
Some non-uniqueness results in the Calderon inverse problem with local
or disjoint data
Abstract: In dimension 3 or higher, the anisotropic Calderon
inverse problem amounts to recovering a Riemannian metric on a compact
connected manifold with boundary from the knowledge of the Dirichlet to
Neumann operator (modulo diffeomorphisms that fix the boundary). In
this talk, I will prove that there is non uniqueness in the Calderon
problem when :
1) the Dirichlet and Neumann data are measured on the same proper
subset of the boundary provided the metric is only Holder continuous.
2) the Dirichlet and Neumann data are measured on distinct subsets of
the boundary (for smooth metrics).
This is a joint work with N. Kamran (McGill) and F. Nicoleau (Nantes).
Wednesday, October 30,
13:30-14:30,
Burnside Hall 920
Jiuzhou
Huang (McGill University)
Title:
A normal mapping method for the isoperimetric inequality
Abstract: In this talk, I will introduce a proof of the
isoperimetric inequality by using the normal mapping. I'll illustrate
this method via Brendle's paper "The isoperimetric inequality for a
minimal submanifold in Euclidean space".
Wednesday, November
6,
13:30-14:30,
Burnside Hall 920
Spiro
Karigiannis (University
of Waterloo)
Title:
Towards higher dimensional Gromov compactness in $G_2$ and $Spin(7)$
manifolds
Abstract: Let $(M, \omega)$ be a compact symplectic manifold.
If
we choose a compatible almost complex structure $J$ (which in general
is not integrable) then we can study the space of $J$-holomorphic maps
$u : \Sigma \to (M, J)$ from a compact Riemann surface into $M$. By
appropriately “compactifying” the space of such
maps, one
can obtain powerful global symplectic invariants of $M$. At the heart
of such a compactification procedure is understanding the ways in which
sequences of such maps can degenerate, or develop singularities.
Crucial ingredients are conformal invariance and an energy identity,
which lead to to a plethora of analytic consequences, including: (i) a
mean value inequality, (ii) interior regularity, (iii) a removable
singularity theorem, (iv) an energy gap, and (v) compactness modulo
bubbling.
Riemannian manifolds with closed $G_2$ or $Spin(7)$ structures share
many similar properties to such almost Kahler manifolds. In particular,
they admit analogues of $J$-holomorphic curves, called associative and
Cayley submanifolds, respectively, which are calibrated and hence
homologically volume-minimizing. A programme initiated by
Donaldson-Thomas and Donaldson-Segal aims to construct similar such
“counting invariants” in these cases. In 2011, a
somewhat
overlooked preprint of Aaron Smith demonstrated that such submanifolds
can be exhibited as images of a class of maps $u : \Sigma \to M$
satisfying a conformally invariant first order nonlinear PDE analogous
to the Cauchy-Riemann equation, which admits an energy identity
involving the integral of higher powers of the pointwise norm $|du|$. I
will discuss joint work with Da Rong Cheng (Chicago) and Jesse Madnick
(McMaster) in which we establish the analogous analytic results of
(i)-(v) in this setting. arXiv:1909.03512
Wednesday, November
13,
13:30-14:30,
Burnside Hall 920
François
Fillastre (Université de Cergy-Pontoise,
UMI-CRM)
Title:
Equivariant Minkowski problem in Minkowski space
Abstract: We present a class of convex bodies, which are
invariant under the action of affine deformations of cocompact lattices
of SO(n,1). They appear naturally in general relativity. Classical
geometric problems can be brung into that setting, and their intrinsic
formulations are on compact hyperbolic manifolds rather than on the
round sphere. In dimension (2+1), those convex sets are related to the
tangent space of Teichmueller space.
Wednesday, November
20,
13:30-14:30,
Burnside Hall 920
Fengrui
Yang (McGill University)
Title:
W2,1 regularity for solutions of the Monge Ampere equation
Abstract: In this talk, I will introduce the thoughts and
ideas
of how to prove W2,1 regularity for Monge Ampere equation. This comes
from a paper written by Guido De Philippis and Alessio Figalli.
Wednesday, November
27,
13:30-14:30,
Burnside Hall 920
Vladmir
Sicca Goncalves (McGill University)
Title:
The Yamabe Invariant of a Measurable Set
Abstract: In this talk I'll present the construction of a
Yamabe
invariant for an arbitrary measurable set in an asymptotically
Euclidean manifold. If time allows, I'll also hint how that can help
solve the prescribed scalar curvature problem in this setting. The talk
is based on the paper "Yamabe Classification and Prescribed Scalar
Curvature in the Asymptotically Euclidean Setting" by Dilts and Maxwell
(2015).
WINTER 2019
Thursday,
January 24,
14:00-15:00,
joint seminar at Concordia University LB 921-4
Almut
Burchard
(University
of
Toronto)
Title:
A geometric stability result for
Riesz-potentials
Abstract: Riesz'
rearrangement inequality
implies
that integral functionals (such as the Coulomb energy of a
charge distribution) that are defined by a pair interaction potential
(such as the Newton potential) which decreases with distance
are maximized (under appropriate constraints)
only by densities that are radially decreasing about some
point. I will describe recent and ongoing work
with Greg Chambers on the stability of this inequality for the special
case of the Riesz-potentials in n dimensions (given by the kernels
|x-y|^-(n-s), for densities that
are uniform on a set of given volume. For 1<s<n we
bound
the square of the symmetric difference of a set from a ball
by the difference in energy of the corresponding uniform
distribution from that of the ball.
Wednesday,
January 30,
13:30-14:30,
Burnside Hall 1104
Wubin
Zhou (Tongji
University, China)
Title:
The Existence of Constant Scalar Curvature PMY Type Kähler
Metrics
Abstract: Let
M be a compact Kähler manifold and N
consist some points of M. In this talk, we will discuss the existence
of constant scalar curvature PMY type Kähler metrics on
non-compact Kähler manifold M-N.
Wednesday,
February 6,
13:30-14:30,
Burnside Hall 1104
Pengfei
Guan (McGill
University)
Title:
Some open problems in geometric analysis
Abstract: We will
discuss some open problems in geometric analysis, related to
isoperimetric type inequalities, isometric embedding problems, rigidity
problems, evolution of hypersurfaces, and regularity of nonlinear
elliptic and parabolic equations. Main emphasis is the role of
nonlinear PDEs in geometric settings.
Wednesday,
February 20,
13:30-14:30,
Burnside Hall 1104
Edward
Chernysh (McGill
University)
Title: A
global compactness result for p-Laplace equations
with critical nonlinearities
Abstract: In this talk,
we will discuss a representation
theorem of Mercuri-Willem (2010) for Palais-Smale
sequences involving critical p-Laplace equations in smoothly
bounded domains with negative parts vanishing at infinity.
Wednesday,
February 27,
13:30-14:30,
Burnside Hall 1104
Changyu
Ren (Jilin
University, China)
Title:
An inequality for C^2 estimates to k-Hessian equations (Case
k=3,n=5)
Abstract: In this talk,
I will introduce an inequality about
the C^2 estimates to k-Hessian equations. We can show the
inequality is true in case k=3, n=5. For general cases,
further
discussions are needed.
Wednesday,
March 13,
13:30-14:30,
Burnside Hall 1104
Jiuzhou
Huang (McGill
University)
Title: A
subsolution method in Monge-Ampere equation
Abstract: In this talk,
I will give a brief introduction
to a subsolution method in Guan-Spruck's
paper "Boundary-value Problems on S^n for surfaces of constant
Gauss Curvature.
Thursday,
March 14,
15:00-16:00,
Burnside Hall 1214
Valentino
Tosatti (Northwestern
University)
Title:
Higher order estimates for collapsing Ricci-flat metrics
Abstract: I will
discuss the problem of obtaining uniform
C^k estimates for solutions of a degenerating family of complex
Monge-Ampere equations where the ellipticity is degenerating along the
fibers of a fibration. Geometrically, the solutions of this
family
of PDE give Ricci-flat Kahler metrics on a Calabi-Yau
manifold with a holomorphic fibration onto a lower-dimensional
space. I will describe new estimates that prove a uniform C^{2,alpha}
bound in general, and C^k bounds for all k when the smooth fibers are
all pairwise isomorphic. This is joint work with H.-J. Hein.
Thursday,
March 21, 15:00-16:00,
Burnside Hall 1214
Sergio
Almaraz (Fluminense
Federal University,
Brazil)
Title: The
mass of asymptotically
hyperbolic manifolds with non-compact boundary
Abstract: We define
a mass-type geometric
invariant for Riemannian manifolds asymptotic to the hyperbolic
half-space and prove a positive mass theorem for spin manifolds. This
is a joint work with Levi Lima (UFC-Brazil).
Wednesday,
March 27,
13:30-14:30,
Burnside Hall 1104
Saikat
Mazumdar (McGill
University)
Title: Q-curvature,
Paneitz operator and
a maximum principle
Abstract: In this
talk, I will discuss the
higher-order version of the Yamabe problem:
"Given a compact Riemannian manifold (M,g), does there exist a metric
conformal
to g with constant Q-curvature"?
The behaviour of Q-curvature under conformal changes of the metric is
governed
by certain conformally covariant powers of the Laplacian. The problem
of prescribing the Q-curvature in a conformal class then amounts to
solving a nonlinear elliptic
PDE involving the powers of Laplacian called the GJMS operator. In
general the
explicit form of this GJMS operator is not explicitly known. This
together with
a lack of maximum principle for polyharmonic operators makes the
problem challenging. In this talk, I will mainly focus on the
biharmonic case and survey some
recent developments.
Wednesday,
April 10,
13:30-14:30,
Burnside Hall 1104
Liangming
Shen (McGill
University)
Title: A
compactness result along a general continuity path in the study of
Kahler-Einstein problem on Fano manifolds
Abstract: Recent
years, Tian and CDS
proved the folklore Yau-Tian-Donaldson conjecture based on the study in
the continuity path of conical Kahler-Einstein metrics.
After that G. Szekelyhidi showed that similar work could be established
along Aubin's continuity path. In this talk I will consider a more
general continuity path mixed with conic singularities and a torsion
term. I will focus on the compactness along the continuity path and
show the geometric structure of the limit space. If time permits I will
briefly discuss how these results lead to a new proof of
Yau-Tian-Donaldson conjecture based on this general continuity path.
This is joint with Feng and Ge.
Wednesday,
April 17,
13:30-14:30,
Burnside Hall 1104
Fengrui
Yang (McGill
University)
Title: Isoperimetric
inequalities for
quermassintegrals of k-star shaped domains
Abstract: This
content comes from a
lecture note of Prof. Guan. Here I will briefly introduce the idea of
how to solve Isoperimetric inequalities for quermassintegrals of k-star
shaped domains.
Wednesday,
May 1st,
13:30-14:30,
Burnside Hall 1104
Ronan
Conlon
(Florida
International University)
Title: Classification
Results for Expanding and Shrinking gradient Kahler-Ricci solitons
Abstract: A complete
Kahler metric g on a
Kahler manifold M is a "gradient Kahler-Ricci soliton" if there exists
a smooth real-valued function f:M-->R with \nabla f holomorphic
such
that Ric(g)-Hess(f)+\lambda g=0 for \lambda a real number. I will
present some classification results for such manifolds. This is joint
work with Alix Deruelle (Université Paris-Sud) and Song Sun
(UC
Berkeley).
Friday,
May 3,
13:30-14:30,
Burnside Hall 1104
, joint
with the Analysis seminar
Luca
Martinazzi (University
of
Padua, Italy)
Title: News
on the Moser-Trudinger
inequality: from sharp
estimates to the Leray-Schauder degree
Abstract: The
existence of critical points
for the Moser-Trudinger inequality for large energies has been open for
a long time. We will first show how a collaboration with G. Mancini
allows to recast the Moser-Trudinger inequality and the existence of
its extremals (originally due to L. Carleson and A. Chang) under a new
light, based on sharp energy estimate. Building upon a recent subtle
work of O. Druet and P-D. Thizy, in a work in progress with O. Druet,
A. Malchiodi and P-D. Thizy, we use these estimates to compute the
Leray-Schauder degree of the Moser-Trudinger equation (via a suitable
use of the Poincaré-Hopf theorem), hence proving that for
any
bounded non-simply connected domain the Moser-Trudinger inequality
admits critical points of arbitrarily high energy. In a work in
progress with F. De Marchis, O. Druet, A. Malchiodi and P-D. Thizy, we
expect to use a variational argument to treat the case of a closed
surface.
Wednesday,
May 15,
13:30-14:30,
Burnside Hall 1104
Bruno
Premoselli
(ULB,
Brussels)
Title: Existence
of infinitely many solutions for the
Einstein-Lichnerowicz system
Abstract: We will
consider in this talk the
Einstein-Lichnerowicz
system of equations. It originates in General Relativity as a way to
determine initial-data sets for the evolution problem. This system
takes
the form of a strongly coupled, supercritical, nonlinear system of
elliptic PDEs. We will investigate its blow-up properties and show
that,
under some assumptions on the physics data, it possesses a non-compact
family of solutions. This family of solutions will be constructed by
combining toplogical methods with a finite-dimensional reduction
approach; due to the non-variational structure of the system, the
latter
has to be carried on in strong spaces and relies of a priori blow-up
estimates that we shall describe.
FALL 2018
Wednesday,
August 8,
13:30-14:30, Burnside Hall 920
Qun Li
(Wright State University)
Title:
The Constant Rank Theorems in Complex Geometry
Abstract:
In this talk, we will present some constant rank
results on the Hessian of solutions to some fully nonlinear equations
in complex variables. We will then discuss some geometric applications
including the connections to some Hermitian curvature flows.
Wednesday,
August 15,
13:30-14:30, Burnside
Hall 920
Yiyan
Xu
(Nanjing University)
Title:
Classification of shrinking Ricci solitons
Abstract:
We will discuss some classification results for
shrinking Ricci solitons. In particular, we will present new proofs of
some known results from the equation point of view.
Wednesday,
August 22,
13:30-14:30, Burnside
Hall 920
Yu Yuan
(University of Washington)
Title: Hessian
estimates for convex solutions to quadratic
Hessian equations
Abstract:
We present Hessian estimates for semi-convex
solutions to quadratic Hessian equations by compactness arguments. This
is based on a new strip argument, a known constant rank theorem, and
also a joint work with Chang on Liouville type results for quadratic
Hessian equation in general dimensions. This is joint work with
McGonagle and Song.
Thursday,
September 20,
14:30-15:30, Burnside
Hall 1104
Liangming
Shen (McGill
University)
Title: The
Kahler-Ricci flow with log canonical
singularities
Abstract: In
this talk we will introduce how
to construct solutions to the Kahler-Ricci flow with log canonical
singularities. First, as Song-Tian did for klt singularites, we will
transform the flow equation to a Monge-Ampere type equation with
singularites. Then we will establish a potential estimate based on
approximations with respect to several parameters. Finally if we have
time we will briefly discuss the high order estimate and relations to
the minimal model program. This work is joint with A. Chau,
H. Ge
and K. Li.
Thursday,
September 27,
14:30-15:30, Burnside
Hall 1104
Casey
Kelleher (Princeton
University)
Title: Symplectic
curvature flow revisited
Abstract:
We continue studying a
parabolic flow of
almost Kähler structures introduced by Streets and Tian which
naturally extends Kähler-Ricci flow onto symplectic
manifolds. In the system of primarily the symplectic form, almost
complex structure, Chern torsion and Chern connection, we establish new
formulas for the evolutions of canonical quantities, in particular
those related to the Chern connection. Using this, we give an extended
characterization of fixed points of the flow originally performed by
Streets and Tian.
Thursday,
October 4,
14:30-15:30, Burnside
Hall 1104
Jérôme
Vétois (McGill
University)
Title: Compactness
of sign-changing solutions to critical
elliptic equations with bounded negative part
Abstract:
In this talk, we will look at
the question of compactness of
sign-changing solutions to a class of critical elliptic
Schrödinger equations on a closed Riemannian manifold. We will
present a sharp compactness result for the set of sign-changing
solutions with bounded negative part. We obtained this result in
dimensions greater than or equal to 7 when the potential function is
below the geometric threshold of the conformal Laplacian. The whole set
of sign-changing solutions is non-compact in general. We will also
discuss constructions of counterexamples in the case of the sphere in
dimensions less than or equal to 6 and for potentials above the
geometric threshold in higher dimensions. This is a joint work with
Bruno Premoselli (ULB, Bruxelles).
Thursday, October
11,
14:30-15:30, Burnside Hall 1104
Fengrui
Yang (McGill
University)
Title:
(Zhejiang
University of Technology)
Title:
Curvature estimates for minimal hypersurfaces
Abstract:
In this talk, I will present
the main ideas and techniques of paper by Schoen-Simon-Yau "Curvature
estimates for minimal hypersurfaces". I will mainly focus on how they
obtain a number of new estimates for the curvature of stable minimal
hypersurface M which is immersed in a Riemannian manifold N.
Thursday, October
18,
14:30-15:30, Burnside Hall 1104
Chuanqiang
Chen (Zhejiang
University of Technology)
Title: Smooth
solutions to the $L_p$-Dual Minkowski problem
Abstract:
In this talk, we consider the
$L_p$-dual Minkowski problem. By
studying the a priori estimates and curvature flows, we establish the
existence theorem of the smooth solutions. This is a joint work with
Yong Huang, and Yiming Zhao.
Thursday, October
25,
14:30-15:30, Burnside Hall 1104
Saikat
Mazumdar
(McGill
University)
Title:
Compactness results for elliptic equations with critical growth and
Hardy weight
Abstract:
In this talk we will consider
a class of elliptic PDEs
with
Hardy weight and
Sobolev critical growth, which are in general non-compact due to scale
invariance.
We want to arrive at suitable conditions which would ensure the
compactness and
this in turn will help establish the existence of solutions to these
equations. We will
start by describing the blow-up behaviour of a sequence of
approximating solutions
approaching our PDE and obtain optimal control on such a sequence. Next
we will
look at the interaction of the various terms in the Pohozaev identity
and calculate
the blow-up rates. The compactness theorems will follow from this.
We will see that the location of the singularity, be it in the interior
of the domain
or on its boundary, affects the analytical properties of the equation
and makes the
two situations quite different. When the singularity is in the
interior, then a lower
order perturbation suffices for high dimensions, while the curvature of
the boundary
plays a crucial role if the singularity is on the boundary for high
dimensions.
This is a joint work with Nassif Ghoussoub (UBC) and
Frédéric Robert (Université
de Lorraine).
Thursday,
November 1,
14:40-15:40, Burnside
Hall 1104
Wubin
Zhou (Tongji
University)
Title:
The Futaki Invariant for Poincaré-Mok-Yau type
Kähler
metrics
Abstract:
In this talk, we recall the
definition of
PMY(Poincaré-Mok-Yau) type Kähler metric and then
define
its Futaki invariant which is an obstruction to the existence of
constant scalar curvature Kähler metrics. Also we will give
some
explicit PMY type metrics on some noncompact Kähler manifolds
with
non-vanishing Futaki invariant. This work is joint with Jixiang Fu.
Thursday, November
8,
14:40-15:40, Burnside Hall 1104
Shaodong
Wang
(McGill
University)
Title:
A compactness theorem for boundary Yamabe problem in the scalar-flat
case
Abstract:
In this talk, I will present
some recent results on the
compactness of the solutions to the Yamabe problem on manifolds with
boundary. The compactness of Yamabe problem was introduced by Schoen in
1988. There have been a lot of works on this topic later on. This is a
joint work with Sergio Almaraz and Olivaine Queiroz.
Thursday, November
15,
14:40-15:40, Burnside Hall 1104
Jiuzhou
Huang (McGill
University)
Title:
Two different skills in double normal derivative estimates for
Monge-Ampere equations
Abstract:
In this talk, I will
illustrate two different skills in
double
normal derivative estimates for Monge-Ampere equation. The materials
are Caffarelli-Nirenberg-Spruck's classical paper: “The
Dirichlet
Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampere
Equations” and Trudinger's lecture “Lecture on
Nonlinear
Elliptic Equations of Second Order”. I will focus on
illustrating
the basic ideas behind their estimates.
Thursday, November
22,
14:40-15:40, Burnside Hall 1104
Vladmir
Sicca
(McGill
University)
Title:
The Lichnerowicz Equation for Einstein's General Relativity
Abstract:
A solution to Lichnerowicz
equation gives a Riemannian
manifold
that can be an initial condition for Einstein's equation in General
Relativity. In this talk I will introduce how Lichnerowicz equation
fits in the context of General Relativity and present an existence
result for the equation proved by Hebey, Pacard and Pollack. The core
of the result relies on the mountain pass lemma, which will also be
presented in the talk.
Thursday,
December
6,
14:40-15:40, Burnside Hall 1104
Peter
Hintz
(MIT)
Title:
Stability of Minkowski space and polyhomogeneity of the metric
Abstract:
I will explain a new proof of
the non-linear stability of
the
Minkowski spacetime as a solution of the Einstein vacuum equation. The
proof relies on an iteration scheme at each step of which one solves a
linear wave-type equation globally. The analysis takes place on a
compactification of R^4 to a manifold with corners whose boundary
hypersurfaces correspond to spacelike, null, and timelike infinity. I
will describe how the asymptotic behavior of the metric can be deduced
from the structure of simple model operators at these boundaries. Joint
work with András Vasy.
WINTER 2018
Monday,
January 8,
16:00-17:00, Burnside Hall 719A
Tristan Collins
(Harvard University)
Title: Sasaki-Einstein
metrics and K-stability
Abstract: I
will discuss the connection between
Sasaki-Einstein metrics and algebraic geometry in the guise of
K-stability. In particular, I will give a differential geometric
perspective on K-stability which arises from the Sasakian view point,
and use K-stability to find infinitely many non-isometric
Sasaki-Einstein metrics on the 5-sphere. This is joint work with G.
Szekelyhidi.
Monday,
January 15,
16:00-17:00, Burnside Hall 719
Brent Pym
(University of Edinburgh)
Title: Geometry
and quantization of Poisson Fano manifolds.
Abstract:
Complex Poisson manifolds and the noncommutative algebras that
"quantize" them appear in many parts of mathematics, but their
structure and classification remain quite mysterious, especially in the
positively curved case of Fano manifolds. I will survey recent
breakthroughs on several foundational conjectures in this area, which
were formulated by Artin, Bondal, Kontsevich and others in the 80s and
90s. For instance, we will see that the curvature of a Poisson manifold
has a strong effect on the singularities of its associated foliation,
and that the remarkable transcendental numbers known as multiple zeta
values arise naturally as universal constants in the corresponding
quantum algebras.
Thursday,
January 18,
16:00-17:00, Burnside Hall 1205
Philip Engel
(Harvard University)
Title: Cusp
Singularities
Abstract:
In 1884, Klein initiated the study of rational double points (RDPs), a
special class of surface singularities which are in bijection with the
simply-laced Dynkin diagrams. Over the course of the 20th century, du
Val, Artin, Tyurina, Brieskorn, and others intensively studied their
properties, in particular determining their adjacencies---the other
singularities to which an RDP deforms. The answer: One RDP deforms to
another if and only if the Dynkin diagram of the latter embeds into the
Dynkin diagram of the former. The next stage of complexity is the class
of elliptic surface singularities. Their deformation theory, initially
studied by Laufer in 1973, was largely determined by the mid 1980's by
work of Pinkham, Wahl, Looijenga, Friedman and others. The exception
was a conjecture of Looijenga's regarding smoothability of cusp
singularities---surface singularities whose resolution is a cycle of
rational curves. I will describe a proof of Looijenga's conjecture
which connects the problem to symplectic geometry via mirror symmetry,
and summarize some recent work with Friedman determining adjacencies of
a cusp singularity.
Monday,
January 22,
16:00-17:00, Burnside Hall 1205
Alex Waldron
(Stony
Brook University)
Title: Yang-Mills
flow in dimension four
Abstract: Among
the classical geometric evolution
equations, YM flow is the least nonlinear and best behaved.
Nevertheless, curvature concentration is a subtle problem when the base
manifold has dimension four. I'll discuss my proof that finite-time
singularities do not occur, and briefly describe the infinite-time
picture.
Thursday,
January 25,
16:00-17:00, Burnside Hall 1205
Kiumars Kaveh (University
of Pittsburgh)
Title: Convex
bodies in algebraic geometry and symplectic geometry
Abstract:
We start by discussing some basic facts about
asymptotic behavior of semigroups of lattice points (which is
combinatorial in nature). We will see how this allows one to assign
convex bodies to projective algebraic varieties encoding important
"intersection theoretic" data. Applying inequalities from convex
geometry to these bodies (e.g. Brunn-Minkowski) one immediately obtains
Hodge inequalities from algebraic geometry. This is in the heart of
theory of Newton-Okounkov bodies. It generalizes the extremely fruitful
correspondence between toric varieties and convex polytopes, to
arbitrary varieties. We then discuss connection with symplectic (and
Kahler) geometry and in particular regarding these bodies as images of
moment maps for Hamiltonian torus actions. For "spherical varieties"
(or "multiplicity-free spaces") these constructions become very
concrete and they bring together algebraic geometry, symplectic
geometry and representation theory. For the most part the talk is
accessible to anybody with just a basic knowledge of algebra and
geometry.
Wednesday,
January 31,
13:30-14:30, Burnside Hall 920
Bruno Premoselli
(ULB,
Brussels)
Title: Examples
of Compact Einstein
four-manifolds with
negative
curvature
Abstract:
We construct new examples of closed, negatively curvedEinstein
four-manifolds. More precisely, we construct Einstein metrics
of negative sectional curvature on ramified covers of compact
hyperbolic
four-manifolds with symmetries, initially considered by Gromov and
Thurston. These metrics are obtained through a deformation procedure.
Our candidate approximate Einstein metric is an interpolation between a
black-hole Riemannian Einstein metric near the branch locus and the
pulled-back hyperbolic metric. We then deform it into a genuine
solution
of Einstein's equations, and the deformation relies on an
involved
bootstrap procedure. Our construction yields the first example of
compact Einstein manifolds with negative sectional curvature which are
not locally homogeneous. This is a joint work with J. Fine (ULB,
Brussels).
Wednesday,
February 7,
13:30-14:30, Burnside Hall 920
Gantumur Tsotgerel
(McGill)
Title: Some
scaling estimates in Besov and Triebel-Lizorkin
spaces
Abstract: This
is continuation of my previous talk,
where we presented elliptic estimates for operators with rough
coefficients. The whole theory depended on certain scaling properties
of functions. Here
we
will discuss ways to establish those properties.
Wednesday,
February 14,
13:30-14:30, Burnside Hall 920
Luca Capogna
(Worcester Polytechnic Institute)
Title: A
Liouville type theorem in
sub-Riemannian geometry, and applications to several complex variables
Abstract:
The Riemann mapping theorem
tells us that any simply connected planar domain is conformally
equivalent to the disk. This provides a classification of simply
connected domains via conformal maps. This classification fails in
higher dimensional complex spaces, as already Poincare' had proved that
bi-discs are not bi-holomorphic to the ball. Since then, mathematicians
have been looking for criteria that would allow to tell whether two
domains are bi-holomorphic equivalent. In the early 70's, after a
celebrated result by Moser and Chern, the question was reduced to
showing that any bi-holomorphism between smooth, strictly pseudo-convex
domains extends smoothly to the boundary. This was established by
Fefferman, in a 1974 landmark paper. Since then, Fefferman's result has
been extended and simplified in a number of ways. About 10 years, ago
Michael Cowling conjectured that one could prove the smoothness of the
extension by using minimal regularity hypothesis, through an argument
resting on ideas from the study of quasiconformal maps. In its simplest
form, the proposed proof is articulated in two steps: (1) prove that
any bi-holomorphism between smooth, strictly pseudoconvex domains
extends to a homeomorphisms between the boundaries that is
1-quasiconformal with respect to the sub-Riemannian metric associated
to the Levi form; (2) prove a Liouville type theorem, i.e. any
$1-$quasiconformal homeomorphism between such boundaries is a smooth
diffeomorphism. In this talk I will discuss recent work with Le Donne,
where we prove the first step of this program, as well as joint work
with Citti, Le Donne and Ottazzi, where we settle the second step, thus
concluding the proof of Cowling's conjecture. The proofs draw from
several fields of mathematics, including nonlinear partial differential
equations, and analysis in metric spaces.
Friday,
February 16,
13:30-14:30, Burnside Hall 920
Loredana Lanzani
(Syracuse)
Title: Harmonic
Analysis techniques in Several Complex Variables
Abstract: This
talk concerns the application of relatively
classical tools from real harmonic analysis (namely,
the T(1)-theorem for spaces of homogenous type) to the novel
context of several complex variables. Specifically, I will present
recent joint work with E. M. Stein (Princeton U.) on the extension to
higher dimension of Calderon's and
Coifman-McIntosh-Meyer's seminal results about the Cauchy integral for
a Lipschitz
planar curve (interpreted as the boundary of a Lipschitz domain
$D\subset C$). From the point of view of complex analysis, a
fundamental feature of the 1-dimensional Cauchy kernel: $H(w,
z)=\frac{1}{2\pi i}\frac{dw}{w-z}$ is that it is holomorphic (that is,
analytic) as a function of $z\in D$. In great contrast with the
one-dimensional theory, in higher dimension there is no obvious
holomorphic analogue
of H(w, z). This is because of geometric obstructions (the
Levi
problem), which in dimension 1 are irrelevant. A good candidate kernel
for the higher dimensional setting was first identified by Jean Leray
in the context of a $C^\infty$-smooth, convex domain D: while
these conditions on D
can be relaxed a bit, if the domain is less than C^2-smooth
(never
mind Lipschitz!) Leray's construction becomes conceptually problematic.
In this talk I will present (a), the construction of the
Cauchy-Leray kernel and (b), the L^p(bD)-boundedness of the
induced singular integral operator under the weakest currently known
assumptions on the domain's regularity -- in the case of a planar
domain these are akin to Lipschitz boundary, but in our
higher-dimensional context the assumptions we make are in fact optimal.
The proofs rely in a fundamental way on a suitably adapted version of
the so-called ``T(1)-theorem technique'' from real harmonic analysis.
Time permitting, I will describe applications of this work to complex
function theory - specifically, to the Szego and Bergman projections
(that is, the orthogonal projections of L^2 onto,
respectively,
the Hardy and Bergman spaces of holomorphic functions). References:
[C] Calderon A. P, Cauchy integrals on Lipschitz curves and related
operators, Proc. Nat. Acad. Sci. 74 no. 4, (1977) 1324-1327.
[CMM] Coifman R., McIntosh A. and Meyer Y., L'integrale de Cauchy
definit un operateur borne sur L^2 pour les courbes Lipschitziennes,
Ann. of Math. 116 (1982) no. 2, 361-387.
[L] Lanzani, L. Harmonic Analysis Techniques in Several Complex
Variables, Bruno Pini Mathematical Analysis Seminar 2014, 83-110, Univ.
Bologna Alma Mater Studiorum, Bologna.
[LS-1] Lanzani L. and Stein E. M., The Szego projection for domains in
C^n with minimal smoothness, Duke Math. J. 166 no. 1 (2017), 125-176.
[LS-2] Lanzani L. and Stein E. M., The Cauchy Integral in C^n for
domains with minimal smoothness, Adv. Math. 264 (2014) 776-830.
[LS-3] Lanzani L. and Stein E. M., The Cauchy-Leray Integral:
counter-examples to the L^p-theory, Indiana Math. J., to appear.
Wednesday,
February 21,
13:30-14:30, Burnside Hall 920
Benoît Pausader
(Brown)
Title: Stability
of
Minkowski
space for the
Einstein equation with a
massive scalar field
Abstract: This
is joint work with A.
Ionescu. We consider
the stability
of the Minkowski space for the Einstein model equations with a matter
model given by a massive scalar field. This problem was already studied
under more stringent conditions by LeFloch-Ma and Q. Wang.
After apropriate parametrization, this is a quasilinear problem
involving a wave equation and a Klein-Gordon equation for which one
proves a small data-global existence result. Part of the complication
comes from the fact that, as per the constraint equation, the ``initial
data'' has a rather poor behavior at infinity, and that we do not
specify a priori the main term in the fall-off decay (e.g. the data is
not necessarily Schwartschild outside a bounded ball).
Wednesday,
February 28,
13:30-14:30, Burnside Hall 920
Jessica Lin (McGill)
Title: Regularity
Estimates for
the Stochastic
Homogenization of Elliptic Nondivergence Form Equations
Abstract: I
will present some regularity
estimates related
to the stochastic homogenization for nondivergence form equations. In a
joint work with Scott Armstrong, we show that in the stochastic
homogenization for linear uniformly elliptic equations in random media,
solutions actually exhibit improved regularity properties in light of
the homogenization process. In particular, we show that with extremely
high probability, solutions of the random equation have almost the same
regularity as solutions of the deterministic homogenized equation. The
argument is similar to the proof of the classical Schauder estimates,
however it utilizes the random structure of the problem to obtain
improvement.
Wednesday,
March 21,
13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill)
Title: An
inverse curvature type
hypersurface in space form
Abstract: We
introduce a new type of hypersurface
flow for bounded starshaped domains in space form. An interesting
property of this type of flow is monotonicity for corresponding
quermassintegrals. The focus is the long time existence and convergence
of the flow. We will discuss some recent progress and open problems
arising from the regularity estimates
Wednesday,
March 28,
13:30-14:30, Burnside Hall 920
Guohuan
Qiu (McGill)
Title: Interior
estimates for Special Lagrangian equations
and Scalar curvature equations
Abstract: I
will reveal some connections between special
Lagrangian equations and Scalar curvature equation. Then I will discuss
how to get interior curvature estimates for Scalar curvature equation
in dimension three.
Wednesday,
April 4,
13:30-14:30, Burnside Hall 920
Fengrui
Yang (McGill)
Title:
The Dirichlet problem for Hessian equation
Abstract: In
this talk, I will present the main ideas and
techniques of the classical paper, "The Dirichlet Problem for nonlinear
Second-Order elliptic Equations, Functions of the eigenvalues of the
Hessian". I will mainly focus on the estimate of double normal second
derivatives.
Wednesday,
April 25,
13:30-14:30, Burnside Hall 920
Rohit
Jain
(McGill)
Title: Regularity
Estimates for the Penalized Parabolic
Boundary Obstacle Problem
Abstract: We
will discuss regularity estimates for the
solution to the time dependent penalized boundary obstacle problem.
We will obtain using geometric arguments Lipschitz estimates
in
time and Holder Regularity in space independent of the permeability
constant of interest in the context of semipermeable membrane
theory.
Wednesday,
May 2,
13:30-14:30, Burnside Hall 920
Jérôme
Vétois (McGill)
Title: Examples
of singular solutions to an elliptic
equation with a sign-changing non-linearity
Abstract:
We will examine the possible behaviors of singular
solutions to an elliptic equation with a sign-changing non-linearity in
a punctured ball. I will present new existence results of radial
solutions with prescribed behavior. This is a joint work with Florica
Cîrstea and Frédéric Robert.
Thursday,
May 31,
10:00-11:00, Burnside Hall 920
Siyuan
Lu
(Rutgers)
Title: Exterior
Dirichlet problem for Monge-Ampere equation
Abstract: We
consider exterior Dirichlet problem for
Monge-Ampere equation with prescribed asymptotic behavior. Based on
earlier work by Caffarelli and Li, we complete the characterization of
existence and nonexistence of solutions in terms of their asymptotic
behaviors. This is a joint work with Y.Y. Li.
Wednesday,
June 13,
13:30-14:30, Burnside Hall 920
Laurent
Moonens
(Paris-Sud)
Title:
Differentiation along rectangles
Abstract: Lebesgue’s
differentiation theorem
states that, when $f$ is a locally integrable function in Euclidean
space, its average on the ball $B(x,r)$ centered at $x$ with radius
$r$, converges to $f(x)$ for almost every $x$, when $r$ approaches
zero.
Many questions arise when the family of balls $\{B(x,r)\}$ is replaced
by a \emph{differentiation basis} $\mathcal{B}=\bigcup_x \mathcal{B}_x$
(where, for each $x$, $\mathcal{B}_x$ is, roughly speaking, a
collection of sets shrinking to the point $x$). In this case, one looks
for conditions on $\mathcal{B}$ such that the average of $f$ on sets
belonging to $\mathcal{B}_x$ are known to converge to $f(x)$ for a.e.
$x$, when those sets shrink to the point $x$.
Many interesting phenomena happen when sets in $\mathcal{B}$ have a
\emph{rectangular} shape (Lebesgue’s theorem may or may not
hold
in this case, depending on the geometrical properties of sets in
$\mathcal{B}$). In this talk, we shall discuss some of the history
around this problem, as well as recent results obtained with E.
D’Aniello and J. Rosenblatt in the planar case, when the
rectangles in $\mathcal{B}$ are only allowed to lie along a fixed
sequence of directions.
FALL 2017
Wednesday, August 2,
13:30-14:30, Burnside Hall 920
Joshua Ching
(Sydney)
Title:
Singular solutions to nonlinear elliptic equations with gradient
dependency
Abstract:
Let
$N \geq 2$ be the dimension. Let
$\Omega \subseteq \mathbb{R}^N$ be a domain containing the origin. We
consider non-negative $C^1(\Omega \setminus \{ 0\})$ solutions to the
following elliptic equation: ${\rm div} (|x|^{\sigma} |\nabla
u|^{p-2}\nabla u)=|x|^{-\tau} u^q |\nabla u|^m$ in $\Omega \setminus \{
0 \}$, where we impose appropriate conditions on the parameters
$m,p,q,\sigma,\tau,N$. We study these solutions from several
perspectives including existence, uniqueness, radial symmetry,
regularity and asymptotic behaviour. In
the model case where $p=2$
and $\sigma=\tau=0$, we impose the conditions $q>0$,
$m+q>1$ and
$0<m<2$. Here, we provide a sharp classification result
of the
asymptotic behaviour of these solutions near the origin and infinity.
We also provide corresponding existence results in which we emphasise
the more difficult case of $m \in (0,1)$ where new phenomena arise. A
key step in these proofs is
to obtain gradient estimates. Using a technique of Bernstein's and some
other ideas, we find a new gradient estimate that is independent of the
domain and is applicable in a more general setting than the model case.
Via these gradient estimates, we will show a Liouville-type result that
extends a theorem of Farina and Serrin (2011). Time permitting, we will
also look at further applications of this gradient estimate. In
this talk, we present
results from Ching and Cîrstea (2015, Analysis &
PDE),
results from my PhD thesis as well as ongoing research.
Wednesday,
August 2,
14:45-15:45, Burnside Hall 920
Laurent Moonens
(Paris-Sud)
Title: Continuous
solutions for divergence-type equations associated to elliptic systems
of complex vector fields
Abstract: In
this talk, we shall discuss a characterization, obtained with T.H.
Picon, of all the distributions $F \in \calD’(\Omega)$ for
which one can locally solve by a \emph{continuous} vector field $v$ the
divergence-type equation
$$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F$$ where
$\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of
linearly independent vector fields with smooth complex coefficients
defined on $\Omega \subset \R^{N}$. In case where $(L_1,\dots, L_n)$ is
the usual gradient field on $\R^N$, we recover a classical result for
the divergence equation, obtained previously by T. De Pauw and W.F.
Pfeffer.
Wednesday,
August 9,
13:30-14:30, Burnside Hall 920
Florica Cîrstea
(Sydney)
Title: Nonlinear
elliptic equations with isolated singularities
Abstract: In
this talk, I will review recent developments on isolated singularities
for various classes of nonlinear elliptic equations, which could
include Hardy-Sobolev type potentials. In particular, we shall look at
fully classifying the behaviour of all positive solutions in different
contexts that underline the interaction of the elliptic operator and
the nonlinear part of the equation. We also provide sharp results on
the existence of solutions with singularities, besides optimal
conditions for the removability of all singularities. I will discuss
results obtained with various collaborators including T.-Y. Chang
(University of Sydney) and F. Robert (University of Lorraine).
Wednesday,
August 16,
13:30-14:30, Burnside Hall 920
Chao Xia
(Xiamen
University)
Title:
Uniqueness of stable capillary hypersurfaces in a ball
Abstract: Capillary
hypersurfaces in a ball $B$ is minimal or CMC hypersurfaces
whose boundary intersects $\partial B$ at a constant angle. They are
critical points of some energy functional under volume
preserving variation. The study of stability of capillary hypersurfaces
in $B$ was initiated by Ros-Vergasta and Ros-Souam in 90's. An open
problem is whether any immersed stable capillary hypersurfaces in a
ball in space forms are totally umbilical. In this talk, we
will give a complete affirmative answer. We remark that the related
uniqueness result for closed hypersurfaces is due to Barbosa-Do
Carmo-Eschenburg. The talk is based on a joint work with Guofang Wang.
Wednesday,
August 23,
13:30-14:30, Burnside Hall 920
Xinan Ma (University
of Science and Technology of China)
Title: The
Neumann problem of special Lagrangian equations with supercritical phase
Abstract: In this talk, we
establish the global $C^2$ estimates of the Neumann problem of special
Lagrangian equations with supercritical phase and the existence theorem
by the method of continuity, we also mention the complex version. This
is the joint work with Chen chuanqiang and Wei wei.
Wednesday,
September 20,
13:30-14:30, Burnside Hall 920
Pengfei Guan
(McGill)
Title: Gauss
curvature flows and Minkowski type problems
Abstract: We discuss a class of
isotropic flows by
power of Gauss curvature of convex hypersurfaces. For each flow, there
is an entropy associated to it, and it is monotone decreasing.
For
this entropy, there is an unique entropy point. The flow
preserves the enclosed volume. The main
question is to control the entropy point. This was done for standard
flows in joint works with Lei Ni, and Ben Andrews and
Lei Ni. For isotropic flows, under appropriate assumptions, one prove
that the entropy point will keep as origin. From
there, one may deduce regularity and convergence. The self-similar
solutions are the solutions to corresponding
Minkowski type problem. Similar results were
also obtained by Bryan-Ivaki-Scheuer via inverse type flows.
Wednesday,
September 27,
13:30-14:30, Burnside Hall 920
Gantumur Tsogtgerel (McGill)
Title:
Elliptic estimates for operators with rough coefficients
Abstract:
We will discuss a possible approach to establish elliptic estimates for
operators with barely continuous coefficients in a Sobolev-Besov and
Triebel-Lizorkin scale. The result would obviously be not new
but the proposed approach is
relatively elementary and therefore of interest. Please be warned that
this is an ongoing project, and the talk is going to be more of a
discussion than a polished lecture.
Wednesday,
October 4,
13:30-14:30, Burnside Hall 920
Jerome Vetois
(McGill)
Title:
Blowing-up solutions for critical elliptic equations on a closed
manifold
Abstract: In this talk,
we will look at the question of
existence of blowing-up solutions for smooth perturbations of positive
scalar curvature-type equations on a closed manifold. From a result of
Druet, we know that in dimensions different from 3 and 6, a necessary
condition for the existence of blowing-up solutions is that the limit
equation agrees with the Yamabe equation at least at one blow-up point.
I will present new existence results in situations where the limit
equation is different from the Yamabe equation away from the blow-up
point. I will also discuss the special role played by the dimension 6.
This is a joint work with Frederic Robert.
Wednesday,
October 11,
13:30-14:30, Burnside Hall 920
Shaya Shakerian
(University of British Columbia)
Title: Borderline
Variational problems for fractional
Hardy-Schrödinger operators
Abstract: In this talk,
we investigate the existence of
ground state solutions associated to the fractional
Hardy-Schrödinger operator on Euclidean space and its bounded
domains. In the process, we extend several results known about the
classical Laplacian to the non-local operators described by its
fractional powers. Our analysis show that the most important parameter
in the problems we consider is the intensity of the corresponding Hardy
potential. The maximal threshold for such an intensity is the best
constant in the fractional Hardy inequality, which is computable in
terms of the dimension and the fractional exponent of the Laplacian.
However, the analysis of corresponding non-linear equations in
borderline Sobolev-critical regimes give rise to another threshold for
the allowable intensity. Solutions exist for all positive linear
perturbations of the equation, if the intensity is below this new
threshold. However, once the intensity is beyond it, we had to
introduce a notion of Hardy-Schrödinger Mass associated to the
domain under study and the linear perturbation. We then show that
ground state solutions exist when such a mass is positive. We then
study the effect of non-linear perturbations, where we show that the
existence of ground state solutions for large intensities, is
determined by a subtle combination of the mass (i.e., the geometry of
the domain) and the size of the nonlinearity of the
perturbations.
Wednesday,
October 18,
13:30-14:30, Burnside Hall 920
Shaodong Wang
(McGill)
Title: A
compactness theorem for Yamabe problem on
manifolds with boundary
Abstract: In this talk,
I will present a compactness result on
Yamabe problems on manifolds with boundary. This is from a paper by
Zheng-chao Han and Yanyan Li.
Wednesday,
October 25,
13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill)
Title: One-dimensional
convex integration
Abstract: Convex
integration theory was introduced by M.Gromov in his thesis
dissertation in 1969 which is a powerful tool for solving differential
relations. An important application of the convex integration theory is
that it can recover the Nash-Kuiper result on C^{1} isometric
embeddings. I will briefly mention the history of h-principle and
rigidly theorem for isometric embedding. Then some details about
one-dimensional convex integration.
Friday,
November 10,
13:30-14:30, Burnside Hall 920
Daniel Pollack
(University of Washington)
Title: On
the geometry and topology of initial data sets
with horizons
Abstract: One of the central and
most fascinating objects which
arise in general relativity are black holes. From a mathematical point
of view this is closely related to questions of "singularities" and
"horizons" which arise in the study of the Einstein equations. We will
present a number of results which relate the presence (or absence) of
horizons to the topology and geometry of the "exterior region" of an
initial data set for Einstein equations. Time permitting we will also
connect these results with previous work of Galloway and Schoen on the
topology of the black holes themselves. This is joint work with Lars
Andersson, Mattias Dahl, Michael Eichmair and Greg Galloway.
Wednesday,
November 15,
13:30-14:30, Burnside Hall 920
Fengrui Yang
(McGill)
Title: The
Dirichlet Problem for Monge-Ampere equation.
Abstract: In this talk, I will
present the main ideas and
techniques of the classical paper, ' The Dirichlet Problem for
nonlinear Second-Order Elliptic Equations I, Monge-Ampere Equation'.
Firstly, I will give a brief introduction of the history of estimating
third-order derivatives of Monge-Ampere equation, and then focus on the
proof of boundary C2 and C2,a estimates.
Wednesday,
November 22,
13:30-14:30, Burnside Hall 920
Saikat Mazumdar
(University
of British Columbia)
Title: Blow-up
analysis
for a critical elliptic equation with vanishing singularity
Abstract: In this talk,
we will examine the asymptotic
behavior of a sequence of ground state solutions of the Hardy-Sobolev
equations as the singularity vanishes in the limit. If this
sequence is uniformly bounded in L-infinity, we obtain a
minimizing solution of the stationary Schrödinger equation
with
critical growth. In case the sequence blows up, we obtain C0 control on
the blow up sequence, and we localize the point of singularity and
derive precise blow up rates.
Wednesday,
November 29,
13:30-14:30, Burnside Hall 920
Edward Chernysh
(McGill)
Title: Weakly
Monotone Decreasing Solutions to
an Elliptic Schrödinger System
Abstract: In this talk
we study positive super-solutions to an
elliptic Schrödinger system in R^n for n\geq3. We
give conditions guaranteeing the non-existence of positive
solutions and introduce weakly monotone decreasing functions. We
establish lower-bounds on the decay rates of positive solutions and
obtain upper-bounds when these are weakly monotone
decreasing.
Wednesday,
December 6,
13:30-14:30, Burnside Hall 920
Rohit
Jain (McGill)
Title: The Two-Penalty
Boundary Obstacle Problem
Abstract: Inspired by a
problem of fluid flow through a
semi-permeable membrane we study optimal regularity estimates for
solutions as well as some structural properties of the free boundary
for a two-penalty boundary obstacle problem. This is ongoing work with
Thomas Backing and Donatella Danielli.
WINTER
2017
Wednesday,
January 25,
13:30-14:30, Burnside Hall 920
Gantumur
Tsogtgerel
(McGill)
Title:
A prescribed scalar-mean curvature problem
Abstract:
In this talk, we will be concerned with a
problem of
prescribing scalar curvature and boundary mean curvature of a compact
manifold with boundary. This is an ongoing work motivated by the study
of the Einstein constraint equations on compact manifolds with
boundary, and builds on the results of Rauzy and of Dilts-Maxwell.
Wednesday,
February 1st,
13:30-14:30, Burnside Hall 920
Mohammad
Najafi Ivaki
(Concordia)
Title: Harnack
estimates for curvature flows
Abstract: I
will discuss
Harnack estimates for curvature flows in the Riemannian and Lorentzian
manifolds of constant curvature and that "duality" allows us to obtain
a certain type of inequalities, "pseudo"-Harnack inequalities.
Wednesday, February
8, 13:30-14:30, Burnside Hall 920
Jerome
Vetois
(McGill)
Title:
Decay
estimates and symmetry of solutions to elliptic systems in R^n
Abstract: In this
talk, we will look at a class of coupled nonlinear Schrödinger
equations in R^n. I will discuss a notion of finite energy solutions
for these systems and I will present some recent qualitative results on
these solutions.
Wednesday, February
22, 13:30-14:30, Burnside Hall 920
Guohuan
Qiu
(McGill)
Title:
Rigidity of closed self-similar solution to
the Gauss curvature flow
Abstract: In the
seminar, I will present Choi and Daskalopoulos's recent
[arXiv:1609.05487v1] rigidity result about Gauss curvature flow. They
proved that a convex closed solution to the Gauss curvature flow in R^n
becomes a round sphere after rescaling.
Wednesday, March
8, 13:30-14:30, Burnside Hall 920
Siyuan
Lu
(McGill)
Title:
Minimal hypersurface and boundary behavior of
compact manifolds with nonnegative scalar curvature
Abstract: In the study
of boundary behavior of compact Riemannian manifolds with nonnegative
scalar curvature, a fundamental result of Shi-Tam states that, if a
compact manifold has nonnegative scalar curvature and its boundary is
isometric to a strictly convex hypersurface in the Euclidean space,
then the total mean curvature of the boundary of the manifold is no
greater than the total mean curvature of the corresponding Euclidean
hypersurface. In this talk, we give a supplement to Shi-Tam's result by
considering manifolds whose boundary includes the outermost minimal
hypersurface of the manifold. Precisely speaking, given a compact
manifold \Omega with nonnegative scalar curvature, suppose its boundary
consists of two parts, \Sigma_h and \Sigma_o, where \Sigma_h is the
union of all closed minimal hypersurfaces in \Omega and \Sigma_o is
isometric to a suitable 2-convex hypersurface \Sigma in a Schwarzschild
manifold of positive mass m, we establish an inequality relating m, the
area of \Sigma_h, and two weighted total mean curvatures of \Sigma_o
and $ \Sigma, respectively. This is a joint work with Pengzi Miao from
Miami.
Wednesday, March
16, 2:00pm-3:3:00pm, Burnside 1234
Yuanwei
Qi
(University of Central Florida)
Title:
Traveling Wave of Gray-Scott model:
Existence, Multiplicity and Stability.
Abstract:
In this talk, I shall present some recent
works I have
done with my collaborators in rigorously proofing the existence of
traveling wave solution to the Gray-Scott model, which is one of the
most important models in Turing type of pattern formation after the
experiments in early 1990s to validate his theory. We shall also
discuss some interesting features of traveling wave solutions. This is
a joint work with Xinfu Chen.
Wednesday, March
22, 13:30-14:30, Burnside Hall 920
Rohit
Jain
(McGill)
Title: Regularity
estimates for Semi-permeable membrane Flow
Abstract: We
study a boundary value problem modeling flow
through the semi-permeable
boundary
$\Gamma$ with finite thickness $\lambda$ and an applied fluid pressure
$\phi(x)$. We study
optimal regularity estimates for the solution as
well as asymptotic
estimates as $\lambda \to 0$.
Wednesday, March
29, 13:30-14:30, Burnside Hall 920
Kyeongsu
Choi
(Columbia)
Title:
Free boundary problems in the Gauss curvature
flow
Abstract:
We will discuss the optimal C^{1,1/(n-1)}
regularity
of the Gauss curvature flow with flat sides, and the C^{\infty}
regularity of the flat sides.
Moreover, we will study connections between the free boundary problems,
the classification to the self-shrinkers, and the prescribed curvature
measure equations.
Wednesday,
April 5,
13:30-14:30, Burnside Hall 920
Shaodong
Wang
(McGill)
Title: Infinitely many
solutions for cubic
Schrödinger equation in dimension 4
Abstract:
In this talk, I will present some recent results in
the existence of blow-up solutions to a cubic Schrödinger
equation
on the standard sphere in dimension four. This is a joint work with
Jerome Vetois.
Friday,
April 7,
13:30-14:30, Burnside Hall 920
Xinliang
An (University of
Toronto)
Title:
On Gravitational Collapse in General Relativity
Abstract:
In the process of gravitational collapse,
singularities may form, which are either covered by trapped surfaces
(black holes) or visible to faraway observers (naked singularities). In
this talk, I will present four results with regard to gravitational
collapse for Einstein vacuum equation.
The first is a simplified approach to
Christodoulou’s monumental result which
showed that
trapped surfaces can form dynamically by the focusing of gravitational
waves from past null infinity. We extend the methods of
Klainerman-Rodnianski, who gave a simplified proof of this result in a
finite region.
The second result extends the theorem of Christodoulou by allowing for
weaker initial data but still guaranteeing that a trapped surface forms
in the causal domain. In particular, we show that a trapped surface can
form dynamically from initial data which is merely large in a
scale-invariant way. The second result is obtained jointly with
Jonathan Luk.
The third result answered the following questions: Can a ``black
hole’’
emerge from a point? Can
we find the boundary (apparent horizon) of a ``black
hole’’
region?
The fourth result extends
Christodoulou’s famous
example on formation of naked singularity for Einstein-scalar field
system under spherical symmetry. With numerical and analytic tools, we
generalize Christodoulou’s result and
construct an
example of naked singularity formation for Einstein vacuum equation in
higher dimension. The fourth result is obtained jointly with Xuefeng
Zhang.
Wednesday,
April 19,
13:30-14:30, Burnside Hall 920
Ben
Weinkove
(Northwestern)
Title:
The Monge-Ampere equation, almost complex
manifolds and geodesics
Abstract:
I will discuss an existence theorem for
the
Monge-Ampere equation in the setting of almost complex manifolds. I
will describe how techniques for studying this equation can be used to
prove a regularity result for geodesics in the space of Kahler metrics.
This is joint work with Jianchun Chu and Valentino Tosatti.
Wednesday,
April 26,
13:30-14:30, Burnside Hall 920
Chen-Yun
Lin (University
of Toronto)
Title:
An embedding theorem: differential analysis behind massive data
analysis
Abstract:
High-dimensional data can be difficult to analyze. Assume data are
distributed on a low-dimensional manifold. The Vector Diffusion Mapping
(VDM), introduced by Singer-Wu, is a non-linear dimension reduction
technique and is shown robust to noise. It has applications in
cryo-electron microscopy and image denoising and has potential
application
in time-frequency analysis. In this talk, I will present a theoretical
analysis of the effectiveness
of the VDM. Specifically, I will discuss parametrisation of the
manifold
and an embedding which is equivalent to the truncated VDM. In the
differential geometry language, I use eigen-vector fields of the
connection Laplacian operator to construct local coordinate charts that
depend only on geometric properties of the manifold. Next, I use the
coordinate charts to embed the entire manifold into a
finite-dimensional
Euclidean space. The proof of the results relies on solving the
elliptic
system and provide estimates for eigenvector fields and the heat kernel
and their gradients.
FALL
2016
Wednesday,
September 21,
13:30-14:30, Burnside Hall 920
Pengfei
Guan
(McGill)
Title:
A volume preserving flow and the
isoperimetric problem in warped product spaces with general base
Abstract:
A flow was introduced
in a previous work
to handle
the isoperimetric problem in sapce forms. We propose to study a similar
normalized hypersurface flow in the more general ambient setting of
warped product spaces with general base. This flow preserves the volume
of the bounded domain enclosed by a graphical hypersurface, and
monotonically decreases the hypersurface area. As an application, the
isoperimetric problem in warped product spaces is solved for such
domains. This is a join work with Junfang Li and Mu-Tao Wang.
Wednesday,
September 28, 13:30-14:30, Burnside Hall
920
Dylan
Cant
(McGill)
Title:
A Curvature flow and application to an
isoperimetric inequality
Abstract:
Long time existence and convergence to a
circle is
proved for radial graph solutions to a mean curvature type curve flow
in warped product surfaces (under weak assumption on the warp product
of surface). This curvature flow preserves the area enclosed by the
curve, and this fact is used to prove a general isoperimetric
inequality applicable to radial graphs in warped product surfaces under
weak assumption on the warp potential.
Wednesday,
October 5, 13:30-14:30, Burnside Hall 920
Rohit
Jain
(McGill)
Title:
Geometric Methods in Obstacle-Type Free
Boundary Problems I
Abstract:
Obstacle-type free boundary problems
naturally appear as mathematical
models in science and engineering with some particular motivations
arising
from contact problems in elasticity, options pricing in financial
mathematics, and phenomenological models in superconductor physics. The
first talk will focus on geometric methods that have been used to study
regularity estimates in Obstacle-Type Free Boundary Problems. The
regularity theory for obstacle-type problems (and other type of free
boundary problems as well) was much inspired by the regularity theory
for
minimal surfaces. We will discuss the basic existence, uniqueness and
regularity questions in the classical obstacle problem. We will point
out
generalizations and current problems of interest in this field of
research. In the second talk we will focus on an obstacle-type problem
arising in stochastic impulse control theory that appeared first as a
model for cash management and portfolio optimization under transaction
costs. Here the underlying theory for the obstacle problem has to be
suitably modified to consider obstacle problems with an implicit and
nonlocal obstacle. Regularity estimates will be presented and natural
directions for future research discussed.
Wednesday,
October 12, 13:30-14:30, Burnside Hall 920
Rohit
Jain
(McGill)
Title:
Geometric Methods in Obstacle-Type Free
Boundary Problems II
Abstract:
We will continue studying Geometric
Methods in Obstacle-Type Free Boundary
Problems. In the second talk we will focus on an obstacle-type problem
arising in stochastic impulse control theory that appeared first as a
model for cash management and portfolio optimization under transaction
costs. Here the underlying theory for the obstacle problem has to be
suitably modified to consider obstacle problems with an implicit and
nonlocal obstacle. Regularity estimates for the solution and the free
boundary will be presented.
Wednesday,
October 19, 13:30-14:30, Burnside Hall
920
Guohuan
Qiu
(McGill)
Title:
Hessian estimate for the Sigma-2 Equation in
dimension Three (After Michah Warren and Yu Yuan)
Abstract:
Heinz derived a Hessian bound for the two
dimensional
Monge-Ampere equation by using Uniformization Theorem. Sigma-2=1 in
three dimension can be viewed as a equation of a special lagranian
graph in C^3. Which is also a three dimensional minimal surface in R^6.
Michah Warren and Yu Yuan used this observation and Michael-Simon's
sobolev inequalities on generalized submanifolds of R^n to prove a
priori interior Hessian estimates for Sigma_2 =1 in three dimension. We
will go through their proof in this seminar.
Wednesday,
November 2, 13:30-14:30, Burnside Hall 920
Siyuan
Lu
(McGill)
Title:
Isoperimetric inequality in warped product
manifold.
Abstract:
We consider isoperimetric inequality in
warped product
manifold. We discuss two results by Montiel and Bray-Morgan. The paper
by Montiel shows that under natural assumption of the warped function,
a star shaped constant mean curvature hypersurface must be a coordinate
slice. The paper by Bray-Morgan shows that under stronger assumption of
the warped function, isoperimetric domain must be a coordinate slice.
Thursday,
November 10, 14:30-15:30, Burnside Hall 920
Tatiana
Toro
(University of Washington)
Title:
Almost minimizers with free boundary
Abstract:
In recent work with G. David, and ongoing
work with G.
David and M. Engelstein, we study almost minimizer for functionals
which yield a free boundary, as in the work of Alt-Caffarelli and
Alt-Caffarelli-Friedman. The almost minimizing property can be
understood as the defining characteristic of a minimizer in a problem
which explicitly takes noise into account. In this talk we will discuss
regularity results for these almost minimizers and as well as the
structure of the corresponding free boundary. A key ingredient in the
study of the 2-phase problem is the existence of almost monotone
quantities.
Wednesday,
November 16, 13:30-14:30, Burnside Hall
920
Siyuan
Lu
(McGill)
Title:
Isoperimetric inequality in warped product
manifold II.
Abstract:
We will continue to discuss the
isoperimetric
inequality in warped product manifold. We'll focus on Bray-Morgan's
result using comparison to obtain the isoperimetric inequality without
the assumption of starshapedness.
Wednesday,
December 14, 13:30-14:30, Burnside Hall
920
Pengzi
Miao
(University of Miami)
Title:
Boundary effect of scalar curvature
Abstract:
Manifolds with nonnegative scalar
curvature arise naturally as
maximal slices of physical spacetimes in general relativity. When the
manifold is noncompact, there are the Riemannian positive mass theorem
and
Penrose inequality which give global results on how scalar curvature
affects the manifold geometry near infinity. When the manifold is
compact,
it models bounded domains in such spacetime slices and how the scalar
curvature affects its boundary geometry is tied to the quasi-local mass
problem. In this talk, I will survey known results on boundary behavior
of
compact manifolds with nonnegative scalar curvature, and if time
permits,
I will discuss related open questions.
Previous
Talks