## Geometric
Analysis Seminar

Organizers: Pengfei Guan and Jerome Vetois.

Seminars are usually held on Wednesdays, 13:30-14:30, in Burnside Hall
Room 920

## FALL 2017

**Wednesday, August 2,****
13:30-14:30, Burnside Hall 920**

**Joshua Ching **(University
of 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 **(University
of 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 C**î**rstea
**(University
of 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 University)

**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
University)

**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 University)

**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 University)

**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
University)

**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
University)

**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 University)

**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 University)

**Wednesday, January 31,****
13:30-14:30, Burnside Hall 920**

**Bruno Premoselli**** **(Université Libre de Bruxelles)

**Wednesday, February 14,****
13:30-14:30, Burnside Hall 920**

**Luca Capogna**** **(Worcester Polytechnic Institute)

**Friday, February 16,****
13:30-14:30, Burnside Hall 920**

**Loredana Lanzani ****(Syracuse)**** **

**Wednesday, February 21,****
13:30-14:30, Burnside Hall 920**

**Benoît Pausader ****(Brown University)**

**WINTER 2017**

**Wednesday, January 25,
13:30-14:30, Burnside Hall 920**

**Gantumur Tsogtgerel**
(McGill University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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
Christodoulouâ€™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
holeâ€™â€™
emerge from a point? Can
we find the boundary (apparent horizon) of a ``black
holeâ€™â€™
region?
The fourth result extends
Christodoulouâ€™s famous
example on formation of naked singularity for Einstein-scalar field
system under spherical symmetry. With numerical and analytic tools, we
generalize Christodoulouâ€™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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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 University)

**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**