Organizers: Pengfei Guan and Jerome Vetois. Seminars are usually held on Wednesdays, 13:30-14:30, in Burnside Hall Room 920.

**Wednesday, September 9, 13:30-14:30, Burnside
920 ****M. Moller** (Princeton and ICTP, Trieste)

Gluing
of Solutions to Nonlinear PDEs of Mean Curvature Type

**Wednesday, September 16, 13:30-14:30,
Burnside 920 ****Guohuan** **Qiu** (McGill and University
of Science and Technology of China)

Title: The Neumann problem
for Hessian equations.

Abstract: In this seminar, we consider the
existence of the Neumann problem for Hessian equations. We solved
this problem which is also a conjecture raised by N.S. Trudinger in
1986. First, we will briefly review the history of Neumann problem
and the standard steps about solving this nonlinear problems using
continuity method. Then we will discuss main obstacles to solving
Neumann problems of fully nonlinear problems before. Last I will talk
about details of our method to overcome these difficulties which
mainly lies in C^2 estimates. (This is joint work with my advisor
Xinan Ma in USTC.)

**Wednesday, September 23, 13:30-14:30,
Burnside 920 ****Chao Xia** (McGill University)

Title:
Inverse anisotropic curvature flow and Minkowski's
inequality.

Abstract: In this talk, I will first discuss the
inverse anisotropic mean curvature flow from a star-shaped
hypersurface and show such flow exists for long time and converges to
a rescale Wulff shape. Next I will discuss general inverse
anisotropic curvature flows from convex hypersurfaces. As an
application,we prove Minkowski's inequality for mixed volumes.

**Wednesday, September 30, 13:30-14:30,
Burnside 920 ****Jerome Vetois** (McGill University)

Title:
Blow-up phenomena for linear perturbations of the Yamabe
equation.

Abstract: We will consider the question of the stability
of solutions to the Yamabe equation under the effect of linear
perturbations. We will see that the linear potential of the Yamabe
equation plays a critical role for this question in dimensions four
and above. I will present results which show the existence of blow-up
phenomena for this equation, looking at different explosion profiles
(one-peak solutions, multi-peak solutions, isolated and non-isolated
explosion points).

**Monday, October 5, 13:30-14:30, Burnside 920
(Joint seminar in Analysis and Geometric Analysis) ****Biao Ou**
(University of Toledo)

Title: An equality for the geodesic
curvature of certain curves on a two-dimensional Riemann surface.

Abstract: We prove an equality for the geodesic curvature
function of certain closed curves in a local domain of a
two-dimensional Riemannian surface. We address its connection to the
local Gauss-Bonnet theorem. We also show that the equality leads to a
four-vertex theorem for simple and closed curves on a two-dimensional
Riemannian surface with a constant Gauss curvature.

**Wednesday, October 28, 13:30-14:30, Burnside
920 ****Pengfei Guan** (McGill)

Title: Flow by powers of
the Gauss curvature.

Abstract: We discuss a joint work with Ben
Andrews and Lei Ni on the Gauss curvature flow by powers. We prove
that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under
the flow by any power $\alpha>\frac{1}{n+2}$ of the Gauss
curvature converge (after rescaling to fixed volume) to a limit which
is a smooth, uniformly convex self-similar contracting solution of
the flow. Under additional central symmetry of the initial body we
prove that the limit is the round sphere.

**Wednesday, November 11, 13:30-14:30, Burnside
920 ****Shaodong** **Wang** (McGill)

Title:
Infinitely many solutions for the Schrodinger equations in R^n with
critical growth

Abstract

**Wednesday, November 18, 13:30-14:30, Burnside
920 ****Shaodong** **Wang** (McGill)

Title:
Infinitely many solutions for the Schrodinger equations in R^n with
critical growth (part II)

Abstract

**Wednesday, Nov. 25, 13:30-14:30, Burnside 920
****Teng** **Fei** (MIT)

Title: Some new solutions to
the Strominger system.

Abstract: The Strominger system is a
system of PDEs derived by Strominger in his study of compactification
of heterotic strings with torsion. It can be thought of as a
generalization of Ricci-flat metrics on non-Kahler Calabi-Yau
3-folds. We present some new solutions to the Strominger system on a
class of noncompact Calabi-Yau 3-folds constructed by twistor
technique. These manifolds include the resolved conifold
Tot(O(-1,-1)->P^1) as a special case.

**Wednesday, December 2, 13:30-14:30, Burnside
920 ****Pierre-Damien Thizy** (University of Cergy-Pontoise)

Title: Schrodinger-Poisson systems in closed manifolds.

Abstract: The Schrodinger-Poisson system we investigate in this
talk arises when we look for standing waves solutions of the full
(time dependent) Schrodinger-Maxwell system in the electrostatic
case. After a short introduction, we will give recent
stability/instability results of the set of the solutions to this
system with respect to small perturbations of the "phase"
(temporal frequency). In the process of this talk, we will make the
connection of these results with some existence, nonexistence,
uniqueness and multiplicity results concerning the positive solutions
to this system.

**Wednesday, December 9, 13:30-14:30, Burnside
920 ****Junfang** **Li** (University of Alabama)

Title:
An integral formula with geometric applications in Riemannian and
Pseudo-Riemannian manifolds.

Abstract: In this talk, we will
present a recent joint work with Chao Xia. We first prove a general
integral formula for bounded domains in Riemannian manifolds. This
formula includes Reilly's integral formula and the recent work of
Qiu-Xia as special cases. In the second part of the talk, we will
apply this formula to prove 1) Heitz-Karcher type inequalities, 2)
Minkowski inequality, 3) two almost Schur type of Theorems. All these
geometric inequalities hold for the so-called substatic Riemannian
manifolds which consists of a large family Riemannian manifolds
including all the space forms. We note that Heitze-Karcher inequality
naturally leads to an Alexandrov rigity theorem for substatic warped
product spaces. Thus we recovered S. Brendle's recent work by a
completely different approach. The results in this talk are focused
on Riemannian manifolds, however it has deep roots from
Pseudo-Riemannian spaces.

**Wednesday, January 27, 13:30-14:30, Burnside
920 ****Siyuan** **LU** (McGill)

Title: Interior C^2
estimate of Monge-Ampere equation in 2-dimension.

Abstract:
Heinz's work and related development.

**Wednesday, February 3, 13:30-14:30, Burnside
920 ****Guohuan** **Qiu** (University of Science and
Technology of China & McGill)

Title: On degenerate case of
prescribed curvature measure problems.

Abstract: We discuss a
problem that curvature measure of radial compact C^1,1 Hypersurface
is prescribed by a given nonnegative smooth function f on sphere. And
we mainly focus on the regularity issue about this problem. This is
well studied problems when the given function is positive. But even
in prescribed mean curvature problem, no regularity results can
permit f to be zero anywhere to our knowledge. While its nature to
consider the case when f can touch zero somewhere on sphere, and it
will cause degeneracy in the fully nonlinear problems. In this talk,
We will talk about our recent development on this topic.

**Wednesday, February 10, 13:30-14:30, Burnside
920 ****Siyuan** **LU** (McGill)

Title: Interior C^2
estimate of Monge-Ampere equation in 2-dimension II.

Abstract:
Continuation of the discussion of Heinz's work and related
development.

**Wednesday, February 17, 13:30-14:30, Burnside
920 ****Frederic Robert** (Universite de Lorraine, France)

Title: Sharp asymptotic profiles for singular solutions to an
elliptic equation with a sign-changing nonlinearity

Abstract: We
consider a positive solution to a nonlinear elliptic equation on a
punctured ball. The linear part is the classical Laplacian. When the
nonlinear part is positive and critical, this is similar to the
classical problem studied by Caffarelli-Gidas-Spruck. When the
nonlinear part is negative and a pure power, the problem is
associated to a natural convex functional and the singularities are
completely understood. In the present work, we mix the two
nonlinearities. We show the existence of several potential behaviors.
Two of them are natural extensions of the case of constant-sign
nonlinearity. Two other behaviors are arising from the interaction of
the two nonlinearity. In this talk, I will describe all the possible
behaviors and I will show how the methods of apriori analysis in
nonlinear elliptic problems are helping understanding this problem.
This is joint work with Florica Cirstea (Sydney)

**Wednesday, March 9, 13:30-14:30, Burnside 920
****Siyuan** **Lu** (McGill)

Title: Curvature
estimates for embedded surfaces into Riemannian manifolds

Abstract:
We discuss the a priori mean curvature estimate for embedded surfaces
in Riemannian manifold. By a modified Heinz's interior C^2 estimate
and a Shi-Tam inequality, we prove that the mean curvature of
strictly convex surface is bounded. This bound is independent of the
structure of the ambient space and the location of the surface, which
is main contribution of the work.

**Wednesday, April 6, 13:30-14:30, Burnside 920
****Saikat** **Mazumdar** (Universite de Lorraine,
France)

Title: HIGHER ORDER ELLIPTIC PROBLEMS WITH CRITICAL
SOBOLEV GROWTH ON A COMPACT RIEMANNIAN MANIFOLD: BEST CONSTANTS AND
EXISTENCE

Abstract: We investigate the existence of solutions to
a nonlinear elliptic problem involving the critical Sobolev exponent
for a Polyharmomic operator on a Riemannian manifold M. We first show
that the best constant of the Sobolev embedding on a manifold can be
chosen as close as one wants to the Euclidean one, and as a
consequence derive the existence of minimizers when the energy
functional goes below a quantified threshold. Next, higher energy
solutions are obtained by Coron’s topological method, provided that
the minimizing solution does not exist and the manifold satisfies a
certain topological assumption. To perform the topological argument,
we obtain a decomposition of Palais-Smale sequences as a sum of
bubbles and adapt Lions’s concentration-compactness lemma.

**Wednesday, April 13, 13:30-14:30, Burnside
920 ****Dima** **Jakobson** (McGill University)

Title:
Zero and negative eigenvalues of the Yamabe operator.

Abstract.
This is joint work with Rod Gover, Asma Hassannezhad and Michael
Levitin. After recalling old results (joint with Canzani, Gover and
Ponge) about conformal invariants that arise from nodal sets of
eigenfunctions lying in a kernel of a conformally covariant operator
on a compact manifold, we show that for the conformal Laplacian, zero
is _not_ an eigenvalue for generic Riemannian metrics. We proved
previously that the Yamabe operator can have an arbitrarily large
number of negative eigenvalues on any compact manifold of dimension
$n \geq 3$; we show that if the number of negative eigenvalues
increases for a sequence of metrics, then that sequence cannot
satisfy certain natural pre-compactness assumptions, and hence cannot
have "convergent subsequences." If time permits, we shall
discuss related results for operators on graphs.

**Wednesday, May 4, 13:30-14:30, Burnside 920
****Panagiota** **Daskalopoulos** (Columbia University)

Title: Ancient solutions to geometric flows

Abstract: We will
discuss ancient solutions to geometric flows such as the Mean
Curvature flow, Ricci flow and Yamabe flow. We will discuss the
classification of ancient or eternal solutions as well as the
construction of ancient solutions from the gluing is solitons.

**Thursday, July 7, 13:30-14:30, Burnside Hall
920 ****Yuxin** **Ge, **(university of Toulouse)

Title:
On the comformally compact Einstein manifolds

Abstract: In this
talk, I will discuss 4-dimensional conformally compact Einstein
manifolds and in particular the compactness result of these manifolds
and some relations to the conformal invariants at infinity.

**Wednesday, July 27, 13:30-14:30, Burnside
Hall 920 ****Pierre Damien Thizy **(University of
Cergy-Pontoise)

Title: Blow-up analysis for the Moser-Trudinger
equation in dimension 2

Abstract: We
will first introduce the Moser-Trudinger elliptic equation with
critical exponential non-linearity and give some variational
motivations to study it. Then, we will give the main results in the
literature concerning this equation. At last we will give our result
obtained with Olivier Druet about the blow-up analysis for this
equation. These results answer questions asked by Adimurthi-Struwe,
Druet, Martinazzi-Malchiodi, and Del Pino-Musso-Ruf.