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

**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: TBA

Abstract: TBA
.

**Wednesday, October 12, 13:30-14:30,
Burnside Hall 920 **

**Rohit Jain **(McGill University)

Title: TBA

Abstract: TBA
.

**Wednesday, October 19, 13:30-14:30,
Burnside Hall 920 **

**Guohuan Qiu **(McGill University)

Title: TBA

Abstract: TBA
.

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

**Siyuan Lu **(McGill University)

Title: TBA

Abstract: TBA
.

**Thursday, November 10, 14:30-15:30,
Burnside Hall 920 **

**Tatiana Toro **(University of Washington)

Title: TBA

Abstract: TBA
.

**Wednesday, December 7, 13:30-14:30,
Burnside Hall 920 **

**Gantumur Tsogtgerel **(McGill University)

Title: TBA

Abstract: TBA
.

**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.