ABSTRACTS
Spyros Alexakis (University of Toronto)
The Penrose Inequality for perturbations of the Schwarzschild
exterior
The Penrose inequality asserts lower bounds on the ADM mass of
a black hole exterior region in terms of the areas of sections
of the event horizon, or (alternatively) of marginally outer trapped
surfaces. Very well-known proofs of the above (due to Huisken-Ilmanen
and Bray) are know in the Riemannian case, which corresponds to
space-times that admit a time-reflection symmetry. We give a proof
of the inequality for single black hole exterior regions which
are a priori assumed to be close to the Schwarzschild exterior,
locally on a slab.
Bernard Bonnard (Université de
Bourgogne)
The contrast problem by saturation in medical NMR imaging
The objective of this talk is to use optimal control techniques
to improve in medical NMR imaging the contrast between two chemical
species. In the first part we discuss the computation of the ideal
contrast using the Maximum principle and shooting-continuation
numerical methods. In the second part we analyze the problem of
computing a robust magnetic field taking into account the B0 and
B1 homogeneities of the applied magnetic fields. The final section
is devoted to open problems in particular some theoretical questions
about the analysis of the Hamiltonian differential equations with
meromorphic singularities.
Simon Brendle (Stanford University)
Rotational symmetry of Ricci solitons
Ricci solitons play a central role in the work of Hamilton and
Perelman, in that they serve as local models for singularity formation.
We give a classification of all 3-dimensional steady Ricci solitons
in dimension 3 which are kappa-noncollapsed: any such soliton
is either flat or isometric to the Bryant soliton (up to scaling)
Guy Bouchitté (Université
de Toulon et du Var)
A new duality framework for non convex optimization
In this talk I will develop a duality theory for classical problems
of the Calculus of Variations of the kind
$$ J(\Omega) := \inf \left\{\int_\Omega (f(\nabla u) + g(u))\,dx
+ \int_{\Gamma_1} \gamma(u) \, dH^{d-1} \ ,\ u=0 \ \hbox{on $\Gamma_0$}
\right\} \, $$
where $g, \gamma$ are possibly non convex functions with suitable
growth conditions and $f$ is a convex intergrand on $\R^d$. Here
$(\Gamma_0 , \Gamma_1)$ is a partition of $\partial \Omega$. A
challenging issue is to characterize the global minimizers of
such a problem and the stability of the minimal value (with respect
for instance to small deformations of the domain $\Omega$).
We present a duality scheme in which the dual problem reads quite
nicely as a linear programming problem. The solvability of this
dual problem is a major issue. It can be achieved in the one dimensional
case and in higher dimensions under special assumptions on $f,g$.
Applications are given for a class of free boundary problems.
Sun-Yung Alice Chang (Princeton University)
Sobolev trace inequality on manifolds
I will report on some recent joint work with Antonio Ache in
which we study 4-th order Sobolev trace inequalities. These inequalities
arise naturally as a term in the log determinant formula of conformal
Laplace and Robin operators on 4-manifolds with boundary. In the
special case of Euclidean 4-balls, this generalizes the classical
Lebedev-Milin inequality on 2-balls.
Fernando Codá Marques (Instituto
Nacional de Matemática Pura e Aplicada) (joint with Andre
Neves)
Min-max Theory and Applications II
The search for closed geodesics is a classical question in geometry.
We will survey the field and explain how min-max theory has been
used to find closed geodesics or minimal hypersurfaces. We will
then talk about our latest result where we showed that manifolds
with positive Ricci admit an infinite number of minimal embedded
hypersurfaces.
Alessio Figalli (University of Texas at
Austin)
A transportation approach to random matrices
Optimal transport theory is an efficient tool to construct change
of variables between probability densities. However, when it comes
to the regularity of these maps, one cannot hope to obtain regularity
estimates that are uniform with respect to the dimension except
in some very special cases (for instance, between uniformly log-concave
densities).
In random matrix theory the densities involved (modeling the distribution
of the eigenvalues) are pretty singular, so it seems hopeless
to apply optimal transport theory in this context. However, ideas
coming from optimal transport can still be used to construct approximate
transport maps (i.e., maps which send a density onto another up
to a small error) which enjoy regularity estimates that are uniform
in the dimension. Such maps can then be used to show universality
results for the distribution of eigenvalues in random matrices.
The aim of this talk is to give a self-contained presentation
of these results.
Nassif Ghoussoub (University of British
Columbia)
On the Hardy-Schrödinger operator with a singularity
on the boundary
I will consider borderline variational problems involving the
Hardy-Schr\"odinger $L_\gamma:=-\Delta - \frac{\gamma}{|x|^2}$
operator on a domain $\Omega \subset {\bf R}^n$.
The classical Hardy inequality says that $L_\gamma$ is a non-negative
operator
as long as $\gamma \leq \frac{(n-2)^2}{4}$. The situation is much
more interesting when $0\in \partial \Omega$.
For one, the operator could then be non-negative for $\gamma$
up to $ \frac{n^2}{4}$. The problem of whether the Dirichlet boundary
problem $L_\gamma u=\frac{u^{2^*(s)-1}}{|x|^s}$ on $\Omega$.
%\[\hbox{$-\Delta u- \frac{\gamma}{|x|^2}u=\frac{u^{2^*(s)-1}}{|x|^s}$
\quad on $\Omega$}\]
has positive solutions, is closely related to whether the best
constants in the
Caffarelli-Kohn-Nirenberg inequalities are attained. Here $2^*(s)=\frac{2(n-s)}{n-2}$
and $s\in [0, 2)$.
Recently, C.S. Lin et al. showed that this is indeed the case
when $\gamma < \frac{(n-2)^2}{4}$ under the condition that
the mean curvature of the domain at $0$ is negative, extending
previous work by Ghoussoub-Robert who dealt with the case $\gamma
=0$.
The case when $\frac{(n-2)^2}{4}\leq \gamma <\frac{n^2}{4}$
turned out to be much more interesting and quite delicate. A detailed
analysis
of $L_\gamma$ shows that, surprisingly, $\gamma=\frac{n^2-1}{4}
$ is another critical threshold for the Hardy-Schr\"odinger
operator, beyond which a ``positive
mass theorem" --in the spirit of Shoen-Yau --is required.
\\
This is joint work with Frederic Robert from the Universit\'e
of Nancy.
Pengfei Guan (McGill University)
New mean curvature estimates for immersed hypersurfaces
We establish two types of mean curvature estimates for immersed
hypersurfaces. The first is the estimate for immersed compact
hypersurfaces in a general ambient Riemannian manifold. It's a
generalization of a classical result for Weyl's isometric embedding
problem, here the estimate is obtained for degenerate cases in
any dimensions for general ambient space. The second estimate
is for non-compact embedded convex hypersurfaces in $R^{n+1}$.
This new estimate yields a rigidity theorem for codimension one
shrinking Ricci solitons.
Gerhard Huisken (Universität Tübingen)
Mean curvature flow with surgery
Francesco Maggi (University of Texas)
A general compactness theorem for Plateau's problem
Plateau's problem (minimizing area among surfaces spanning a
given boundary curve) is one of the most basic questions in the
Calculus of Variations. To give a precise mathematical formulation
of this problem one needs to specify notions of surface, area
and boundary, and the properties of solutions depend crucially
on these choices. For example, by solving Plateau's problem (on
surfaces in R3) in the framework of the theory of currents one
finds solutions which are always smooth away from their boundary,
in contrast to what is occasionally observed on real soap films
spanning specific boundary curves. Various alternative formulations
have been proposed in the years, starting with the pioneering
work by Reifenberg, and ending up with more recent contributions
by David, De Pauw, Harrison and Pugh, and others. We provide here
a compactness principle which is applicable to different formulations
of Plateau's problem in codimension one and which is exclusively
based on the theory of Radon measures and elementary comparison
arguments. Exploiting some additional techniques in geometric
measure theory, we can use this principle to give a different
proof of a theorem by Harrison and Pugh and to answer a question
raised by Guy David about "sliding minimizers". This
is a joint work with Camillo De Lellis and Francesco Ghiraldin
(U. Zurich).
Aaron Naber (Northwestern University)
Einstein Manifolds and the Codimension Four Conjecture
In this talk we discuss the recent solution of the codimension
four conjecture. Roughly, this tells us that if we study a noncollapsing
limit of Einstein manifolds (M^n,g_i)->(X,d), or more generally
just manifolds with bounded Ricci curvature, then X is smooth
away from a closed set of codimension four. Using the quantitative
stratification one can use this to prove a priori L^p estimates
for the curvature |Rm| on Einstein manifolds for all p<2, and
L^2 bounds in dimension four. Another application is the proof
of Anderson's finite diffeomorphism conjecture. This is joint
work with Jeff Cheeger.
Andre Neves (Imperial College London)
(joint with Fernando Coda-Marques)
Min-max Theory and Applications I
Alexander Nabutovsky (University of
Toronto)
Curvature-free estimates for solutions of variational problems
in Riemannian geometry
In my survey talk I will discuss when it is possible to give
curvature-free upper bounds for lengths/areas/volumes of the ``simplest"
solutions of some classical variational problems on Riemannian
manifolds. These stationary objects will include three simple
periodic geodesics on Riemannian 2-spheres, periodic geodesics,
minimal hypersurfaces, and different geodesics between a fixed
pair of points in closed Riemannian manifolds. I am going to mention
several open problems.
Ovidiu Savin (Columbia University)
Higher regularity for certain thin free boundaries.
We discuss the higher regularity of certain ``thin" free
boundary problems in which the free boundary has codimension two.
We focus on the thin obstacle problem and thin one-phase free
boundary problem. This is a joint work with D. De Silva.
Richard Schoen (Stanford University and
UC, Irvine)
Variational problems related to sharp eigenvalue estimates
on surfaces
The problem of finding metrics on surfaces of a fixed area with
maximum lowest eigenvalue has been much studies, but is still not
well understood in general. There are analogous questions for surfaces
with boundary. In this talk we will describe the problems, summarize
the state of the subject, and present some of our recent results
(partially joint with A. Fraser) on the problem.
SHORT TALKS
Abbas Moameni, Carleton University
A characterization for solutions of the Monge-Kantorovich mass
transport problem
I will present a measure theoretical approach to study the solutions
of the Monge-Kantorovich optimalmass transport problems. This
approach together with Kantorovich duality provide a tool tostudy
the support of optimal plans for the mass transport problem involving
general cost functions. I also talk about a criterion for the
uniqueness.
Adrian Tudorascu, West Virginia
University
Lagrangian solutions for the Semi-Geostrophic Shallow Water system
in physical space
Coauthors: Mikhail Feldman, University of Wisconsin-Madison
SGSW is a third level specialization of Navier-Stokes (via Boussinesq,
then Semi-Geostrophic),and it accurately describes large-scale,
rotation-dominated atmospheric flow under the extra-assumption
that the horizontal velocity of the fluid is independent of the
vertical coordinate. The Cullen-Purser stability condition establishes
a connection between SGSW and Optimal Transport by imposing semi-convexity
on the pressure; this has led to results of existence of solutions
in dual space (i.e., where the problem is transformed under a
non-smooth change of variables). In this talk I will present very
recent results on existence and weak stability of solutions in
physical space (i.e., in the original variables) for general initial
data, the very first of their kind. This is based on joint work
with M. Feldman (UW-Madison).
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