## Montreal Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia,
McGill or Universite de Montreal

For suggestions, questions etc. please contact Dmitry Jakobson
(jakobson@math.mcgill.ca), Iosif Polterovich
(iossif@dms.umontreal.ca) or
Alina Stancu (alina.stancu@concordia.ca)

## SUMMER/FALL 2017

** Monday, June 12, 13:30-14:30, McGill, Room 1205**

** Siran Li** (Oxford, CRM, McGill)

Fluid Dynamics, Isometric Embeddings, and the Theory of Compensated
Compactness

** Abstract:**
In this talk we shall discuss some linkages between the PDEs of fluid dynamics
and the Gauss-Codazzi-Ricci equations for isometric embeddings of Riemannian
manifolds. First, we prove the existence of isometric embeddings of certain
negatively curved 2-dimensional surfaces into $\mathbb{R}^3$ via a ''fluid
dynamic formulation'' of the Gauss-Codazzi equations. The key technique is
the method of compensated compactness, previously used by Lax, DiPerna,
Morawetz and others to show the existence of solutions to hyperbolic
conservation laws and transonic gas dynamics. Second, we give global and
intrinsic proofs for the weak rigidity of isometric embeddings of
Riemannian/semi-Riemannian manifolds, using the generalised compensated
compactness theorems recently established in the geometric settings. Third,
we discuss an elementary proof for the existence of infinitely many ''wild''
solutions to the Euler equations, previously constructed by De Lellis,
Szekelyhidi and others via convex integrations. Our proof relies on a direct
dynamic analogue with the isometric embeddings and the celebrated results by
Nash and Gromov. The talk is based on joint works with G.-Q. Chen (Oxford),
M. Slemrod (Wisconsin-Madison), Dehua Wang (Pittsburgh), and Amit Acharya
(Carnegie Mellon).

** Friday, June 16, 13:30-14:30, McGill, Room 1214**

** Francois Nicoleau** (Nantes)

Local inverse scattering at a fixed energy for radial
Schrodinger operators and localization of the Regge poles.

** Abstract:**
We study inverse scattering problems at a fixed energy for radial
Schrodinger operators on $\R^n$, $n \geq 2$. First, we consider the
class $\mathcal{A}$ of potentials $q(r)$ which can be extended
analytically in $\Re z \geq 0$ such that $\mid q(z)\mid \leq C \ (1+
\mid z \mid )^{-\rho}$, $\rho > \frac{3}{2}$. If $q$ and $\tilde{q}$ are
two such potentials and if the corresponding phase shifts $\delta_l$
and $\tilde{\delta}_l$ are super-exponentially close, then
$q=\tilde{q}$. Secondly,
we study the class of potentials $q(r)$ which can be split into
$q(r)=q_1(r) + q_2(r)$ such that $q_1(r)$ has compact support and $q_2
(r) \in \mathcal{A}$.
If $q$ and $\tilde{q}$ are two such potentials, we show that for any
fixed $a>0$,
${\ds{\delta_l - \tilde{\delta}_l \ = \ o \left( \frac{1}{l^{n-3}} \
\left( {\frac{ae}{2l}}\right)^{2l}\right)}}$ when $l \rightarrow
+\infty$ if and only if $q(r)=\tilde{q}(r)$ for almost all $r \geq a$.
The proofs are close in spirit with the celebrated Borg-Marchenko
uniqueness theorem, and rely heavily on the localization of the Regge
poles. This is a joint work with Thierry Daude (Universite de
Cergy-Pontoise).

** Monday, July 17, 13:30-14:30, Univ. de Montreal, Room TBA**

** David Sher** (DePaul University)

Title TBA

** Monday, July 17, 14:30-15:30, Univ. de Montreal, Room TBA**

** Magda Khalile** (Univ. Paris Sud)

Title TBA

** Friday, September 15, Time and Room TBA**

** Eli Liflyand** (Bar Ilan University)

Title TBA

** Monday, September 18, Time and Room TBA**

** Alex Iosievich** (Rochester University)

Title TBA

## WINTER 2017

** Friday, February 17, Concordia, 13:30, Room LB 921-04 **

**Mohammad Najafi** (T.U. Wien and Concordia)

On the classification of ancient solutions to curvature flows
on the sphere

** Abstract:**
An ancient solution to a curvature flow is a solution that exists backwards
in time forever. In this talk, I will discuss the classification of ancient
solutions to curvature flows on the sphere. I will prove that the sphere
exhibits a very strong rigidity: any ancient solution of a curvature flow
which satisfies a backwards in time uniform bound on mean curvature must be
stationary or a family of shrinking geodesic sphere. The main tools are
geometric, employing the maximum principle, a rigidity result in the sphere
and an Alexandrov reflection argument.

** Monday, February 20, McGill, 13:30, Burnside 920**

**Melissa Tacy** (ANU)

L^p estimates for eigenfunctions on manifolds with boundary

** Abstract:**
Measuring the L^p mass of an eigenfunction allows us to determine its
concentration properties. On a manifold without boundary such estimates
follow from short time properties of the wave or semiclassical
Schrödinger propagators. However the presence of a boundary opens the
possibility for multiple reflections even in short time. This will lead
to greater concentration of the eigenfunction (displayed by higher L^p
norms). It is known, for example, that the whispering gallery modes show
this higher concentration. In this talk I will introduce a method of
studying the boundary L^p problem semiclassically by considering an
exact solution to the boundary problem and an approximate solution to
the ambient Helmholtz equation.

** Wednesday, February 22, McGill, 14:30-15:30, Burnside 1234**

**Victor Ivrii** (Toronto)

Spectral asymptotics for fractional Laplacian

** Abstract:**
Consider a compact domain with the smooth boundary in the Euclidean space.
Fractional Laplacian is defined on functions supported in this domain as a
(non-integer) power of the positive Laplacian on the whole space restricted
then to this domain. Such operators appear in the theory of stochastic
processes. It turns out that the standard results about distribution of
eigenvalues (including two-term asymptotics) remain true for fractional
Laplacians. There are however some unsolved problems.

** Friday, March 17, McGill, 13:30, Burnside 920**

**Asilya Suleymanova** (Humbolt)

Spectral geometry of manifolds with conic singularities

** Abstract:**
n 1966 Mark Kac asked his famous question “Can one hear the shape of a drum?”
that corresponds to the following mathematical problem. Suppose we are
given a sequence of eigenvalues of some geometrical operator on a manifold.
What information about geometry of the manifold can be derived from the
sequence of eigenvalues? If a manifold is allowed to have singularities,
then one can ask a similar question “Can one hear the presence of a
singularity?”. In the talk we investigate this problem for manifolds with
conic singularities. Our main tool is small-time asymptotic expansion of the
heat trace.

We begin the talk with an overview of the heat kernel of Laplace operator
on a manifold. In case of a compact smooth manifold the heat trace expansion
gives some geometrical information such as dimension, volume and total
scalar curvature of the manifold. In case of a manifold with conic
singularities the heat trace expansion is more difficult than in the smooth
case. Using Singular Asymptotics Lemma of Bruening and Seeley we obtain
information about a singularity from the heat trace expansion.

** 2017 Nirenberg lectures in Geometric Analysis**

** March 24-28, 2017, CRM**

Announcement

** Friday, March 31, 13:30, Concordia, LB 921-04**

**Dmitry Faifman** (Toronto)

The intrinsic volumes of indefinite quadratic forms

** Abstract:**
The Euclidean intrinsic volumes - the last two of which are volume
and surface area - admit many equivalent descriptions: as integrals of
curvature, or through Crofton-type formulas, or by their characterization
as the unique continuous valuations invariant under rigid motions, which is
Hadwiger's now classical result. We will recall some notions of the theory
of convex valuations, and then discuss how those approaches can be
generalized when the Euclidean quadratic form is replaced by a
non-degenerate quadratic form of indefinite signature. We will see in
particular how the non-vanishing of a certain Selberg-type integral
implies the existence of Crofton formulas in the indefinite case.

** Friday, April 7, 13:30-14:30, Burnside 920**

**Xinliang An** (Toronto)

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.

** Friday, April 21, 13:30-14:30, Burnside 920**

**Jonathan Ben-Artzi** (Cardiff)

Spectral methods in ergodic theory: obtaining uniform ergodic theorems

** Abstract:**
Von Neumann's original proof of the ergodic theorem for one-parameter
families of unitary operators relies on a delicate analysis of the spectral
measure of the associated flow operator and the observation that over
long times only functions that are invariant under the flow make a
contribution to the ergodic integral. In this talk I shall show that for a
specific class of generators -- namely vector fields -- the spectral
measure is rather simple to understand via a Fourier transform. This allows
us to obtain a uniform ergodic theorem (on an appropriate subspace). The
analysis is performed in both Sobolev and weighted-Sobolev spaces. These
results are closely related to recent results on the 2D Euler equations,
and have applications to other conservative flows, such as those governed
by the Vlasov equation (modeling plasmas and galactic dynamics,
for instance).

** Wednesday, May 17, 2017. 13:30-14:30, Burnside 920**

** Alena Erchenko ** (Penn State)

A flexibility program in dynamical systems and first results.
** Abstract:**
We introduce the flexibility program proposed by A. Katok and discuss
first results. We show the flexibility of the entropy with respect to the
Liouville measure and topological entropy for geodesic flow on negatively
curved surfaces with fixed genus and total area (joint with A. Katok). Also,
we point out some restrictions which come from additionally fixing a
conformal class of metrics (joint with T. Barthelme). If time permits, we
describe a flexibility result for Lyapunov exponents for smooth expanding
maps on a circle of fixed degree.

## FALL 2016

** Friday, September 16, 13:30-14:30, McGill, Burnside 920**

**Jeff Galkowski** (CRM/McGill)

A Quantitative Vainberg Method for Black Box Scattering

** Abstract: **
We give a quantitative version of Vainberg’s method relating pole
free regions to
propagation of singularities for black box scatterers. In particular, we
show that there is a
logarithmic resonance free region near the real axis of size t with
polynomial bounds on the
resolvent if and only if the wave propagator gains derivatives at rate t.
Next we show that if
there exist singularities in the wave trace at times tending to infinity
which smooth at rate t, then
there are resonances in logarithmic strips whose width is given by t. As
our main application
of these results, we give sharp bounds on the size of resonance free
regions in scattering on
geometrically nontrapping manifolds with conic points. Moreover, these
bounds are generically
optimal on exteriors of nontrapping polygonal domains.

** Monday, October 3, UdeM, 14:00-15:00, U. Montreal,
Pav. Andre Aisenstadt, Room 5448**

**Alexandre Girouard** (Laval University)

Discretization and Steklov eigenvalues of compact manifold with boundary

** Abstract: **
Let M be a compact n-dimensional Riemannian manifolds with cylindrical
boundary, Ricci curvature bounded below by k and injectivity radius bounded
below by r. We introduce a notion of discretization of the manifold M,
leading to a graph with boundary which is roughly isometric to M , with
constant depending only on k, r, n. In this context, we prove a uniform
spectral comparison inequality between the Steklov eigenvalues of the
manifold M and those of its discretization. Some applications to the
construction of sequences of surfaces with boundary of fixed length and
with large Steklov eigenvalues are given. In particular, we obtain such a
sequence for surfaces with connected boundary. These applications are based
on the construction of graph-like surfaces which are modeled using
sequences of graphs with good expansion properties. This talk is based on
joint work with Bruno Colbois (Neuchatel) and Binoy Raveendran.

** Friday, October 28, Concordia, 14:00, LB 921-4**

**Scott Rodney** (Cape Breton university)

The Poincare Inequality and the $p$-Laplacian

** Abstract: **
In this talk I will explore a necessary and sufficient condition for the
validity of a Poincar\'e - type inequality.
This work will be presented in the context of the degenerate Sobolev
spaces $W^{1,p}_Q(\Omega)$ and the Degenerate
$p$-Laplacian given by $\Delta_p u =
\text{Div}\Big( \Big|\sqrt{Q(x)}\nabla u(x)\Big|^{p-2} Q(x)
\nabla u\Big)$ where
$Q(x)$ is a non-negative definite $L^1_{\text{loc}}$ matrix valued function.

** Special Analysis/Applied seminar**

** Friday, October 28, McGill, 15:30-16:30, Burnside 1120**

** Daniel Han-Kwan** (Ecole Polytechnique)

The quasineutral limit of the Vlasov-Poisson system

**Abstract: ** We shall review a few recent results about the
quasineutral limit of the Vlasov-Poisson system.
This is a singular limit in which the Poisson equation degenerates and
which gives rise to singular non-linear PDEs that are ill-posed in general.

** Friday, November 4, McGill, Burnside 920, 13:30-14:30**

**Carlos Perez** (BCAM, Bilbao)

Rough Singular Integrals: A1 theory

** Abstract:**
In this mostly expository talk we will discuss new results for rough
Singular Integral Operators involving $A_1$ weights. These are convolution
operators whose kernels do not satisfy the standard regularity conditions
(Lipschitz or Dini). Being more precise we will discuss estimates for
these operators in the context of weighted $L^p$ spaces with weights
satisfying the $A_1$ condition. These results are known since the 80's
(work of Duoandikoetxea y Rubio de Francia) for $A_p$ weights but we found
quantitative estimates in the context of the special class of $A_1$. Most
probably these results are far from satisfactory and we will discuss some
open problems.

This is a joint work with I. Rivera-Rios and L. Roncal.

** Monday, November 7, McGill, Burnside 920, 14:00-15:00**

**Steve Lester** (CRM and McGill)

Quantum unique ergodicity for half-integral weight automorphic forms

** Abstract:**
Given a smooth compact Riemannian manifold (M, g) with no boundary an
important problem in Quantum Chaos studies the distribution of L^2 mass of
eigenfunctions of the Laplace-Beltrami operator in the limit as the
eigenvalue tends to infinity. For M with negative curvature Rudnick and
Sarnak have conjectured that the L^2 mass of all eigenfunctions
equidistributes with respect to the Riemannian volume form; this is known
as the Quantum Unique Ergodicity (QUE) Conjecture. In certain arithmetic
settings QUE is now known. In this talk I will discuss the analogue of QUE
in the context of half-integral weight automorphic forms. This is based on
joint work Maksym Radziwill.

** Friday, November 11, Burnside 920, 13:00-14:00**

**Michael Hitrik** (UCLA)

Spectra for non-selfadjoint operators and integrable dynamics

**Abstract: ** Spectra for non-selfadjoint operators often display
fascinating features, from lattices of eigenvalues for operators of
Kramers-Fokker-Planck type to eigenvalues for operators with analytic
coefficients in dimension one, concentrated to unions of curves. In this
talk, we shall discuss spectra for non-selfadjoint perturbations of
selfadjoint semiclassical operators in dimension 2, assuming that the
classical flow of the unperturbed part is completely integrable. We give
complete asymptotic expansions for all individual eigenvalues in
suitable regions of the complex spectral plane, close to the edges of
the spectral band. It turns out that those eigenvalues have the form of
the "legs in a spectral centipede" and are generated by suitable
rational flow-invariant Lagrangian tori. This is joint work with
Johannes Sjostrand.

** Friday, November 11, Burnside 920, 14:00-15:00**

**Katya Krupchyk** (UC Irvine)

$L^p$ bounds on eigenfunctions for operators with double
characteristics

** Abstract:**
Starting with the pioneering works of Hormander and Sogge, the
question of establishing uniform and, more generally, $L^p$ estimates
for eigenfunctions of elliptic self-adjoint operators in the high energy
limit has been of fundamental significance in spectral theory and
applications. In this talk, after a brief introduction to this circle of
questions, we shall discuss $L^p$ bounds on the ground states for a
class of semiclassical operators with double characteristics, including
some Schrodinger operators with complex-valued potentials. Sharp
bounds are obtained under the assumption that the quadratic
approximations along the double characteristics are elliptic. This is a
joint work with Gunther Uhlmann.

** Monday, November 21, 13:30-14:30, Burnside 920**

**Matthieu Leautaud** (Paris and Montreal)

Quantitative unique continuation and intensity of waves in the shadow
of an obstacle
** Abstract:** The question of global unique continuation is the
following: Does the observation of the wave intensity on a little subdomain
during a time interval (0,T) determine the total energy of the wave? In an
analytic context, this question was solved in 1949 by the well-know
Holmgren-John theorem; in the "smooth case", it was finally tackled by
Tataru-Robbiano-Zuily-Hörmander in the nineties. After a review of these
results, we shall describe the quantitative unique continuation estimate
associated to the qualitative theorem of Tataru-Robbiano-Zuily-Hörmander,
that is, give the optimal logarithmic stability result. In turn, this
estimate yields the optimal a priori bound on the penetration of waves
into the shadow region, as well as the cost of approximate controls for
the wave equation. This is joint work with Camille Laurent.

** Monday, December 12, 13:00-14:00, Burnside 920**

**Thierry Daude** (Cergy-Pontoise)

Non-uniqueness results for the anisotropic Calderon problem with data
measured on disjoint sets.

** Abstract:**
In this talk, we shall give some simple counterexamples to uniqueness
for the anisotropic Calderon problem on Riemannian manifolds with boundary
when the Dirichlet and Neumann data are measured on disjoint sets of the
boundary. We provide counterexamples in the case of two and three
dimensional Riemannian manifolds with boundary having the topology of
circular cylinders in dimension two and toric cylinders in dimension three.
The construction could be easily extended to higher dimensional Riemannian
manifolds. This is joint work with Niky Kamran (McGill University) and
Francois Nicoleau (Université de Nantes).

## SUMMER 2016

** Thursday, June 2, 13:30-14:30, Concordia, Room LB 921-04
**

**Vlad Yaskin** (University of Alberta, Edmonton)

Stability results for sections of convex bodies

** Abstract:**
(pdf)
Let $K$ be a convex body in $\mathbb R^n$.
The parallel section function of $K$ in the direction $\xi\in S^{n-1}$ is
defined by
$$
A_{K,\xi}(t)=\mathrm{vol}_{n-1}(K\cap \{\xi^{\perp}+t\xi\}), \quad
t\in \mathbb R.
$$
If $K$ is origin-symmetric (i.e. $K=-K$), then Brunn's theorem implies
$$
A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t)
$$
for all $\xi\in S^{n-1}$.
The converse statement was proved by Makai, Martini and \'Odor.
Namely, if $A_{K,\xi}(0) =\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all
$\xi\in S^{n-1}$, then $K$ is origin-symmetric.
We provide a stability version of this result. If $A_{K,\xi}(0)$ is
close to $\max_{t\in \mathbb R} A_{K,\xi}(t)$ for all $\xi\in
S^{n-1}$, then $K$ is close to $-K$.
Joint work with Matthew Stephen.

** Monday, August 1, 13:30-14:30, McGill, Burnside 920 (room
to be confirmed) **

**Fedor Pakovich** (Ben Gurion University)

On semiconjugate Rational Functions

** Abstract: ** pdf

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