## Montreal Analysis Seminar

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

## 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)
Title TBA

## 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
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.