## Montreal Analysis Seminar

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

## 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, (date and time to be confirmed), Room TBA
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.