## 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)
Title TBA

Friday, November 4, Time and Room TBA
Carlos Perez (BCAM, Bilbao)
Title TBA

Monday, November 7, Time and Room TBA
Steve Lester (CRM and McGill)
Title TBA

Friday, November 11, Burnside 920, 13:00-14:00
Michael Hitrik (UCLA)
Title TBA

Friday, November 11, Burnside 920, 14:00-15:00
Katya Krupchyk (UC Irvine)
Title TBA

## WINTER 2017

Wednesday, March 15, Time and Room TBA
Alexander Olevskii (Tel Aviv)
Title TBA

## 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.