2018-19 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 (dmitry.jakobson@mcgill.ca), Iosif Polterovich (iossif@dms.umontreal.ca) or Alina Stancu (alina.stancu@concordia.ca)

FALL 2018

Friday, November 23, 13:30-14:30, McGill, Burnside Hall, Room 1104
Herve Lombaert (ETS Montreal/Inria Sophia-Antipolis)
Spectral Correspondence and Learning of Surface Data - Example on Brain Surfaces
Abstract: How to analyze complex shapes, such as of the highly folded surface of the brain? In this talk, I will show how spectral representations of shapes can benefit neuroimaging and, more generally, problems where data fundamentally lives on surfaces. Key operations, such as segmentation and registration, typically need a common mapping of surfaces, often obtained via slow and complex mesh deformations in a Euclidean space. Here, we exploit spectral coordinates derived from the Laplacian eigenfunctions of shapes and also address the inherent instability of spectral shape decompositions. Spectral coordinates have the advantage over Euclidean coordinates, to be geometry aware and to parameterize surfaces explicitly. This change of paradigm, from Euclidean to spectral representations, enables a classifier to be applied directly on surface data, via spectral coordinates. The talk will focus, first, on spectral representations of shapes, with an example on brain surface matching, and second, on the learning of surface data, with an example on automatic brain surface parcellation.
Friday, October 26, 14:00-15:00, McGill, Burnside Hall, Room 1120 (note the new room!)
K. Luli (UC Davis)
Variational Problems on Arbitrary Sets
Abstract: Let E be an arbitrary subset of R^n. Given real valued functions f defined on E and g defined on R^n, the classical Obstacle Problem asks for a minimizer of the Dirichlet energy subject to the following two constraints: (1) F = f on E and (2) F >= g on R^n. In this talk, we will discuss how to use extension theory to construct (almost) solutions directly. We will also explain several recent results that will help lay the foundation for building a complete theory revolving around the belief that any variational problems that can be solved using PDE theory can also be dealt with using extension theory.
Friday, October 19, 13:30-14:30, McGill, Burnside Hall, Room 1104
Hans Christianson (UNC Chapel Hill)
Quantum Ergodic properties for eigenfunctions on triangles
Abstract: In this talk we will discuss one aspect of the idea of classical-quantum correspondence in planar domains called quantum ergodicity. For a planar domain, the classical problem is to imagine the domain is a billiard table, while the quantum problem is to imagine it is a drum. Waves on a drum head tend to follow along billiard trajectories, so if the billiard trajectories are sufficiently chaotic, we expect the waves to spread out. This is called quantum ergodicity. We will informally discuss some of the subtle history of this topic, where detailed information about the billiard flow is essential for the quantum problem. We will then move on to a very unsubtle result which hints at quantum ergodicity for triangular domains without using any dynamical systems information at all. This result follows from a very short proof using nothing more than integrations by parts.
Friday, October 5, 13:30-14:30, McGill, Burnside Hall, Room 1104
J. De Simoi (Toronto)
On spectral rigidity for generic symmetric convex billiards
Abstract: In this talk I will present a most recent result in the setting of length- spectral rigidity for convex billiards. In a joint work with A. Figalli and V. Kaloshin we show that there exists an open and dense set S of smooth convex axially-symmetric planar domains so that every non-isometric deformation of some D in S necessarily changes the length of a periodic orbit of the billiard in D.
Friday, September 21, 13:30-14:30, McGill, Burnside Hall, Room 1104
Javad Mashreghi (Laval)
Polynomial approximation in super-harmonically weighted Dirichlet spaces
Abstract: Taylor polynomials are natural objects for approximation in function spaces. Indeed, it works in several function spaces, e.g., Hardy and Bergman spaces. However, it also fails in some cases and a remedy is needed, e.g., disc algebra and weighted Dirichlet spaces. We show that in the latter, Taylor polynomials may diverge. However, by properly adjusting the last coefficient we produce a convergent sequence in local Dirichlet spaces. We also show that in super-harmonically weighted Dirichlet spaces, Fejer averages provide a convergent sequence. Joint work with T. Ransford.
Monday, September 17, 13:30-14:30, McGill, Burnside Hall, Room 1104
Matthew de Courcy-Ireland (Princeton)
Monochromatic waves at shrinking scales
Abstract: We study an ensemble of random functions on a compact Riemannian manifold. These random functions have been proposed by Berry as a model for high-frequency Laplace eigenfunctions in chaotic settings. We prove that, with high probability, they are evenly distributed in the mean square sense on shrinking geodesic balls. The rate of shrinking comes within a logarithmic factor of the optimal wave scale, and equidistribution occurs simultaneously over all possible centers for the ball.

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