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 Galia Dafni (galia.dafni@concordia.ca)

SUMMER 2019

Joint seminar with Geometric Analysis
Friday, August 9, 13:30-14:30, McGill, Burnside Hall, Room 1104
Ali Aleyasin (Stony Brook)
Singular and degenerate Monge-Ampere equations
Abstract: It is well known that several non-linear elliptic partial differential equations have applications in various fields of geometry and analysis, including but not limited to the Calabi problem, the Weyl, and Minkowski problems, and optimal transport. An important class of such non-linear equations are the real and complex Monge-Amp\`ere equations. Although the case of strictly elliptic equations with smooth source term has been rather well-understood, the behaviour of solutions in the vicinity of possible singularities or degeneracies of the source term is far from being understood. This corresponds to the vanishing or blowing up of the prescribed curvature in the Weyl problem. In this talk, I will present an application of a differential geometric approach to the study of certain singularities and degeneracies of elliptic complex Monge-Amp\`ere equations. This approach will allow new estimates for solutions to be derived. I shall also outline how the idea works in the case of several other important geometric partial differential equations.

WINTER 2019

Friday, June 7, 13:30-14:30, McGill, Burnside Hall, Room 1104
Felix Finster (Regensburg)
A positive mass theorem for static causal fermion systems
Abstract: After a short general introduction to causal fermion systems and to the positive mass theorem for asymptotically flat Riemannian manifolds, I will explain how to describe an asymptotically flat manifold by a static causal fermion system. The total mass of a static, asymptotically flat causal fermion system is defined abstractly, and it is shown that this definition reduces to the ADM mass in a certain limiting case. I will also outline the proof that for a minimizer of the causal action principle, the total mass is always non-negative. I am reporting on joint work with Andreas Platzer.

Friday, May 31, Concordia, 13:30-14:30, Concordia, Library Building, Room LB 921-4
Kinvi Kangni (Universite Felix Houphouet-Boigny)
Spherical Grassmannian on a reductive Lie group
Abstract: Let $G$ be a locally compact group, $K$ a compact subgroup of $G$ and $\delta$ an arbitrary class of irreducible unitary representations of $K$. The $p$-$\delta$-spherical Grassmannian $\mathcal{G}_{p,\delta}$ is an equivalence class of spherical functions of type $\delta$-positive of height $p$.
In this talk, we construct some elements of $\mathcal{G}_{p,\delta}$ on reductive Lie group using a generalized Abel transform.
And if the discret serie is not empty,we give a extension of Paley Wiener theorem using a compact Cartan subgroup of $G.$

Joint seminar with Geometric Analysis
Friday, May 3, 13:30-14:30, McGill, Burnside Hall, Room 1104
Luca Martinazzi (University of Padua, Italy)
News on the Moser-Trudinger inequality: from sharp estimates to the Leray-Schauder degree
Abstract: The existence of critical points for the Moser-Trudinger inequality for large energies has been open for a long time. We will first show how a collaboration with G. Mancini allows to recast the Moser-Trudinger inequality and the existence of its extremals (originally due to L. Carleson and A. Chang) under a new light, based on sharp energy estimate. Building upon a recent subtle work of O. Druet and P-D. Thizy, in a work in progress with O. Druet, A. Malchiodi and P-D. Thizy, we use these estimates to compute the Leray-Schauder degree of the Moser-Trudinger equation (via a suitable use of the Poincare-Hopf theorem), hence proving that for any bounded non-simply connected domain the Moser-Trudinger inequality admits critical points of arbitrarily high energy. In a work in progress with F. De Marchis, O. Druet, A. Malchiodi and P-D. Thizy, we expect to use a variational argument to treat the case of a closed surface.

Friday, April 5, 13:30-14:30, McGill, Burnside Hall, Room 1104
Michael Levitin (University of Reading)
Sharp eigenvalue asymptotics for Steklov problem on curvilinear polygons
Abstract: I will discuss a work in progress (joint with Leonid Parnovski, Iosif Polterovich and David Sher) on Steklov (or Dirichlet-to-Neumann map) eigenvalue asymptotics for curvilinear polygonal domains in R^2. The results are unexpected, and the asymptotics depends on the arithmetic properties of the angles of the polygon. There are also connections to classical probleems of hydrodynamics (the sloping beach problem and the sloshing problem) and to the Laplacian on quantum graphs.

Friday, March 29, 13:30-14:30, McGill, Burnside Hall, Room 1104
Ilia Binder (Toronto)
Computability and Complexity in Complex Analysis and Complex Dynamics.
Abstract: In this talk, I will discuss recent advances in the computability of various objects arising in Complex Dynamics and Theory of Univalent functions. After a brief introduction to the general Computability and Complexity Theory, I will talk about the computational properties of polynomial Julia sets and conformal maps. I will also consider the computability questions related to the boundary extensions of conformal maps. Based on joint work with M. Braverman (Princeton), C. Rojas (Universidad Andres Bello), and M. Yampolsky (University of Toronto).

Friday, March 22, 13:30-14:30, McGill, Burnside Hall, Room 1104
Mikhail Karpukhin (Irvine)
Applications of algebra and topology to isoperimetric eigenvalue inequalities.
Absract: Spectrum of the Laplace-Beltrami operator is one of the fundamental invariants of a Riemannian manifold. Finding the optimal isoperimetric inequalities for its eigenvalues is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. One of the main attractions of this problem is the variety of methods employed to study it. In the present talk we demonstrate this feature and, in particular, outline the connections to the theory of minimal surfaces, algebraic geometry and topology. These include recent applications of moduli spaces and cobordism theory. The talk is based on joint works with V. Medvedev, N. Nadirashvili, A. Penskoi and I. Polterovich.

Friday, March 15, 13:30-14:30, McGill, Burnside Hall, Room 1104
Siran Li (CRM, McGill)
Some Problems On Harmonic Maps from B^3 to S^2.
Abstract: Harmonic map equations are an elliptic PDE system arising from the minimisation of Dirichlet energies between two manifolds. In this talk we present some recent works concerning the symmetry and stability of harmonic maps. We construct a new family of ''twisting'' examples of harmonic maps and discuss the existence, uniqueness and regularity issues. In particular, we characterise the singularities of minimising general axially symmetric harmonic maps, and construct non-minimising general axially symmetric harmonic maps with arbitrary 0- or 1-dimensional singular sets on the symmetry axis. Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$ to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data for $p>=2$. (Joint work with Prof. Robert Hardt.)

Friday, March 1, 13:30-14:30, McGill, Burnside Hall, Room 1104
Alexandre Girouard (Laval)
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Abstract: The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the Riemannian metric on the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires wild perturbations of the metric in arbitrarily small neighborhoods of the boundary. In recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the corresponding eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. The proof of is based on a generalized Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.

Joint seminar, Analysis and Number Theory
Monday, February 25, McGill, 13:30-14:30, McGill, Burnside Hall, Room 1104
Dmitry Logachev (UFAM Manaus)
Anderson t-motives - a parallel world to abelian varieties, in finite characteristic.
Abstract: Formally, Anderson t-motives (generalizations of Drinfeld modules) are some modules over a ring of non-commutative polynomials in two variables over a complete algebraically closed field of finite characteristic. Surprisingly, it turns out that their properties are very similar to the properties of abelian varieties (more exactly, of abelian varieties with multiplication by an imaginary quadratic field). For example, we can define Tate modules of Anderson t-motives, Galois action on them, lattices, modular curves, L-functions etc. Nevertheless, this analogy is far to be complete. There is no functional equation for their L-functions; notion of the algebraic rank is not known yet; 1 - 1 correspondence between Anderson t-motives and lattices also is known only for Drinfeld modules. A survey of the theory of Anderson t-motives and statements of some research problems will be given.

Friday, February 15, McGill, 13:30-14:30, McGill, Burnside Hall, Room 1104
Javad Mashreghi (Laval)
Carleson measures for the Dirichlet space
Abstract: We show that a finite measure $\mu$ on the unit disk is a Carleson measure for the Dirichlet space $\mathcal{D}$ if it satisfies the one-box condition $\mu(\, S(I) \,) = O(\, \varphi(|I|) \,)$, where $\varphi$ such that $\varphi(x)/x$ is integrable. We also show that the integral condition on $\varphi$ is sharp.

Friday, February 8, McGill, 13:30-14:30, McGill, Burnside Hall, Room 1104
Michael Lipnowski (McGill)
Geometry of the smallest 1-form Laplacian eigenvalue on hyperbolic manifolds.
Abstract: I’ll describe a relationship between the smallest 1-form Laplacian eigenvalue and surface complexity on hyperbolic manifolds. We'll then speculate on "surface theft", a prospect based on this relationship for proving good lower bounds on the smallest 1-form Laplacian eigenvalue for big congruence arithmetic hyperbolic 3-manifolds M. This is joint work with Mark Stern.

Thursday, January 24, 14:00-15:00, Concordia, Room LB 921-4
Almut Burchard (Toronto)
A geometric stability result for Riesz-potentials
Abstract: Riesz' rearrangement inequality implies that integral functionals (such as the Coulomb energy of a charge distribution) that are defined by a pair interaction potential (such as the Newton potential) which decreases with distance are maximized (under appropriate constraints) only by densities that are radially decreasing about some point. I will describe recent and ongoing work with Greg Chambers on the stability of this inequality for the special case of the Riesz-potentials in n dimensions (given by the kernels |x-y|^-(n-s)), for densities that are uniform on a set of given volume. For 1< s < n, we bound the square of the symmetric difference of a set from a ball by the difference in energy of the corresponding uniform distribution from that of the ball.

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.