## Montreal Analysis Seminar

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

## WINTER 2014

Friday, January 10, 13:30-14:30, Burnside 920
Igor Khavkine (University of Trento)
Presymplectic current and the inverse problem of the calculus of variations
Abstract: The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. I present a reformulation of the problem: a variational formulation can be recovered (at least for a subsystem) if and only if the PDE system admits a compatible presymplectic current. The presymplectic current represents a certain cohomology class in the on-shell variational bicomplex associated to the PDE. This construction (following a recent remark due to Bridges, Hydon and Lawson) is a generalization to PDEs of an old observation of Henneaux in the context of ordinary differential equations (ODEs). No constraints on the differential order or number of dependent or independent variables are assumed. [arXiv:1210.0802]

Friday, January 24, 13:30-14:30, Burnside 920
Daniel Elton (University of Lancaster)
Some Analytic Aspects of Dirac Operators in R^n
Abstract: Roughly speaking a Dirac operator is a first order differential operator whose square is a second order operator such as the Laplacian. Such operators were first considered in the 1920s in models of relativistic quantum particles with spin (such as electrons) but have since turned up in other areas of physics (eg., models of graphene) and mathematics (especially differential geometry). In this talk we will review some mathematical aspects of Dirac operators, principally concentrating on such operators in R^n which arise in applications in physics.
Friday, January 31, 13:30-14:30, Burnside 920
Line Baribeau (Laval)
The Spectral Nevanlinna-Pick Problem
Abstract: Let U denote the unit disc, and $\Omega$ the set of complex $n\times n$ matrices with spectral radius less than 1. Given $z_1,\dots,z_m\in U$ and $W_1,\dots, W_m\in \Omega$, does there exist a holomorphic map $F\colon U\to \Omega$ such that $F(z_i)=W_i$ for $i=1,2,\dots,m$? The problem of finding necessary and sufficient conditions on the interpolating data for the existence of $F$ is called the Spectral Nevanlinna-Pick Problem. I will survey some of the approaches used to attack this still unsolved problem, which involve several complex variables and set-valued analytic functions.
Friday, February 7, 13:30-14:30, Burnside 920
Gerasim Kokarev (Munich, LMU)
Multiplicity bounds for Schrodinger eigenvalues on Riemannian surfaces
Abstract: A classical result by Cheng in 1976, improved later by Besson and Nadirashvili, says that the eigenvalue multiplicities of the Schrodinger operator with a smooth potential on a Riemannian surface are bounded in terms of the eigenvalue index and the genus of a surface. I will talk about a recent solution to the question asking whether similar bounds hold for singular potentials. The key ingredient in the proof is based on the study of the Caratheodory prime ends of nodal domains.
Friday, February 21, 13:30-14:30, Burnside 920
Dmitry Jakobson (McGill)
Conformal invariants on weighted graphs
Abstract: This is joint work with Thomas Ng, Matthew Stevenson and Mashbat Suzuki. We define the moduli space of conformal structures on a finite connected simple graph G. We propose a denition of conformally covariant operators on graphs, motivated by conformally covariant operators on Riemannian manifolds. We provide examples of such operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, motivated by recent results of Canzani, Gover, Jakobson and Ponge.
Friday, March 7, 13:30-14:30, Concordia, Library building, Room 921-04
Vassili Nestoridis (University of Athens)
Universality and regularity of the integration operator
Abstract: pdf
Friday, March 14, 13:30-14:30, Burnside 920
Peter Herbrich (Dartmouth)
Magnetic Schrodinger Operators and Mane's Critical Value
Abstract: The talk will deal with lifted magnetic fields on covers of closed manifolds. In particular, the spectra of corresponding periodic magnetic Schrodinger operators can be related to Mane's critical energy values of the corresponding classical Hamiltonian systems. Namely, if the covering transformation group is amenable, then the bottom of the spectrum is bounded from above by Mane's critical value. In the special case of abelian covers, the spectral analysis reduces to the study of shifted magnetic potentials on the compact quotient which parallels the behaviour of Mane's critical value of the corresponding classical systems. The talk will finish with examples of magnetic fields on homogeneous spaces, which facilitate comparisons between the classical and the quantum data.
Friday, April 4, 13:30-14:30, Burnside 920
Leonid Parnovski (University College, London)
Asymptotic behaviour of the spectral function of multidimentional almost-periodic Schroedinger operators.
Abstract: We study the spectral function (the kernel of the spectral projection) of Schroedinger operators with almost-periodic potentials and obtain the complete asymptotic expansion, both on and off the diagonal. This expansion is obtained using the method of gauge transform. This is a joint result with Roman Shterenberg.
Monday, April 14, 13:30-14:30, Burnside 920
Renato Calleja (IIMAS-UNAM)
Construction of quasi-periodic response solutions for forced systems with strong damping
Abstract: I will present a method for constructing quasi-periodic response solutions (i.e. quasi-periodic solutions with the same frequency as the forcing) for over-damped systems. Our method applies to non-linear wave equations subject to very strong damping and quasi-periodic external forcing and to the varactor equation in electronic engineering. The strong damping leads to very few small divisors which allows to prove the existence by using a contraction mapping argument requiring very weak non-resonance conditions on the frequency. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.
Friday, May 9, 13:30-14:30, Burnside 920
Hans Christianson (UNC, Chapel Hill)
From resolvent estimates to damped waves
Abstract: The damped wave equation is a prototype nonself-adjoint PDE, having both advective (wave) component and a diffusive (damping) component. The associated stationary problem then has both real and imaginary parts, leading to complex spectrum. The size of the imaginary parts of the eigenvalues gives information about the rate of energy decay. The interplay between the advection and the location of the diffusion is expressed in terms of geometric control theory. In this talk, I will survey a number of recent and historical results, both with geometric control and lack of geometric control, and explain how resolvent estimates on scattering manifolds can be "glued" into damped wave problems. At the end, if there is time, I will briefly explain some potential new directions in "advection assisted diffusion".

Monday, May 12, 13:30-14:30, Burnside 920
Jan Cannizzo (Ottawa)
Invariant measures on graphs and subgroups
Abstract: We will give an introduction to the theory of so-called invariant random graphs and invariant random subgroups. We will discuss several recent results, among them the result (joint with Vadim Kaimanovich) that the boundary action of an invariant random subgroup is conservative, and also discuss a number of open problems.
Friday, May 23, 13:30-14:30, Burnside 920
Jared Wunsch (Northwestern)
The wave trace for conic manifolds
Abstract: I will discuss joint work with Austin Ford on the trace of the half-wave operator for manifolds with conic singularities. We compute the leading order singularities along closed geodesics that are allowed to interact with the cone points.

Friday, June 13, 10:00-11:00, Burnside 1234
Robert Kusner (Univ of Massachusetts, Amherst)
Linear area growth for CMC surfaces in S^3
Abstract: In 1970, J. Hersch showed that the first eigenvalue lambda_1 of the Laplacian is maximized for the round metric on S^2 among all metrics with fixed area. Different generalizations were obtained by Li, Yang and Yau a decade later. Using an identity of Reilly, Choi and Wang obtained a lower bound on lambda_1 for any embedded minimal surface in a 3-manifold in terms of a (positive) lower bound on the Ricci curvature. By the Riemann-Roch theorem, this gives an upper area bound for minimal surfaces in S^3 which is linear in the genus. We sketch similar area bounds for CMC surfaces in S^3 and speculate on whether they correspond to similar lambda_1 bounds.

## ANALYSIS-RELATED TALKS ELSEWHERE, WINTER 2014

McGill Mathematical Physics seminar
Wednesday, January 8, Burnside 1120, 16:00-19:00
Tristan Benoist (ENS)
Repeated non demolition measurements and wave function collapse
Abstract: One difficulty in quantum optic experiments is to measure a system without destroying it. For example to count a number of photons usually one would need to convert each photons into an electric signal. To avoid such destruction one can use non demolition measurements. Instead of measuring directly the system, quantum probes interact with it and are then measured. The interaction is tuned such that a set of system states are stable under the measurement process. This situation is typically the one of Serge Haroche's group experiment which inspired the work I will present. In their experiment they used atoms as probes to measure the number of photons inside a cavity without destroying them. With Denis Bernard and Michel Bauer we studied the convergence of such repeated non demolition measurements. We proved that the wave function collapse is obtained as a martingale almost sure convergence. This convergence is exponentially fast. We found an explicit rate depending on the limit system state. We studied the dependency of this rate with respect to the choice of probes. We also proved that the limit does not depend on the initial state but only on the measurement record. These results are useful in the evaluation of the performance of a given measurement method. In this talk I will start with a reminder on the description of measurements in quantum physics. I will then explain Serge Haroche's group experiment principles. Based on this example I will present the repeated non demolition measurement model and show the relation between wave function collapse and martingale convergence. Finally I will explain how we proved the exponential speed of convergence, its rate and the stability of a system state estimation.
McGill Mathematical Physics seminar
Friday, January 10, Burnside 1120, 16:00-19:00
Tristan Benoist (ENS)
Markovian continuous indirect measurement models for wave function collapse
Abstract: Quantum trajectories are non linear stochastic differential equations widely used in the description of continuous measurements of quantum systems. With Clement Pellegrini we were interested in finding when these trajectories would reproduce the wave function collapse corresponding to von Neumann postulate of quantum physics. Following the work I had done with Denis Bernard and Michel Bauer, we have found a non demolition condition for these quantum trajectory equations. We have proved an equivalence between a stability property for a set of system states and a martingale property of the system density matrix diagonal elements. Adding some non degeneracy conditions, using martingale convergence theorem, we have found that in the long time limit the state collapse and the distribution of the limit state is the one of von Neumann projection postulate. Using martingale change of measure we have also proved that the convergence is exponential and have found an explicit rate depending on the limit state. Moreover we have been able to show that if one start a computation with an estimate state, then the limit state is the same as the true one. In this talk I will start with a presentation of quantum trajectories both as a model given by constrains imposed by quantum physics and as an approximation of true physical situations of continuous measurement. I will then present the results we obtain with Clement Pellegrini on wave function collapse, exponential convergence rate and estimation stability.
Universite de Montreal Analysis seminar
Wednesday, January 15, 11:30, Universite de Montreal, Pav. Andre-Aisenstadt, 2920, chemin de la Tour, SALLE 5340
Kirill Datchev (MIT)
Taux de decroissance quantique pour des varietes a bouts hyperboliques
Abstract: Mathematiquement, les taux de decroissance quantique apparaissent comme des parties imaginaires des poles du prolongement meromorphe des fonctions de Green. Lorsque l'energie croit, les taux de decroissance sont lies aux proprietes du flot geodesique et a la structure a l'infini. Une pointe possede un infini "petit", ce qui ralentit typiquement la decroissance. Neanmoins, je presenterai une famille d'exemples pour lesquels les taux de decroissance tendent vers l'infini meme en presence d'une pointe. Ceci fait partie d'une investigation plus generale des resonances sur des varietes a bouts hyperboliques.
McGill Mathematical Physics seminar
Wednesday, January 15, Burnside 1120, 16:00-19:00
Claude-Allain Pillet (Toulon)
Non-equilibrium quantum statistical mechanics
CRM-ISM Mathematics colloquium
Friday, January 17, 16:00, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Boris Khesin (Toronto)
Nondegenerate curves and pentagram maps
Abstract: How many classes of closed nondegenerate curves exist on a sphere? We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to the 2D pentagram map defined by R.Schwartz in 1992. We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties. This is a joint work with Fedor Soloviev.
Universite de Montreal Analysis Seminar
Wednesday, January 22, 11:30-12:30, CRM, Room 6214
Joel Fish (IAS)
From Gromov to the Moon
Abstract: I will present some recent applications of symplectic geometry to the restricted three body problem. More specifically, I will discuss how Gromov's original study of pseudoholomorphic curves in the complex projective plane has led to the construction of global surfaces of section, and more generally finite energy foliations, below and slightly above the first Lagrange point in the regularized planar circular restricted three body problem. The talk will be accessible to a general mathematical audience.
Universite de Montreal Analysis seminar
Wednesday, January 29, Time and Room TBA
Gerasim Kokarev (Munich, LMU)
Direct methods in extremal eigenvalue problems
Abstract: Extremal eigenvalue problems is an actively developing area of spectral geometry. I will talk about the problems on Riemannian surfaces related to determining metrics of fixed area which maximise a given Laplace eigenvalue. I will give a short survey on the subject, and outline an approach to extremal problems via the direct method of calculus of variations.
CRM-ISM Mathematics colloquium
Friday, February 7, 16:00, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Charles Epstein (University of Pennsylvania)
Degenerate Diffusions arising in Population Genetics
Abstract: I will speak on recent work, joint with Rafe Mazzeo and Camelia Pop, on the analysis of solutions to a class of degenerate diffusion equations that arise as limits of Markov chain models used in population genetics and mathematical finance. These equations are naturally defined on spaces with rather singular boundaries, like simplices and orthants. In addition to basic existence, uniqueness and regularity results, I will discuss Harnack inequalities and heat kernel estimates.
Nirenberg Lectures in Geometric Analysis
CRM, May 13-16
Alessio Figalli (UT Austin)
Stability results for geometric and functional inequalities
Website

## FALL 2013

The Fall 2013 seminar web page was maintained by Alina Stancu.

2005/2006 Analysis Seminar