## 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 (jakobson@math.mcgill.ca), Iosif Polterovich (iossif@dms.umontreal.ca) or Alina Stancu (alina.stancu@concordia.ca)

## SUMMER 2015

Thursday, June 4, Burnside 920, 13:00-14:00
Junehyuk Jung (KAIST)
Title TBA

## WINTER 2015

Monday, January 5, Burnside 920, 13:00-14:00
Jerome Vetois (Universite de Nice Sophia Antipolis)
Compactness and blow-up phenomena for sign-changing solutions of scalar curvature-type equations

Friday, January 9, Burnside 920, 13:00-14:00
Xiangwen Zhang (Columbia)
ABP estimate on Riemannian manifolds and spacetime Minkowski formulae

Monday, January 12, Burnside 920, 13:00-14:00
Armen Shirikyan (Cergy-Pontoise)
Controllability and mixing for nonlinear differential equations

Monday, January 19, Burnside 920, 13:00-14:00
Jesse Gell-Redman (Johns Hopkins)
Geometric analysis on singular and non-compact spaces
Abstract: The geometric objects which arise naturally in mathematics are frequently and in important cases not smooth. Instead, many have structured ends which look for example like cones, horns, or families of these; a paradigmatic example is the Riemann moduli space of surfaces of genus $g > 1$ with the Weil-Petersson metric, which near its singular locus is approximately Riemannian products of families of horns. This talk concerns geometric analysis on such spaces, especially the analysis of naturally arising differential operators like the Hodge-Laplacian, the Dirac operator, and the D'Alembertian (wave operator). The analysis of such operators is substantially more complex in the non-smooth setting, but we will present tools from microlocal analysis which provide both a clear picture of and a resolution for many problems. We will focus in particular on recent progress in index theory and spectral theory.

Friday, January 23, Burnside 920, 13:00-14:00
Yaiza Canzani (Harvard and IAS)
Geometry and topology of the nodal sets of Schrodinger eigenfunctions
Abstract: In this talk I will present some new results on the structure of the zero sets of Schrodinger eigenfunctions on compact Riemannian manifolds. I will first explain how wiggly the zero sets can be by studying the number of intersections with a fixed curve as the eigenvalue grows to infinity. Then, I will discuss some results on the topology of the zero sets when the eigenfunctions are randomized. This talk is based on joint works with John Toth and Peter Sarnak.
Monday, January 26, Burnside 920, 13:00-14:00
Marcello Porta (Zurich)
Effective dynamics of weakly interacting fermionic systems
Abstract: Systems composed by a large number of particles are often impossible to describe starting from the fundamental laws of motion. For this reason, physicists introduced effective models, much simpler to study than the original many-body systems, that at the same time are expected to capture their main features if the number of particles is large enough. Examples are: the Boltzmann equation, for the dynamics of classical particles in the low density regime, or Thomas-Fermi theory, for the ground state of fermionic quantum systems in the mean-field scaling. In this talk I will discuss a rigorous derivation of the time-dependent Hartree-Fock equation, an effective evolution equation for a system of interacting fermionic particles in the meanfield scaling. I will consider both pure states (zero temperature) and mixed states (positive temperature). With respect to the well understood bosonic case, here the main difference is that fermionic mean-field is naturally coupled with a semiclassical scaling.

Monday, February 2, Burnside 920, 13:00-14:00
Anna Lisa Panati (McGill)
Energy conservation, counting statistics and return to equilibrium
Abstract: We study a microscopic Hamiltonian model describing a finite level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at inverse temperature \beta. Assuming that the coupled system S + R is mixing with respect to the joint thermal state, we study the Full Counting Statistics (FCS) of the energy transfers S \to R and R \to S in the process of return to equilibrium. The first FCS is an atomic probability measure P_{S,\lambda,t} concentrated on the set of energy differences sp(HS)−sp(HS) (HS is the Hamiltonian of S, t is time at which the measurement of the energy transfer is performed, and \lambda is the coupling constant describing the strength of the interaction between S and R). The second FCS P_{R,\lambda,t} is typically a continuous probability measure whose support is the whole real line. We study the large time limit t \to\infty of these two measures followed by the weak coupling limit \lambda\to 0 and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and quantum transfer operator representation of FCS. (joint work with V. Jaksic, J. Panangaden, C-A. Pillet)
Friday, February 13, Burnside 920, 13:00-14:00
Frederic Naud (Avignon)
Nodal lines and domains for Eisenstein series on surfaces
Abstract: Eisenstein series are the natural analog of “plane waves” for hyperbolic manifolds of infinite volume. These non-L^2 eigenfunctions of the Laplacian parametrize the continuous spectrum. In this talk we will discuss the structure of nodal sets and domains for surfaces. Upper and lower bounds on the number of intersections of nodal lines with “generic” real analytic curves will be given, together with similar bounds on the number of nodal domains inside the convex core. The results are based on equidistribution theorems for restriction of Eisenstein series to curves that bear some similarity with the so-called “QER” results for compact manifolds.
Thursday, February 19, 13:30-14:30, Concordia, Library building, Room LB 921-04
Scott Rodney (Cape Breton university)
Degenerate elliptic equations with rough coefficients: recent regularity results and H=W
Abstract: During this talk we will discuss the derivation of a local Harnack inequality for weak solutions to degenerate elliptic quasi-linear equations with rough coefficients and also how this leads to Holder continuity of solutions. Following this, I will describe some related problems involving degenerate Sobolev spaces and H=W.

Friday, February 20, Burnside 920, 13:00-14:00
Patrick Munroe (McGill)
Moments of Eisenstein series on convex co-compact hyperbolic manifolds
Abstract: On infinite-volume hyperbolic manifolds, the Eisenstein series are non-L^2 eigenfunctions of the Laplacian which parametrize the continuous spectrum. In this talk, we will discuss the moments of Eisenstein series at high-energy on convex co-compact manifolds, i.e., on infinite-volume hyperbolic manifolds without cusps. More precisely, a new result about the vanishing of the odd moments will be presented and an explicit limit for the fourth moment on surfaces will be given.
Monday, February 23, Burnside 920, 13:00-14:00
Tomasz Kaczynski (Sherbrooke)
Towards a formal tie between combinatorial and classical vector field dynamics
Abstract: The Forman’s discrete Morse theory is an analogy of the classical Morse theory with, so far, only informal ties. Our goal is to establish a formal tie on the level of induced dynamics. Following the Forman’s 1998 paper on “Combinatorial vector fields and dynamical systems”, we start with a possibly non-gradient combinatorial vector field. We construct a flow-like upper semi-continuous acyclic-valued mapping whose dynamics is equivalent to the dynamics of the Forman’s combinatorial vector field, in the sense that isolated invariant sets and index pairs are in one-to-one correspondence.

Working seminar in Geometric Analysis
Wednesday, February 25, Burnside 920, 13:30-14:30
Yi Wang (IAS and Johns Hopkins)
Isoperimetric inequalities and $Q$-curvature in conformal geometry and CR geometry
Abstract: In this talk, we will discuss the connection between isoperimetric inequalities and Branson's $Q$-curvature on conformally flat manifolds. This is the higher dimensional analog of the classical Fiala-Huber's isoperimetric inequality for surfaces. Also, we notice that the same phenomenon extends to the sub-Riemannian case, with modified Paneitz operator and $Q$-curvature on CR-manifolds. We will talk about recent progress in this direction.
Friday, February 27, Burnside 920, 13:00-14:00
M. Karpukhin (McGill)
Upper bounds for the first Laplace eigenvalue on non-orientable surfaces and real algebraic geometry
Abstract: One of the important questions of spectral geometry is to determine the supremum of the first Laplace eigenvalue over the space of Riemannian metrics of unit volume on a fixed surface M. The celebrated inequality of Yang and Yau guarantees finiteness of this quantity for orientable surfaces. In the present talk we prove the analog of Yang-Yau inequality for non-orientable surfaces. The proof uses interesting concepts of real algebraic geometry.
Monday, March 16, Burnside 920, 13:00-14:00
Victor Kalvin (Concordia)
Moduli spaces of meromorphic functions and determinant of Laplacian
Abstract: The Hurwitz space is the moduli space of pairs (X,f), where X is a compact Riemann surface and f is a meromorphic function on X. We consider the Laplace operator on the flat non-compact singular Riemannian manifold (X, |df|^2). We define a regularized relative determinant of the Laplace operator and obtain an explicit expression for the determinant in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential df. In this talk I will mainly speak about a surgery formula of the type of Burghelea-Friedlander-Kappeler for the relative determinant of the Laplace operator on singular flat surfaces with conical and Euclidean ends. This formula allows to close the conical/Euclidean ends and thus reduces the proof of explicit expression for the relative determinant to the proof of a similar expression for the zeta-regularized determinant of Laplace operator on the (compact) manifold with closed ends. We believe the surgery formula is also of independent interest. The talk is based on a joint work with Alexey Kokotov and Luc Hillairet.
Friday, March 20, Burnside 920, 13:00-14:00
Jun Kitagawa (Toronto)
Generated Jacobian equations and regularity: optimal transport, geometric optics, and beyond
Abstract: Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degenerate ellipticity. Motivated by an example from geometric optics I will talk about the class of Generated Jacobian Equations, recently introduced by Trudinger. This class encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Amp{\e}re equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems in geometric optics. This talk is based on joint works with N. Guillen.
Monday, March 23, 14:00-15:00, UdeM/CRM, Room 4336
Beatrice-Helen Vritsiou (U. of Michigan)
On the Bourgain-Milman inequality: a proof that uses only tools from Convex Geometry
Abstract: The classical Blaschke-Santaló inequality states that the volume product vol(K)vol(K^o) of symmetric convex bodies K, or, more generally, convex bodies K with barycentre at the origin, is (uniquely) maximised by ellipsoids (here vol denotes Lebesgue measure and K^o is the polar body of K). The Bourgain-Milman inequality, proved by Bourgain and Milman in 1987, is an (asymptotic) inverse to the Blaschke-Santalo inequality: it tells us that vol(K)vol(K^o) >= a^n vol(E)vol(E^o) >= (b/n)^n for every n-dimensional convex body K that contains the origin in its interior, where E is any n-dimensional symmetric ellipsoid and a, b are constants independent of the dimension n. We will present an alternative proof of the Bourgain-Milman inequality that uses only convex-geometric tools, and in particular methods developed for the study of isotropic convex bodies. This is joint work with A. Giannopoulos and G. Paouris.
Monday, March 30, 13:30-14:30, Concordia, Library building, Room LB 921-04
Vitali Milman (Tel Aviv)
Some algebraic related structures on families of convex sets

Friday, April 10, Burnside 920, 13:00-14:00
Ray McLenaghan (Waterloo)
Huygens' principle and Hadamard's problem of diffusion of waves
Abstract: Huygens' principle is satisfied by a second order linear hyperbolic partial differential equation if the solution at any point of every Cauchy initial value problem depends only on the data in an arbitrarily small neighbourhood of the intersection of the retrograde characteristic conoid from the point with the initial surface. The ordinary wave equation in an even number of independent variables greater than or equal to four has this property while the wave equation in an odd number of variables does not. In 1923 Hadamard posed the problem, which remains unsolved, of determining up to equivalence all equations which possess the Huygens property. The lecture will describe the history and current status of attempts to solve the problem with emphasis on the physically interesting case of four independent variables. It will include a description of an apparently new non-trivial Huygens equation.
Friday, April 24, 13:30-14:30, Concordia, LB 921-4
Ataollah Askari-Hemmat
On shearlets of L^2(Q_p^2)
Abstract:
Friday, May 1, Burnside 920, 13:00-14:00
Luc Hillairet (Univ. d'Orleans)
On the wave propagation in generalized polygons
Abstract: We study the wave propagation on flat surfaces with conical singularities that generalize polygons. We give a new construction of the wave propagator near the intersection of the direct front and the diffracted front and relate it with the class of singular FIO that was introduced by Melrose-Uhlmann. We apply this result to the computation of the contribution to the wave-trace of any kind of periodic orbit. (joint work with A. Ford and A. Hassell).
Propagation des ondes dans les polygones généralisés
Résumé: On étudie la propagation des ondes sur les surfaces euclidiennes à singularités coniques qui généralisent les polygones. On donne une nouvelle construction du propagateur au voisinage de l'intersection des fronts direct et diffracté et on l'interprète dans la classe des OIF singuliers introduite par Melrose-Uhlmann. On utilise ce résultat pour calculer la contribution à la formule de trace de n'importe quel type d'orbite périodique. (collaboration avec A. Ford et A. Hassell).
Monday, May 4, Burnside 920, 13:00-14:00
Jacopo de Simoi (Toronto)
An integrable billiard close to an ellipse of small eccentricity is an ellipse
Abstract: In 1927 G. Birkhoff conjectured that if a billiard in a strictly convex smooth domain is integrable, the domain has to be an ellipse (or a circle). The conjecture is still wide open, and presents remarkable relations with open questions in inverse spectral theory and spectral rigidity. In the talk we show that a version of Birkhoff's conjecture is true for small perturbations of ellipses of small eccentricity. This is joint work with A. Avila and V. Kaloshin
Friday, May 8, Burnside 920, 13:00-14:00
Alessandro Savo (University of Rome - La Sapienza)
Constant heat flow, Serrin problem and the isoparametric property
Abstract: On a compact domain in a Riemannian manifold, we study the solution of the heat equation having constant unit initial conditions and Dirichlet boundary conditions. The aim of this talk is to discuss the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the " constant flow property", or that they are "perfect heat diffusers". In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? In the talk, we first relate this property with the well-know Serrin problem. Then, we give a precise characterization in terms of the second fundamental form of the boundary and the isoparametric property.

Wednesday, May 13, Burnside 1205, 13:30-14:30
Mike Wilson (Vermont)
Almost-orthogonality: almost as good as orthogonality
Abstract.

Friday, May 15, Univ. de Montreal, Pav. Andre Aisenstadt, Room 5448, 14:00-15:00
Rick Laugesen (UIUC)
Steklov spectral inequalities through quasiconformal mapping
Abstract: Eigenvalues of the Steklov or Dirichlet-to-Neumann operator represent frequencies of vibration of a free membrane whose mass is concentrated at the boundary. They arise also in sloshing problems. We show the disk maximizes various functionals of the Steklov eigenvalues, under normalization of the perimeter and a kind of boundary moment. The results cover the first eigenvalue, spectral zeta function and trace of the heat kernel. Interestingly, the method employs quasiconformal mapping to estimate the distortion of the energy functional (Dirichlet integral).

## ANALYSIS-RELATED TALKS ELSEWHERE, WINTER 2015

CRM/McGill Applied Mathematics seminar
Monday, February 9, 15:00-16:00, McGill, 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.
CRM-ISM Colloquium
Thursday, March 5, McGill University, Burnside Hall, 805 Sherbrooke str. West, Room 920
THE TALK IS CANCELLED Alvaro Pelayo (UC San Diego)
Classical and quantum integrable systems

CRM NIRENBERG LECTURES IN GEOMETRIC ANALYSIS
CRM, March 20-24, Pav. Andre-Aisenstadt, Univ. de Montreal, Salle / Room 6214
Andre Neves (Imperial College London)
• Friday, March 20, 16:00
Min-max Theory and Geometry I
Abstract: I will survey my recent work with Fernando C. Marques, where we used Min-max Theory to solve some long standing open questions in geometry.
This lecture is aimed at a general mathematical audience.
A reception will follow the lecture at the Salon Maurice-L'Abbe, Pavillon Andre-Aisenstadt (room 6245).
• Monday, March 23, 16:00
Min-max Theory and Geometry II
Abstract: I will explain how to use Min-max Theory to solve the Willmore Conjecture dating from 1965. This is joint work with Fernando C. Marques.
• Tuesday, March 24, 14:00
Min-max Theory and Geometry III
Abstract: I will explain how to use Min-max Theory to find infinitely many minimal hypersurfaces in manifolds with positive Ricci curvature. This is joint work with Fernando C. Marques which partially answers a conjecture of Yau from 1982.

• ## SUMMER 2014

Tuesday, August 26, Burnside 920, 13:15-14:15
Michael Wilson (Vermont)
Almost-orthogonality without discreteness or smoothness
Abstract

Tuesday, August 26, Burnside 920, 14:30-15:30
Alexander Stokolos (Georgia Southern)
Geometric maximal function in harmonic analysis
Abstract: Some interesting examples of maximal functions associated with a differentiation bases of convex sets will considered.

## FALL 2014

Please, note that in the Fall 2014, Monday seminars at McGill will be held in Burnside 306 (3rd floor) between 14:30-15:30
Monday, September 15, 14:30-15:30, Burnside 306
Suresh Eswarathasan (McGill)
Perturbations of the Schrodinger equation on negatively curved surfaces
Abstract: In this talk, we will take small perturbations of the semiclassical Schrodinger equation on negatively curved surfaces and consider some of the corresponding long-time quantum evolutions. We will show that, under certain admissibility conditions on the perturbation, these solutions become equidistributed in the semiclassical limit for "typical" perturbations. This is joint work with Gabriel Riviere.
Monday, October 6, 14:30-15:30, Burnside 306
Javad Mashreghi (Laval)
Embedding theorems for the Dirichlet space
Abstract: A finite positive Borel measure $\mu$ on $\mathbb{D}$ is a {\em Carleson measure} for the (classical) Dirichlet space $\mathcal{D}$ if $\|f\|_{L^2(\mu)} \leq C \|f\|_{\mathcal{D}}, \qquad f \in \mathcal{D}.$ Equivalently, we can say that $\mathcal{D}$ embeds in $L^2(\mu)$. We will discuss the geometric characterization of such measures and present a particular "one-box condition".

Friday, October 10, 15:10-16:10, Burnside 1120 (time changed!)
Boaz Slomka (Concordia, CRM)
Covering numbers of convex sets and their functional extension
Abstract: In the first part of the talk we will discuss the notions of classical and fractional covering numbers, mainly in the context of convex bodies. In particular, we will describe an application to Hadwiger's famous covering problem. In the second part of the talk we will focus on the extension of covering numbers to the realms of functions (mainly log-concave). We will present some of their properties as well as related inequalities. Based on joint works with Shiri Artstein-Avidan

Monday, October 27, 14:30-15:30, Burnside 306
Andrei Martinez-Finkelshtein (Almeira)
Phase transitions and equilibrium measures in random matrix models
Abstract: We are interested in the so-called phase transitions in the Hermitian random matrix models with a polynomial potential. Or, in a language more familiar to approximators, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also either as temperature or time. By phase transitions we understand the loss of analyticity of the equilibrium energy. Our main tools are differentiation formulas with respect to the parameters of the problem, and a representation of the equilibrium potential in terms of a hyperelliptic integral. This allows to find a dynamical system that describes the evolution of families of equilibrium measures. On this basis we are able to systematically derive results on phase transitions, such as the local behavior of the system at all kinds of phase transitions. We discuss in depth the case of the quartic external field.
Monday, November 10, 14:30-15:30, Burnside 306
Raphaël Ponge (Seoul National University and Berkeley)
Noncommutative geometry and Vafa-Witten inequality
Abstract: The inequality of Vafa-Witten produces an uniform bound for the first eigenvalue of a Dirac operator with coefficients in a Hermitian vector bundles. It's a remarkable fact that bound does not depend on the vector bundle. In this talk, we will explain how to use the framework of noncommutative geometry to reformulate Vafa-Witten inequality in various new geometric settings such as conformal geometry, noncommutative tori, and some symmetric spaces.

Friday, November 14, 14:30-15:30, Burnside 920
Monica Ludwig (TU Wien)
On the geometric classification of functions
Abstract: A fundamental theorem of Hadwiger classifies all rigid-motion-invariant and continuous functionals on convex bodies that satisfy the inclusion-exclusion principle. Moreover, Hadwiger's theorem characterizes the n+1 intrinsic volumes (volume, surface area, etc.) in Euclidean n-space. Recently, important functions in analysis and probability theory have been characterized by geometric properties and the inclusion-exclusion property, for example, the Fisher information matrix and the operator that associates with a function its optimal Sobolev norm. An overview of these results will be given.
Monday, November 24, 13:30-14:30, Concordia, LB 921-04
Boaz Slomka (Concordia, CRM)
Functional covering numbers
Abstract: Covering and separation (packing) numbers are useful tools in various areas of mathematics. In particular, they play an important role in the theory of convex bodies. In this talk we will introduce natural extensions of these geometric concepts to the realms of functions, and discuss their properties as well as related inequalities. We will mainly consider log-concave functions, as our original motivation is the study of their geometry and interplay with convex bodies. Joint work with Shiri Artstein-Avidan.
Friday, November 28, Burnside 920, 14:30-15:30
Todd Oliynik (Monash University)
Dynamical compact bodies in General Relativity
Abstract: The visible universe contains many different types of dynamical compact bodies including asteroids, comets, planets, stars and even more exotic objects such as neutron stars. In spite of their fundamental importance to astrophysics and cosmology, there are currently very few analytical results available that apply to these dynamical bodies. In particular, even the most basic problem of establishing the (local) existence and uniqueness of solutions that represent gravitating compact bodies was, until very recently, a long standing open problem in General Relativity (GR). In this talk, I will discuss this problem and pay particular attend to the case of elastic matter. After presenting some general background on the dynamics of compact bodies in GR, I will describe, in detail, the initial value formulation for the particular case of elastic matter and outline the analytic difficulties that have hindered progress in understand the initial value problem for this system. I will then summarize recent results obtained in collaboration with Lars Anderson and Bernd Schmidt in which we establish the existence and uniqueness of solutions that represent gravitating dynamical elastic bodies. Time permitting, I will describe some open problems and promising directions for future work.

Tuesday, December 2, Burnside 1205, 13:00-14:00 (time changed!)
Yaiza Canzani (Harvard and IAS)
Pointwise Weyl Law and Universal scaling asymptotics
Abstract: In this talk I will present new off-diagonal remainder estimates for the kernel of the spectral projector of the Laplacian onto frequencies up to $\lambda$. One of the consequences is that the kernel of the spectral projector onto frequencies $(\lambda,\lambda+1]$ has a universal scaling limit as $\lambda \to \infty$ near any non self-focal point. These results have applications to immersions by eigenfunctions, gradient estimates, and to the study of zero sets of random waves. This is joint work with Boris Hanin.

## ANALYSIS-RELATED TALKS ELSEWHERE, FALL 2014

UdeM-McGill Spectral Theory seminar
Thursday, September 11, Universite de Montreal, pav. Andre-Aisenstadt, room 5448, 14:00-15:00
Guillaume Roy-Fortin (UdeM)
Nodal sets and growth exponents of Laplace eigenfunctions on surfaces
Seminar website
UdeM-McGill Spectral Theory seminar
Thursday, September 25, McGill, Burnside 1120, 14:00-15:00
John Toth (McGill)
L^2-restriction lower bounds for Schrodinger eigenfunctions in classically forbidden regions.
Seminar website
UdeM-McGill Spectral Theory seminar
Thursday, October 2, Universite de Montreal, pav. Andre-Aisenstadt, room 5448, 14:00-15:00
Frederic Naud (Avignon)
Sharp Resonances on hyperbolic manifolds
Seminar website
McGill Mathematical Physics mini-course
Jakob Yngvanson (Vienna)
A crash course on thermodynamics and entropy
• Thursday, Oct 9, 9:00-11:00, Burnside 1120
• Friday, Oct 10, 13:00-15:00, Burnside 1120.
• Tuesday, Oct 14, 9:00-11:00

• Montreal Probability seminar and Analysis seminar
Tuesday, October 14, 16:30, McGill, Burnside 1214
Mark Freidlin (University of Maryland)
Long time effects of small perturbations

Mathematical Physics Week
October 27-31, Burnside Hall
More details here.
• October 27, 14:30-15:30, Burnside 306: Andrei Martinez Finkelshtein (Analysis seminar), Phase transitions and equilibrium measures in random matrix models.
• October 28, 15:30 - 16:30, Burnside 1120: Tristan Benoist (McGill), Reservoir engineering.
• October 29, 17:30 - 19:30, Burnside 1120: Vojkan Jaksic (McGill) Non-equilibrium quantum statistical mechanics: State of the art (I).
• October 30, 17:30 - 19:30, Burnside 1120: Vojkan Jaksic (McGill) Non-equilibrium quantum statistical mechanics: State of the art (II).
• October 31, 13:00-14:00, Burnside 1120: Laurent Bruneau (Paris), Mixing properties of one-atom maser.
• October 31, 14:00-15:00, Burnside 1120: Yan Pautrat (Paris and McGill), Open quantum walks.
• October 31, 15:30-16:30, Burnside 1120: Armen Shirikyan (Paris), Limit theorems in statistical hydrodynamics.

• UdeM-McGill Spectral Theory seminar
Thursday, November 6, McGill, Burnside 1120, 14:00-15:00
Annalisa Panati (CRM/McGill)
Spectral and scattering theory for abstract QFT Hamiltonians (joint work with Christian Gerard)
Abstract.
UdeM-McGill Spectral Theory seminar
Thursday, November 13, Universite de Montreal, pav. Andre-Aisenstadt, room 5448, 14:00-15:00
Frederic Rochon (UQAM)
Title TBA

UdeM-McGill Spectral Theory seminar
Thursday, November 20, Universite de Montreal, pav. Andre-Aisenstadt, room 5448, 14:00-15:00
Iosif Polterovich (UMontreal)
Spectral geometry of the Steklov problem
Abstract: The Steklov problem is an elliptic eigenvalue problem with the spectral parameter in the boundary conditions. While this problem shares some common properties with its more well known Dirichlet and Neumann cousins, the Steklov eigenvalues and eigenfunctions have a number of distinctive geometric features. We will discuss some recent advances in the subject, particularly in the study of spectral asymptotics, spectral invariants, eigenvalue estimates, and nodal geometry. The talk is based on a joint survey article with A. Girouard.
Concordia Dynamical Systems seminar
Friday, November 21, 11:45-12:45, LB-921-4 (Library Building)
Maciej P. Wojtkowski (University of Warmia i Mazury, Olsztyn, Poland)
Bi-partitions of the 2-d torus, 1-dimensional tilings, hyperbolic automorphisms and their Markov partitions
Abstract: Bi-partitions are partitions of the 2-dim torus by two parallelograms. They give rise to 2-periodic tilings of the plane, and further to 1-dim tilings which have a host of well known combinatorial properties, e.g. these are Sturmian sequences. When a bi-partition is a Markov partition for a hyperbolic toral automorphism (= Berg partition), the tilings are substitution tilings. Substitutions preserving Sturmian sequences have the remarkable `3-palindrome property''. The number of different substitutions was determined by Seebold '98, and the number of nonequivalent Berg partitions by Siemaszko and Wojtkowski '11. The two formulas coincide. Using tilings we explain the formula by the 3 palindrome property. The coincidence then shows that every combinatorial substitution preserving a Sturmian sequence is realized geometrically as a Berg partition.
UdeM-McGill Spectral Theory seminar
Thursday, November 27, McGill, Burnside Hall, room 1120, 14:00-15:00
Mikhail Karpukhin (McGill)
Regularity theorems for maximal metrics
Abstract: Given a smooth surface M the first eigenvalue of the Laplace operator can be seen as a functional on the space of Riemannian metrics on M of unit volume. One of the important problems of spectral geometry is to find the supremum of this functional. However it is not even clear that the supremum is attained on a smooth metric. In this talk I will present a recent result on the regularity theory of these maximal metrics, which states that under fairly weak condition the maximal metric is a smooth metric with isolated conical singularities. The talk is based on the results of R. Petrides
Mathematical Physics Seminar
Tuesday, December 2, CRM, UdeM, Pavillon Andre-Aisenstadt, 2920, ch. de la Tour, salle 4336, 15:30-16:30
David Ruelle (IHES)
Introduction to hydrodynamic turbulence and non-equilibrium statistical mechanics
Abstract: We present here basic facts about hydrodynamic turbulence and statistical mechanics (equilibrium and non-equilibrium). This introduction should allow to understand the main talk without too much previous knowledge of the relevant areas of physics and mathematics.
Analysis/Mathematical Physics seminar
Wednesday, December 3, McGill, Burnside 708, 16:30-17:30
David Ruelle (IHES)
The Lee-Yang Circle Theorem and some applications
UdeM-McGill Spectral Theory seminar
Thursday, December 4, Universite de Montreal, pav. Andre-Aisenstadt, room 5448, 14:00-15:00
Daniel Valtorta (Polytechnique Lausanne)
Minkowski estimates on critical and nodal sets of solutions to elliptic PDEs
Abstract: Given a nonconstant harmonic function, we obtain Minkowski bounds on its critical and almost critical set. The proof relies on a refined blow-up analysis for harmonic functions based on the properties of Almgren's frequency. With minor modifications, these estimates are valid also for solutions to a very general class of elliptic PDEs. Given the link between harmonic functions and eigenfunctions of the Laplacians, with the necessary modifications these results apply also to nodal and singular sets of eigenfunctions. This is joint work with Aaron Naber.
Analysis/Mathematical Physics seminar
Friday, December 5, McGill, 14:00-15:00, Burnside 920
David Ruelle (IHES)
Non-equilibrium Statistical Mechanics of Turbulence
Abstract: The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system. Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents $\tau_p$ associated with moments of velocity fluctuations. In particular we derive probability distributions at finite Reynolds number for the velocity fluctuations which permit an interpretation of numerical experiments. Specifically, if $p(z)dz$ is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number ${\cal R}$ increases, $\ln p(z)$ passes from a concave to a linear then to a convex profile for large $z$ as observed in Navier-Stokes studies. We show that the central limit theorem applies to the velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents $\tau_p$ and $\zeta_p$.

2005/2006 Analysis Seminar