## 2001/2002 Analysis Seminar

Seminars are usually held on Fridays, 2:30-3:30 PM, at Burnside Hall 920 (McGill University)
We also post information about other talks in Analysis in Montreal

## Fall 2001

### A series of seven lectures

Ilan Vardi (IHES)
Leading digits, lattice points and algebraic numbers
Tuesday, September 4, 11-12am, Burnside 920; 2:30pm-4:00pm, Concordia, Library Bldg, LB-540
Tuesday, September 11, 10-12am, Burnside 920
Tuesday, September 18, 11-12am, Burnside 920; 2:30-4:00pm, Burnside 1234
Tuesday, September 25, 11-12am, Burnside 920; 2:30-4:00pm, Burnside 1234

### A series of four lectures

Li Ma (Tsinghua University, Beijing, China)
Mean Curvature Flow for Lagrangian Submanifolds
(I) Monday, September 10, 3:00, CRM, Pav. André-Aisenstadt, salle 5183
(II) Thursday, September 13, 3:00, CRM, Pav. André-Aisenstadt, salle 5183
(III) Friday, September 14, 3:00, CRM, Pav. André-Aisenstadt, salle 5183
(IV) Monday, September 17, 3:00, CRM, Pav. André-Aisenstadt, salle 5183
Abstract: Special (minimal) Lagrangian submanifolds in a Calabi-Yau manifold are important in Mirror symmetry. To get a minimal Lagrangian submanifold, it is nature to study the mean curvature flow (MCF) for Lagrangian submanifolds. In these lectures, we discuss some basic ingredients like some gradient estimates, singularities, and monotonicity formula (and its variants) for MCF. We will pay more attention to some special cases like the graphical flow and MCF for ruled surfaces.

### Wednesday, September 12, 2:30-3:30, Burnside 920

Alex Gamburd (MSRI)
On the spectral gap and Hausdorff dimension for infinite volume "congruence" surfaces
Abstract: A celebrated theorem of Selberg states that for congruence subgroups of the modular group there are no exceptional eigenvalues below 3/16. We will present a new proof of Selberg's theorem and prove its generalization for infinite index congruence subgroups. For such subgroups with a high enough Hausdorff dimension of the limit set we will establish a spectral gap property and consequently solve a problem of Lubotzky pertaining to expander graphs.

### Wednesday, September 26. 1:30-2:30pm, Burnside 920

N. K. Nikol'skii (Bordeaux and St. Petersburg)
Systemes completes des translates entieres

### Wednesday, September 26, 4:00pm, CRM, Pavillon André-Aisenstadt, salle 6214

Louis Nirenberg (Courant)
A problem on differential forms coming from economics

### Friday, October 5, 1:30-2:30pm, Burnside 1205

Carolyn S. Gordon (Dartmouth)
Isospectral Riemannian manifolds with different local geometry
Abstract: To what extent does the spectrum of the Laplacian of a compact Riemannian manifold determine its geometry? We will address this question by discussing a method for constructing isospectral manifolds with different local geometry. Examples will reveal various curvature properties which are not spectrally determined. For non-compact manifolds, one considers the scattering poles instead of the spectrum (which need no longer be discrete). We construct continuous families of isoscattering metrics given by compact perturbations of the Euclidean metric on R^n.

### Friday, October 5, 4:00pm, UQAM, Pavillon Sherbrooke, 200, rue Sherbrooke O, salle SH-2420

Carolyn S. Gordon (Dartmouth)
Can you hear the shape of a manifold
Abstract: Mark Kac's question "Can you hear the shape of a drum?" asks whether the eigenvalue spectrum of the Laplace operator on a bounded plane domain determines the shape of the domain. We will discuss this question and its analogue for Riemannian manifolds, exhibiting specific "sound-alike" manifolds and comparing their geometry.

### Joint CRM/Analysis Seminar: Friday, October 12, 1:30-2:30pm, Burnside 1205

Francois Ledrappier (Ecole Polytechnique)
Ergodic properties of some linear actions
Abstract: Let $\Gamma$ be a lattice in $SL(2, {\bf R}$ and consider the natural linear action of $\Gamma$ on ${\bf R}^2$> This action preserves Lebesgue measure and is ergodic ([Hedlund]). Moreover, the Lebesgue measure is the only continuous locally finite invariant measure ([Dani]). We discuss the asymptotic distribution of orbits, namely: \proclaim {Theorem } Let $f$ be a continuous even function with compact support on ${\bf R}^2$, $x$ a point in ${\bf R}^2$ with dense $\Gamma$-orbit. Then $${1 \over T} \sum _{\gamma ; \Vert \gamma \Vert \leq T} f(\gamma x) \to {C \over |x|} \int {f(y) \over |y|} dy \quad {\rm as} \quad T \to \infty .$$ We explain the proof of this theorem and of some extensions of it.

### Friday, October 19, 2:30-3:30pm, Burnside 920

A. Vasy (MIT)
The spectral shift function in many-body scattering
Abstract: The spectral shift function in scattering theory is the analogue for the counting function of eigenvalues for the Laplacian on compact manifolds without boundary. I will talk about various properties, including the high energy asymptotics, of the spectral shift function in many-body scattering. This is a joint project with Xue Ping Wang.

### Friday, October 19, 4:00pm, UQAM, Pavillon Sherbrooke, 200, rue Sherbrooke O., salle SH-2420

Paul Gauduchon (Ecole Polytechnique)
Variétés kählériennes ortho-toriques

### CRM-ISM Colloquium/Analysis Seminar: Friday, October 26, 4:00pm, UQAM, Pavillon Sherbrooke, 200, rue Sherbrooke O., salle SH-2420

C. Sogge (Johns Hopkins)
Riemannian manifolds with maximal eigenfunction growth
Abstract: On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$, e.g. only $M = S^2$ (topologically) in dimension $2$ can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which ${\mathcal L}_x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth, thus disproving a naive converse to the main result.

### Thursday, November 1, 3:30-4:30pm, Burnside 920

David Ruelle (IHES)
Perturbation theory for Lyapunov exponents of a toral map: extension of a result of Shub and Wilkinson

### CRM-ISM Colloquium: Friday, November 2, 4:00pm, UQAM, Pavillon Sherbrooke, 200, rue Sherbrooke O., salle SH-2420

David Ruelle (IHES)
Nonequilibrium Statistical Mechanics: Entropy production for quantum spin systems

### Friday, November 9, 2:30-3:30pm, Burnside 920

Reem Yassawi (Trent U.)
Limit measures for permutative cellular automata

### Friday, November 16, 2:30-3:30pm, Burnside 920

Robert McCann (U. Toronto)
Geometrical Inequalities via Optimal Mappings
Abstract: Geometrical inequalities such as the isoperimetric theorem, Brunn-Minkowski inequality and other relations between mixed-volumes are based on domination of the geometric by the arithmetic mean. This intuition can be made precise by exploiting the theory of optimal mappings. After related developments are surveyed, the method is applied to derive a new interpolation theorem extending various Euclidean inequalities to the Riemannian setting.

### Monday, November 19, 2:30-3:30pm, Burnside 920

Yiannis Petridis (McGill)
Weyl's law and the lattice point counting

### Friday, November 23, 2:30-3:30pm, Burnside 920

Wilhelm Schlag (Princeton and Caltech)
On dispersive inequalities for rough and time-dependent potentials

### Monday, November 26, 4:30-5:30pm, Burnside 920

Special seminar
Igor Rivin (Temple U.)
Geometry of Polyhedra

### Friday, November 30, 1:30-2:30, Burnside 920

Joint Probability/Analysis seminar
Luc-Rey Bellet (University of Virginia)
Non-equilibrium steady states in classical statistical mechanics.
Abstract: We study the ergodic properties of anharmonic crystals coupled to free phonon fields at positive temperature. Steady states are constructed and their properties are studied (speed of convergence, entropy production).

## Winter 2002

### Friday, January 11, 2:30-3:30pm, Burnside 920

Jared Wunsch (Stony Brook)
Wave propagation on conic manifolds

### Monday, January 14, 2:30-3:30pm, Burnside 920

Special seminar
Ailana Fraser (Brown Univ.)
Minimal surfaces and curvature
Abstract: A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. While geodesics play an important role in results of this nature in classical Riemannian geometry, minimal surfaces have more recently had striking applications to the study manifolds. This will be a survey talk in which we will describe some results on minimal surfaces and applications to two-convex hypersurfaces and manifolds with positive isotropic curvature.

### Thursday, January 17, 2:30-3:30pm, Burnside 920

Konstantina Trivisa (University of Maryland, College Park)
Hyperbolic Conservation Laws in Continuum Physics
Abstract: Recent advances in the area of Hypebolic Conservation Laws are discussed. Applications and models from Continuum Physics are discribed. In this talk, we present results on the existence and stability of solutions to hyperbolic systems of conservation laws with large initial data. This is essentially an unexplored area of research. The main tools in our analysis are the wave front tracking algorithm and the notion of an entropy functional.

### Monday, January 21, 2:30-3:30pm, Burnside 920

Francis Clarke (Institut universitaire de France et Université de Lyon)
Méthode de Lyapunov, équation Hamilton-Jacobi, et feedbacks en contrôle.

### Thursday, January 24, 2:30-3:30pm, Burnside 920

Joint Number Theory/Analysis seminar
Izabella Laba (UBC)
On finite sets that tile the integers

### Friday, February 1, 2:30-3:30pm, Burnside 920

Dmitry Jakobson (McGill)
Critical points and symmetry properties of eigenfunctions

### Friday, February 8, 2:30-3:30pm, Burnside 920

Victor LeBlanc (U. of Ottawa)
Euclidean Symmetry and Spiral Wave Dynamics
Abstract: Spiral waves occur in many different situations: Belousov-Zhabotinsky chemical reactions, bacteria colonies, and perhaps most importantly, in cardiac tissue where these waves quickly lead to death. These different physical situations where spiral waves are observed are all governed by different mathematical equations. However, there is a common underlying link: these equations are all invariant under the group SE(2) of planar translations and rotations. In this talk, we will show that many of the physically observed dynamics and bifurcations of spiral waves can be understood mathematically in a model-independent manner using an abstract analysis of SE(2)-equivariant dynamical systems. We will then report on some recent extensions of this work which uses the concept of forced symmetry-breaking in order to study the effects of boundaries, inhomogeneities and anisotropy on spiral waves. We will compare these results to experimental observations.

### Friday, February 15, 1:30-2:30pm, Burnside 920

Joint Probability/Analysis seminar
A. Ruzmaikina (U. of Virginia)
Quasi-stationary states of Indy500 model
Abstract: We consider a space configurations of infinitely many particles on the negative real line. The particles in each configuration perform independent identically distributed jumps at each time step and after each time step, the configuration is shifted so the leading particle is at 0. We prove that a stationary measure of this stochastic process can be represented as a measure supported on Poisson processes with densities exp(-a x), where a > 0 is a parameter.

### Friday, February 15, 2:30-3:30pm, Burnside 920

PDE's and algebraic geometry in several complex variables

### CIRGET Seminar at UQAM

Friday, March 1, 11:00am-12:00pm, PK-5115, pavillon Président-Kennedy, UQAM
McKenzie Wang (McMaster)
A Variational Approach to Homogeneous Einstein Metrics
Abstract: An Einstein metric is a Riemannian metric g which satisfies the equation Ric(g) = C g where Ric(g) is the Ricci tensor of g and C is a constant. On a closed manifold, an Einstein metric is a critical point of the Hilbert action which associates to each Riemannian metric of volume 1 the integral of its scalar curvature. The Hilbert action is bounded neither from above nor from below, and it is well-known that it does not satisfy the Palais-Smale condition. A homogeneous Einstein metric is one whose isometry group acts transitively on the manifold. In general, a homogeneous manifold may admit infinitely many inequivalent transitive actions. If we fix a compact Lie group G acting transitively on a homogeneous manifold, then the G-invariant Einstein metrics are critical points of the restriction of the Hilbert action to the space of unit volume G-invariant metrics. In this talk I will present joint work with C. Boehm and W. Ziller regarding the Palais-Smale condition for this restricted functional. Compactness and existence results will be described.

### Friday, March 8, 2:30-3:30pm, Burnside 920

Richard Froese (UBC)
Realizing holonomic constraints

### Monday, March 18, 2:30-3:30pm, Burnside 920

A. Iosevich (U. Missouri)
Average decay of the Fourier transform and applications
Abstract

### Friday, March 22, 2:30-3:30pm, Burnside 920

Joint Probability/Analysis seminar
Donald Dawson (Carleton and McGill)
Reaction Diffusion Equations with Highly Singular Coefficients and Catalytic Super-Brownian Motion
Abstract: There is a well known relationship between a class of reaction diffusion type PDE and super-Brownian motion. In the catalytic case the reaction'', resp. branching'', only takes place in the presence of a catalyst. We consider the case in which the catalyst is singular and/or random. We describe results of joint work with K. Fleischmann and P. Moerters on clumping and extinction phenomena for this process. This includes a description of the development under a space-time-mass rescaling of large mass clumps at spatially rare sites. Although the problems can be formulated in terms of a reaction diffusion equation with singular coefficients, probabilistic methods including the historical process and Brownian snake are essential to the analysis.

### Thursday, April 4, 2:30-3:30pm, Burnside 1214

Nick Varopoulos (Univ. Paris VI & I.U.F.)
Central Limit Theorems in Lipschitz domains

### Friday, April 5, 2:30-3:30pm, Burnside 920

Duong Phong (Columbia)
Energy functionals and stability in K\"ahler geometry

### Friday, April 12, 2:30-3:30pm, Burnside 920

L. Grafakos (U. Missouri)
Uniform bounds for the bilinear Hilbert transforms and the bilinear disc multiplier
Abstract: In the early 1960's A. Calderon wrote the first commutator as an average of a family of bilinear Hilbert transforms and asked the question whether these operators are bounded uniformly in the parameter involved. The boundedness of these operators was established by Lacey and Thiele in 1997 and 1999 but their work left open the question of their uniform boundedness. In this work we will present some ideas of the solution of this problem which in particular answers the original conjecture of Calder\'on. An extension of this theorem involving the boundedness of the bilinear disc multiplier will also be discussed.

### Friday, April 19, CANCELLED

A. Magyar (Georgia Tech)

### Monday, April 22, 1:30-2:30pm, Burnside 920

M. Wilson (U. Vermont)
Littlewood-Paley Estimates for Sums of Almost-Orthogonal Functions
Abstract

### Tuesday, June 4, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

P.V.Paramonov (Moscow State University)
Approximation by polyanalytic functions

### Tuesday, June 18, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

G. Dafni (Concordia)
Fractional Carleson measures and some function spaces involving Hausdorff capacity

### Tuesday, July 2, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

André Boivin (Western Ontario)
Approximation rationelle: globale vs. locale

### Tuesday, July 23, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

Paul M. Gauthier (UdeM)
Un théorème de type Carleman pour les plongements

### Tuesday, July 30, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

Pouryayevali, Mohamad Reza (University of Isfahan)
Approximation and interpolation on Stein manifolds

### Tuesday, August 6, 11:30am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

Paul M. Gauthier (UdeM)
Universalité par rapport à une suite d'automorphismes

### Tuesday, August 13, 11:00am, UdeM, Pavillon Andre-Aisenstadt, salle 5183

A. Boivin (Univ. of Western Ontario)
Sur les séries de Fourier non harmoniques