## 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

For suggestions, questions etc. please contact Galia Dafni
(gdafni@discrete.concordia.ca), Dmitry Jakobson (jakobson@math.mcgill.ca)
or Vojkan Jaksic (jaksic@math.mcgill.ca)

## 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

**Sophia Vassiliadou** (Toronto)
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

1999/2000 Seminars

2000/2001 Seminars

2002/2003 Seminars