## Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia or at McGill

## WINTER 2012

Monday, January 9, 13:30-14:30, Burnside 920
Elliot Lieb (Princeton)
Applications of Reflection Positivity: Graphene and Other Examples
Abstract: Reflection positivity is a useful tool in statistical mechanics and condensed matter physics. A recent application is to the determination of the possible distortions of the hexagonal graphene lattice. Other applications, such as to potential theory, the flux-phase problem, Peierls instability and stripe formation, will also be given.

Friday, January 13, 13:30-14:30, Burnside 920
Alexandre Girouard (Neuchatel)
Spectral stability of the Dirichlet spectrum
THE TALK WAS CANCELLED
Abstract: The goal of this talk will be to make sense of the following idea: If two domains are close, then their Dirichlet eigenvalues and eigenfunctions are close. In particular, I will survey different notions of distance and of convergence of domains leading to convergence of the spectrum. The emphasis will be on wild outer perturbations.
Friday, January 27, 13:30-14:30, Burnside 920
Alex Eskin (University of Chicago)
Sums of Lyapunov exponents of the Teichmueller geodesic flow, the determinant of the Laplacian and the Siegel-Veech constants
Abstract: This is joint work with Maxim Kontsevich and Anton Zorich.
Friday, February 3, 13:30-14:30, Burnside 920
Sam Drury (McGill)
Tournament Matrices and the proof of the Brualdi-Li Conjecture

Friday, February 10, 14:00-15:00, Burnside 920
Boundary behavior of analytic functions on D.
Slides
Abstract: The space of all analytic functions on the open unit disc is "too large". As a manifestation of this claim, there are several results about the "wild behavior" of analytic functions on the boundary. After stating some of these results, we turn into subclasses of Hol(D) in order to get some reasonable behavior at the boundary. In particular, we discuss some fact about the Hardy spaces Hp, their model subspaces, and de Branges-Rovnyak spaces H(b). Our main focus is on the boundary behavior of functions and their derivatives in these spaces.
Monday, February 13, 13:30-14:30, Burnside 920
Sven Bachmann (UC Davis)
Product vacua with boundary states and the classification of gapped phases
Abstract: In this talk, I will introduce a family of simple quantum spin chains with a unique product ground state in the thermodynamic limit and additional degenerate edge states on finite chains. Their spectrum is uniformly gapped in the length of the chain. Despite their superficial simplicity, these models can be chosen as representatives of gapped ground state phases in one dimension. As a relevant example, I will show that one of them is equivalent - in a precise sense - to the spin-1 antiferromagnetic chain known as the AKLT model.
Friday, March 2, 13:30-14:30, Burnside 920
Almut Burchard (Toronto)
(Non)-convergent sequences of symmetrizations
Abstract: It is often useful to approximate the symmetric decreasing rearrangement by a sequence of simpler rearrangements, such as polarization (two-point rearrangement), Steiner symmetrization, or the Schwarz rounding process. In this talk, I will discuss recent results with Marc Fortier on the convergence of random sequence of such sequences. We derive conditions under which such random sequences converge almost surely to the symmetric decreasing rearrangement. For the special case of independent uniformly distributed polarizations on the sphere, the distance from the symmetric decreasing rearrangement typically decreases with 1/n in the number of steps. We also show almost sure convergence for sequences of rearrangements chosen at random from small sets. In particular, full rotational symmetry can be achieved using Steiner symmetrization along a finite set of directions. Finally, we construct examples of Steiner symmetrization along a dense sequence of directions that fail to converge.
Monday, March 5, 13:30-14:30, Burnside 920
Christian Hainzl (Tuebingen)
Effective dynamics of Bose-Einstein condensates of fermion pairs
Joint CIRGET-analysis seminar
Wednesday, March 7, 10:00-11:00
UQAM, Pavillion President Kennedy, PK-4323
Frederick Gardiner (CUNY)
Using double poles to prove Slodkowski's extension theorem for holomorphic motions
Abstract: We show how to derive Slodkowski's extension theorem on extending holomorphic motions by using the minimum Dirichlet principle for measured foliations and the heights mapping for quadratic differentials with double poles.
Friday, March 23, 13:30-14:30, Burnside 920
Gerasim Kokarev (Munich)
Variational aspects of Laplace eigenvalues on Riemannian surfaces
Abstract: I will talk about some variational problems for Laplace eigenvalues on Riemannian surfaces, and will describe direct methods for such problems. I will also explain a related formalism of dealing with singular metrics, and will tell about the regularity issues.
Joint Mathematical Physics/Analysis seminar
Wednesday, March 28, 12:30, Burnside 1214
Daniel Egli (Toronto)
On Blowup in Nonlinear Heat Equations
Abstract: We establish the asymptotics of blowup for nonlinear heat equations with superlinear power nonlinearities in arbitrary dimensions, and we estimate the remainders.
Friday, March 30, 13:30-14:30, Burnside 920
Dmitri Burago (Penn State)
On surfaces in normed spaces
Abstract: We will discuss a few problems in the geometry of surfaces in normed spaces. The questions may look elementary and even naive but in reality they prove to be rather hard and intriguing. We will be particularly interested in ellipticity of the most natural surface area functionals and in intrinsic geometry of surfaces in normed spaces.
Wednesday, April 4, 13:30-14:30, Burnside 920
Alex Gamburd (CUNY)
Expander Graphs, Thin Groups and Superstrong Approximation
Talk will be rescheduled, new date TBA
Joint CIRGET-analysis seminar
Friday, April 13, 13:30-14:30, Burnside 920
Colin Guillarmou (Ecole Normale)
The Chern-Simons line bundle on Teichmuller space.
Monday, April 16, 13:30-14:30, Burnside 920
Semyon Dyatlov (Berkeley)
Semiclassical limits of distorted plane waves
Abstract: On certain complete noncompact Riemannian manifolds, one can define distorted plane waves $E_h(\lambda,\xi)$, where $h>0$ is the semiclassical parameter, $\lambda>0$ corresponds to the energy of the plane wave, and $\xi$ to its direction at infinity. These functions, also known as Eisenstein functions in the case of hyperbolic manifolds, generalize the notion of eigenfunctions of the Laplacian. Without additional strong assumptions on trapping (such as the pressure condition $P(1/2)<0$), the functions $E_h$ could grow rapidly in $h$ for certain values of $\lambda$. However, we show that if one averages in $\lambda\in [1,1+h]$ and $\xi$, then $E_h(\lambda,\xi)$ converge microlocally to a certain limiting measure $\mu_\xi$, under a mild assumption that the trapped set has Liouville measure zero. We further estimate the rate of convergence by a power of $h$ depending on the classical escape rate; as an application, we obtain expansions of the trace and the scattering phase with fractal remainders $\mathcal O(z^\delta)$, with $\delta$ related to the dimension of the trapped set. Joint work with Colin Guillarmou.
Joint Mathematical Physics/Analysis seminar
Wednesday, April 25, 12:30, Burnside 1214
Jacob S. Moller (Aarhus)
Polaron-phonon Scattering
Abstract: The large polaron model, introduced by Herbert Froehlich, describes an electron in a polar crystal interacting with the crystal through optical lattice excitations. The model has a long history, but most effort has gone into understanding its ground state properties. In this talk we will present a framework within which one can address scattering problems, in particular asymptotic completeness. For models of this type, asymptotic completeness is equivalent to the statement that the model has an exact quasi-particle structure. The main result of the talk is asymptotic completeness in the sector corresponding to scattering of an incoming/outgoing polaron on a single optical phonon. The talk is based on joint work with Morten Grud Rasmussen and Wojciech Dybalski.
Monday, April 30, 13:30-14:30, Burnside 920
W. Minicozzi (Johns Hopkins)
The talk will be rescheduled, new date TBA
Monday, April 30, 13:30-14:30, Burnside 920
H. Christianson (UNC)
High frequency resolvent estimates in spherically symmetric media.
Friday, May 4, 13:30-14:30, Burnside 920
C. S. Lin (Taiwan University)
Classification and Nondegeneracy of entire solutions to Toda system.
Abstract
Monday, May 28, 13:00-14:00, U. de Montreal, Pav. Andre Aisenstadt, Room 4186
Stefano Bianchini (SISSA)
The Monge problem for convex costs
Abstract:

Wednesday, June 6, 12:30pm, Burnside 1214
Daniel Ueltschi (University of Warwick)
Introduction to cluster expansions: applications, combinatorics, tree estimates
Friday, June 29, 13:30-14:30, Burnside 920
Junfang Li (University of Alabama)
A priori estimates of prescribing curvature measure problems in Riemann spaces
Abstract: In this talk, we will discuss the prescribing curvature measure problem in Riemann spaces (space forms with constant sectional curvature, 0, -1, or 1.) This will generalize the previous work in Guan-Lin-Ma and Guan-Li-Li from Euclidean space to elliptic space and hyperbolic space. I will focus on the a priori estimates since these are the key steps. We propose a uniform approach for C^0, C^1 estimate. The crucial step is the C^2 estimate. As a result, we will settle down the problem in elliptic space and prove the a priori C^2 estimates for some special cases in hyperbolic space. For example, we will show a uniform C^2 estimate for surfaces in 3 dimensional hyperbolic space. Moreover, the uniform gradient estimate will yield the existence for the prescribing mean curvature measure in all the three Riemann spaces.
Thursday, July 26, time and room TBA
Le Hai Khoi (Nanyang Technological University)
Composition Operators on Dirichlet series
Abstract: We consider some problems for composition operators on a class of entire Dirichlet series with real frequencies in the complex plane whose Ritt order is zero and logarithmic orders are finite. Criteria for action and boundedness of such operators are given.
Monday, July 30, 13:30-14:30, Burnside 920
Frederic Robert (Nancy)
Sign-changing blow-up for scalar curvature type equations
Abstract:
Monday, August 13, 13:30-14:30, Burnside 920
M. del Mar Gonzalez (Barcelona)
A Discrete Bernoulli Free Boundary Problem
Abstract: We consider a free boundary problem for the p-Laplace operator which is related to the so-called Bernoulli free boundary problem. In this formulation, the classical boundary gradient condition is replaced by a condition on the distance between two different level surfaces of the solution. For suitable scalings our model converges to the classical Bernoulli problem; one of the advantages in this new formulation is that one does not need to consider the boundary gradient. We shall study this problem in convex and other regimes, and establish existence and qualitative theory. This is joint work with M. Gualdani and H. Shahgholian.

## ANALYSIS-REALTED TALKS ELSEWHERE, WINTER 2012

McGill-UdeM Spectral Theory Seminar
January 12, 13:00-14:00, McGill, Burnside 1205
Alexander Girouard (Neuchatel)
The Korevaar method : metric measured spaces in spectral geometry
Abstract: In 1993, Korevaar devised a complicated construction allowing to detect concentration of measures on a Riemannian manifold. This method was simplified, and further developed, by Grigor'yan, Netrusov and Yau. The goal of this talk will be to survey this construction as well as its applications. This includes upper bounds for Steklov eigenvalues, as well as upper bounds for conformal eigenvalues.
CRM-ISM colloquium Friday, February 3, 16:00-17:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420
Equivalence relations, random graphs and stochastic homogenization
Abstract: The theory of measured discrete equivalence relations provides a natural setup for studying infinite random graphs. We shall discuss several results related to construction and various approximations of invariant measures of the associated equivalence relations.
February 27, 29 and March 2, CRM
Richard Schoen (Stanford)
Announcement
CIRGET seminar
Friday, March 2, 11:00-12:00
UQAM, Pavillion President Kennedy, PK-5115
Iosif Polterovich (U. Montreal)
Inequalities for Laplace eigenvalues and Topology 101
Abstract: Isoperimetric inequalities for Laplace eigenvalues is a classical subject in spectral geometry. Surprisingly, some of the proofs of these inequalities rely on results in (elementary) topology. This unexpected link will be emphasized in the talk.
CRM-ISM colloquium Friday, March 30, 16:00-17:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420
Dmitri Burago (Penn State)
Boundary rigidity and minimal surfaces: a survey
Abstract: A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric). To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside. This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside. This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc. In a on-going joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first open class of boundary rigid metrics beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one. The approach grew up from another long-term project of studying surface area functionals in normed spaces, which we have been working on for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifolds. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will sweep technical details under the carpet).
A series of lectures by Tadashi Tokieda (Cambridge)
Invitation to simple modeling of complex phenomena
Abstract: The lecture is intended for graduate students and postdocs interested in applied mathematics, but should be understandable for general mathematical audience.
Monday April 2: McGill 14:00-15:00, Burnside 708
Tuesday April 3: McGill 14:30-15:30, Burnside 920
Wednesday April 4: McGill 10:30-11:30, Burnside 1214
CRM-ISM colloquium
Thursday, April 19, 16:00-17:00
Universite de Montreal, Pav. Andre-Aisenstadt, 2920, chemin de la Tour, SALLE 6214
Vitali Milman (Tel Aviv University)
The Reasons Behind Some Classical Constructions in Analysis
Abstract: The talk will be devoted to two goals: to understand how some classical constructions appear (uniquely) from elementary (simplest) properties, and to build operations/algebraic relations which produce specific actions (which are important and needed in Analysis). In this spirit we will consider (and characterize) Fourier Transform, derivation, Laplace Operator, and others. It is obvious from the above that the talk will be also well understood by mathematics PhD students, and they are invited to come and see very novel and unexpected facts about operations you think you know very well.
Friday, April 27, 2012, 14:30-15:30
Young-Heon Kim (UBC)
Analysis and geometry in regularity of optimal maps
Abstract: In this talk, I would like to discuss some recent progress in understanding the regularity of optimal maps arising when mass distributions are matched in most cost effective way. The problem is related to PDE's of Monge-Ampere type, as well as Riemannian geometry. A key notion is curvature of the cost function, now often called Ma-Trudinger-Wang curvature.
CRM thematic semester on Geometric Analysis and Spectral Theory

## FALL 2011

Spectral Theory Seminar
Thursday, August 25, 13:00-14:00
Room 4186, Pav. Andre-Aisenstadt, Universite de Montreal
Leonid Parnovski (University College, london)
Integrated density of states for periodic and almost-periodic Schrodinger operators

Spectral Theory Seminar
Thursday, September 8, 13:00-14:00
Room 4186, Pav. Andre-Aisenstadt, Universite de Montreal
Fedor Nazarov (University of Wisconsin-Madison and Kent State University)
Title to be announced
Friday, September 9, 13:30-14:30, Burnside 920
Fedor Nazarov (University of Wisconsin-Madison and Kent State University)
On a problem by Halmos on invariant subspaces

Friday, September 23, 13:30-14:30, Burnside 920
David Borthwick (Emory)
Resonance asymptotics for hyperbolic manifolds
Abstract: We will review the scattering theory for asymptotically hyperbolic manifolds and discuss the problem of understanding the distribution of resonances. We'll try to explain the ingredients used to obtain upper and lower bounds on the resonance counting function, including some recent progress on sharp estimates of the constant in these bounds.
Monday, September 26, 13:30-14:30, Burnside 920
Liviu Nicolaescu (Notre Dame)
Critical sets of random smooth functions on compact manifolds
Abstract: Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\rightarrow \infty$, where $C_m$ is a positive constant that depends only on the dimension $m$. Moreover, we show that as $m \rightarrow infty$ we have the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$. The proof uses probabilistic ideas to reduce the estimates of $N_L$ to estimates of the spectral function of the Laplacian, and the estimates of $C_m$ to Wigner's semi-circle theorem in random matrix theory.
Friday, September 30, 14:30-15:30, Burnside 920
Rupert Frank (Princeton)
Uniqueness and nondegeneracy of ground states for non-local equations in 1D
Abstract: We prove uniqueness of energy minimizing solutions Q for the nonlinear equation (-\Delta)^s Q + Q - Q^{\alpha+1}= 0 in 1D, where 0 < s < 1 and 0 < \alpha < (4s)/(1-2s) for s < 1/2 and 0 < \alpha < \infty for s \geq 1/2. Here (-\Delta)^s is the fractional Laplacian. As a technical key result, we show that the associated linearized operator is nondegenerate, in the sense that its kernel is spanned by Q'. This solves an open problem posed by Kenig, Martel and Robbiano. The talk is based on joint work with E. Lenzmann.
Wednesday, October 12, 14:45-15:45, Burnside 308
Matthew Barrett (Purdue)
Generalization of a theorem by Clunie and Hayman
Abstract: pdf
Wednesday, October 19, 14:45-15:45, Burnside 308
Yanir Rubinstein (Stanford)
Kahler-Einstein metrics singular along a divisor
Abstract: The simplest example of a Kahler-Einstein (KE) metric is a football. A European football corresponds to a smooth KE metric, while an American one corresponds to a KE metric with conical singularities. The existence of smooth KE metrics on compact Kahler manifolds was proven in the 70's by Aubin and Yau for nonpositive curvature, and in the early 90's by Tian for positive curvature, under some assumptions. In the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface). More recently, Donaldson has suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and has put forward several influential conjectures. In this talk we will describe a proof of Tian's conjecture in the case the divisor is smooth, as well as a proof of the first of Donaldson's conjectures, obtained recently in joint work with T. Jeffres and R. Mazzeo.
Friday, October 21, 14:30-15:30, Burnside 920
Linan Chen (McGill)
The Fundamental Solution to the Wright-Fisher Equation
Abstract: Wright-Fisher equation is used to model demographic evolution in the presence of diffusion. In joint work with Daniel W. Stroock, we treat the Wright-Fisher equation as a perturbation of a model equation which has the same degeneracy at the origin. By relating to the fundamental solution of the model equation, we give a careful analysis to the fundamental solution of the Wright-Fisher equation, with particular emphasis on its short time behavior near the boundary.
Friday, October 28, Concordia, LB 921-4, 10:30-11:30
Composition operators on model subspaces
Abstract: The classical subordination principle of Littlewood, a beautiful result in the complex function theory, can be restated in the language of operator theory. It simply says that a composition operator on the Hardy space $H^p$ is bounded. The literature on composition operators on different function spaces is very extensive, and in a sense, out of control. However, the bounded composition operators on model subspaces of $H^2$ are not completely characterized yet. In this talk, we discuss some recent progress on this topic.
Friday, November 4, 13:30-14:30, Burnside 920
Suresh Eswarathasan (CRM and McGill)
Regular value theorem in the the fractal setting
Abstract: The classical regular value theorem says that if X and Y are smooth manifolds of dimensions n and m, n > m, respectively, and F is a submersion from X to Y at the point F(y), then the level set at this point is either empty or a smooth n-m dimensional submanifold of X. In this talk we shall see that under appropriate assumptions on F, X may be taken to be an arbitrary product type set of a given Hausdorff dimension. Sobolev bounds for generalized radon transforms play a key role and the sharpness examples are based on the theory of distribution of lattice points on polynomial surfaces.
Tuesday, November 15, 13:00-14:00, Burnside 1214
Louis-Pierre Arguin (Universite de Montreal)
On the number of ground states of the Edwards-Anderson spin glass model
Abstract: The Edwards-Anderson model on Z^d corresponds to the Ising model where the couplings are random, e.g. independent standard Gaussians. Unlike the Ising model, the behavior of the model at low temperature is widely misunderstood from a rigorous (and non-rigorous) point of view for d>2. This complexity stems essentially from the presence of both ferromagnetic and antiferromagnetic couplings. In this talk, I will review the possible scenarios at low temperature and explain recent rigorous results on the uniqueness of the ground state (zero-temperature case) on the half-plane for d=2. This is joint work with M. Damron, C. Newman, D. Stein.
Spectral Theory seminar
Thursday, November 17, 13:00-14:00
Room 5183, Pav. Andre-Aisenstadt, Universite de Montreal
Commutator trave identities for operators and pencils
Special seminar
Thursday, November 24, 13:30-14:30, Burnside 708
Louis Nirenberg (Courant)
Some historical and general remarks about partial differential equations
Abstract: This will not be a technical talk or about current research. It will be a bit historical, with some acnedotes, and remarks on PDE.
Special seminar
Thursday, November 24, 15:30-16:30, Burnside 1B36
Nassif Ghoussoub (UBC)
Best constant in the Moser-Onofri-Aubin inequality on the 2-dimensional sphere and its ramifications

Friday, November 25, 13:30-14:30, Burnside 920
V. Vougalter (Cape Town)
On the existence of stationary solutions for some non-Fredholm integro-differential equations
Abstract: We show the existence of stationary solutions for some reaction-diffusion type equations in the appropriate H^2 spaces using the fixed point technique when the elliptic problem contains second order differential operators with and without Fredholm property.
Friday, December 9, 14:30-15:30, Burnside 920
V. Kalvin (Concordia)
Spectral properties of the Laplacian on manifolds with axial analytic asymptotically cylindrical ends
Abstract: We study the Laplacian on manifolds with axial analytic asymptotically cylindrical ends by the complex scaling method. Other known methods, being effective under different geometric assumptions, fail in this setting because it allows for arbitrarily slow convergence of the metric to its limit at infinity. We develop an analog of the celebrated Aguilar-Balslev-Combes-Simon theory of resonances and show that there is a certain connection between spectral theory of Schrodinger operators in R^n and and the analysis on manifolds with axial analytic asymptotically cylindrical ends. In particular, we prove that the Laplacian has no singular continuous spectrum. We introduce resonances as the discrete non-real eigenvalues of non-selfadjoint deformations of the Laplacian by means of the complex scaling. The resonances are identified with the poles of resolvent matrix elements meromorphic continuation. By using the Phragmen-Lindelof principle we prove exponential decay of the eigenfunctions corresponding to the non-threshold eigenvalues of the Laplacian. The rate of decay is prescribed by the distance from the corresponding eigenvalue to the next threshold. Under our assumptions on the behavior of the metric at infinity accumulation of eigenvalues occur. The results on decay of eigenfunctions combined with the compactness argument due to Perry imply that the eigenvalues can accumulate only at thresholds and only from below. The eigenvalues are of finite multiplicity.

## ANALYSIS-RELATED TALKS ELSEWHERE, FALL 2011

CRM-ISM colloquium Friday, September 9, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420, 16:00-17:00
Fedor Nazarov (University of Wisconsin-Madison and Kent State University)
Non-trivial convex bodies with maximal sections of constant volume
Abstract: In late 1960's V. Klee asked if a convex body in $R^d$ ($d>2$) whose maximal sections in all directions have the same volume must be a ball. We give a negative answer by constructing a continuous family of bodies of revolution with this property. This is a joint work with D. Ryabogin and A. Zvavitch.
CRM-ISM colloquium Friday, November 4, Univ. de Montreal, Pav. Andre-Aisenstadt, 2920 chemin de la Tour, Salle 6214, 16:00-17:00
Leonid Chekhov (Steklov Mathematical Institute and Concordia University)
Teichmuller spaces of Riemann surfaces with holes and algebras of geodesic functions
Abstract: Poincare uniformized Riemann surfaces of genus g with s>0 holes represented as a quotient of the hyperbolic upper half-plane under the action of a Fuchsian group admit partitions into ideal triangles (with vertices at the absolute). We provide the fat graph description of the corresponding Teichmuller spaces and of closed geodesics on the Riemann surfaces, as well as describe the Poisson structures on the Teichmuller spaces and on the set of geodesic functions. Automorphisms of these structures are generated by mapping class group transformations. Quantizing these structures we obtain quantum geodesic algebras. For the Riemann surfaces with one and two holes, automorphisms of these quantum algebras can be described in terms of a quantum braid-group action on (finite) subsets of quantum geodesic functions. We use the simplest example of the torus with one hole to illustrate the whole construction.
CRM-ISM colloquium
Friday, November 18, Univ. de Montreal, Pav. Andre-Aisenstadt, 2920 chemin de la Tour, Salle 6214, 16:00-17:00
Tricks in Spectral Theory
Abstract: The spectra of elliptic boundary value problems for PDEs (for example, a Laplacian acting in a domain in a Euclidean space or on a Riemannian manifold) can be solved analytically only in a small number of trivial cases. In this talk, we survey a number of analytic or geometric tricks and methods which allow one to obtain some a priori information about these spectral problems (in particular, the geometrically motivated eigenvalue estimates) or drastically simplify their numerical study.
Concordia Departmental Research Seminar
Friday, November 25, Concordia University, Library Building, 1400 de Maisonneuve O, LB 921-4, SGW, 12:00-13:00
Alexey Kokotov (Concordia)
Polyakov-Alvarez formula and Weil reciprocity law for polyhedra
Abstract: Starting with a short introduction to the spectral theory of 2d smooth compact Riemannian manifolds,I will prove the classical result due to Polyakov (1981) and Alvarez (1983) - the comparison formula for spectral determinants of Laplacians. It turns out that there exists an analog of this result for flat singular 2d manifolds (e. g. boundaries of Euclidean polyhedra or, more generally, 2d simplicial complexes).
As a simple corollary of this new analog of comparison formula one gets a reciprocity law for conformally equivalent polyhedra, which could be alternatively derived from the Weil eciprocity law for harmonic functions with logarithmic singularities.
CRM-ISM colloquium
Friday, November 25, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420, 16:00-17:00
Alex Furman (University of Illinois at Chicago)
Groups with good pedigrees, or superrigidity revisited
Abstract: In the 1970s G.A. Margulis proved that certain discrete subgroups (namely lattices) of such Lie groups as SL(3,R) have no linear representations except from the given imbedding. This phenomenon, known as superrigidity, has far reaching applications and has inspired a lot of research in such areas as geometry, dynamics, descriptive set theory, operator algebras etc. We shall try to explain the superrigidity of lattices and related groups by looking at some hidden symmetries (Weyl group) that they inherit from the ambient Lie group. The talk is based on a joint work with Uri Bader.
Probability Seminar
Thursday, December 1, Burnside Hall 920, 16:30-17:30
Linan Chen (McGill)
On the structure of infinite dimensional Gaussian measures
Abstract: Gaussian measures are often adopted in the study of integration theory in infinite dimensions due to the absence of Lebesgue measure. The general construction of infinite dimensional Gaussian measures, via ~SAbstract Wiener Space~T(AWS), was first introduced by L. Gross and lately reformulated by D. Stroock. While AWS plays an important role in infinite dimensional integral calculus (for example, it provides a rigorous foundation for Gaussian free fields), the structure of AWS itself is of mathematical interest. Under the setting of AWS, we studied probabilistic variations of the classical Cauchy functional equation. In this process, we developed various techniques in infinite dimensional analysis which lead naturally to results about the structure of AWS. This is joint work with D. Stroock.

2005/2006 Analysis Seminar