## Analysis Seminar

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

## WINTER 2011

Monday, January 17, 14:30-15:30, Burnside 920
Michael Monastyrsky (Moscow)
Duality transformations for spin lattice systems and Hecke surfaces
Abstract: I discuss a generalization of famous Kramiers-Wannier duality for Ising Model in the theory of phase transitions and some applications to different problems in mathematics, mainly a construction of special class of Riemann surfaces - Hecke surfaces with Regular graphs, surfaces with Large cusps and so on. All of these problems have some physical origin and show deep interplay between modern mathematics and physics.

Friday, January 28, 14:30-15:30, Burnside 920
Semiclassical measures and dispersion for the Schrodinger equation on the torus.
Abstract: We will present some results concerning the dispersive and regularizing properties of the linear Schrodinger equation on the flat torus. Among these, we prove the following: take a sequence of initial conditions $u_n$, normalized in L^2, and construct the probability measures on the torus, $$\int_0^1 |e^{it\Delta}u_n(x)|^2 dt) dx.$$ Then any weak limit of this sequence is an absolutely continuous measure. We will also compare this type of result with what is known in other geometries.

Friday, February 11, 14:30-15:30, Burnside 920
Vincent Grandjean (Fields Institute)
Geodesics at singular points of singular subspaces: a few striking examples
Abstract: Assume a Riemannian manifold (M,g) is given. Let X be a locally closed subset of M, that is singular at some of its point, that is X is not a submanifold at this point. We can think of singular real algebraic sets, or germs or real analytic sets as a model of the singularities we are interested in dealing with. The smooth part of X comes equipped with a Riemannian metric induced from the ambient one. We would like to understand how do geodesics on the regular part of X behave in a neighbourhood of a singular point. It turns out that very little is known (or even explored) about very elementary singularities (conical, edges or corners). The purpose of specifying such a singular set was to study the propagation of singularities for the wave equation on such a singular "manifold" (Melrose, Vasy, Wunsch,...). In this joint work with Daniel Grieser (Oldenburg, Germany), we want to address a naive and elementary question: Given an isolated surface singularity X in (M,g), can a neighbourhood of the singular point be foliated by geodesics reaching the singular point ? Then can we define an exponential mapping at a such point ? This property is true for conical singularities of any dimension. With D. Grieser, we have exhibited very simple examples of non-conical real surfaces with an isolated singularity, and cuspidal like, in a 3-manifold, such that the geodesics reaching the singular point behave differently according to the considered class.
Joint Analysis/Mathematical Physics seminar
Monday, March 7, 13:30-14:30, Burnside 1205
Christian Hainzl (Tuebingen)
A Spatially Homogeneous and Isotropic Einstein-Dirac Cosmology
Abstract: We consider a spatially homogeneous and isotropic cosmological model where Dirac spinors are coupled to classical gravity. For the Dirac spinors we choose a Hartree-Fock ansatz where all one-particle wave functions are coherent and have the same momentum. If the scale function is large, the universe behaves like the classical Friedmann dust solution. If however the scale function is small, quantum effects lead to oscillations of the energy-momentum tensor. It is shown numerically and proven analytically that these quantum oscillations can prevent the formation of a big bang or big crunch singularity. The energy conditions are analyzed. We prove the existence of time-periodic solutions which go through an infinite number of expansion and contraction cycles. This is joint work with Felix Finster.
Special seminar in Spectral Theory
Thursday, March 10, 13:30-14:30
Universite de Montreal: Pav. Andre Aisenstadt, 5448

Alexei V. Penskoi (Moscow State University, Independent University of Moscow and Bauman Moscow State Technical University)
Extremal spectral properties of Lawson tau-surfaces and the Lame equation
Abstract: Given a closed compact surface, eigenvalues of the Laplace-Beltrami operator are functionals on the space of Riemannian metrics of fixed area on this surface. The question about extremal metrics for these eigenvalues is a difficult problem of a differential geometry.
In this talk we shall describe significant advances is this domain happened during last ten years and last results about extremal metrics on Lawson tori and Klein bottles representing an interesting interplay between extremal metrics, minimal surfaces and the classical Lame equation.
Monday, March 14, 13:30-14:30, Burnside 1205
Luc Hillairet (Nantes)
Semiclassical concentration and eigenvalue branches
Abstract: Consider a real-analytic family of operators $A_t$ that becomes singular when $t$ goes to $0$. We will exhibit relations between concentration of eigenfunctions of the associated semiclassical operator $A_{h_n}$ and the asymptotic behavior of eigenbranches. We will show on different examples how these ideas apply to improve generically on existing spectral results.
Wednesday, March 16, 13:30-14:30, Burnside 1205
Dmitry Dolgopyat (Univ. of Maryland and Fields Institute)
Dissipative perturbations of area preserving flows on surfaces.
Abstract: I review probabilistic phenomena which appear for small dissipative perturbations of area preserving flows on surfaces. In particular, I show that in case of higher genus surfaces such pertubations could lead to an intermittent behavior. This is a joint work with Mark Freidlin and Leonid Koralov.
Friday, March 18, Burnside 920, 14:30-15:30
Tatiana Toro (Univ. of Washington)
Analysis on non-smooth domains
Abstract: In this talk we will discuss the solvability of the Dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients which are perturbations of the Laplacian in rough domains. Our approach requires the development of several tools from harmonic analysis on these domains. The results presented are joint work with E. Milakis and J. Pipher.
Special seminar in Spectral Theory
Thursday, March 24, 13:30-14:30
Universite de Montreal: Pav. Andre Aisenstadt, 5448

Leonid Polterovich (University of Chicago and Tel Aviv University)
Nodal inequalities on surfaces
Friday, March 25, 14:30-15:30, Burnside 920
Alex Stokolos (Georgia Southern)
Solution to some Rudin's Problem
Abstract: I will present the solution to Walter Rudin's problem of the tangential boundary behavior of bounded harmonic functions.The result complements the classical theorems of Fatou, Littlewood, Rudin, Nagel and Stein. This is a joint work with Fausto di Biase, Olaf Svensson and Tom Weiss.
Joint Analysis/Mathematical Physics Seminar
Monday, March 28, 13:30-14:30, Burnside 1205
Milton Jara (IMPA, Rio de Janeiro)
Universality of KPZ equation
Abstract: The KPZ equation was introduced in the '80s by Kardar, Parisi and Zheng as a continuous model for stochastic growth interfaces. Despite its simplicity, a mathematically rigorous formulation of KPZ equation is still lacking. In this talk we introduce the concept of energy solutions of KPZ equation and we prove that under mild assumptions, they appear as the scaling limit of conservative interacting particle systems in equilibrium. Joint work with Patricia Goncalves, U. do Minho.
Friday, April 1, 14:30-15:30, Burnside 920
Andrew Comech (Texas A/M)
Global attractor for Klein-Gordon equation in discrete space-time
Abstract: We consider the U(1)-invariant Klein-Gordon equation in discrete space-time, with the nonlinearity concentrated at one point. We show that solitary waves form the weak global attractor for this equation. That is, for large positive or negative times any finite energy solution converges to the set of all solitary waves. The convergence takes place in localized (weighted) norms. This is a joint work with Alexander Komech, Vienna University and IITP, Moscow
Friday, April 8, 14:30-15:30, Burnside 920
Leonid Friedlander (Arizona)
Determinants of elliptic operators
Abstract: I will give a historical overview of how the notion of the determinant was being developed, starting from Leibniz. Certain problems arise when one goes from matrices to operators acting in infinite-dimensional spaces. I will discuss how to use different regularization techniques to deal with these problems. In particular, I will discuss determinants of elliptic differential operators, and anomalies that are associated with regularization procedures.
Monday, April 11, 13:30-14:30, Burnside 1205
Richard Froese (UBC)
Absolutely continuous spectrum for the Anderson model on a product of a tree with a finite graph
Friday, April 15, 13:30-14:30, Burnside 920
Lia Bronsard (McMaster)
Vortices in Ginzburg-Landau systems
Abstract: The Ginzburg-Landau model is a popular and successful variational principle in physics describing phenomena such as superconductivity, superfluidity, and Bose-Einstein condensation. It is no less remarkable for its mathematical features, in particular the quantized vortices which characterize its minimizing states. In this talk, I will discuss some PDE problems associated with Ginzburg-Landau vortices, which arise in characterizing all solutions which are "locally minimizing" in an appropriate sense (due to De Giorgi). I will compare the results on the original Ginzburg-Landau model with a more complex, two-component Ginzburg-Landau system where more interesting vortex core structures are possible.
Friday, April 29, 13:30-14:30, Burnside 920
Jeremy Tyson (Univ. of Illinois at Urbana-Champaign)
Modulus and Poincare inequalities on Sierpinski carpets Abstract: A carpet is a metric space homeomorphic to the standard Sierpinski carpet. We characterize, within a certain class of examples, carpets supporting curve families of nontrivial modulus and supporting Poincare inequalities. These carpets are the first known examples of compact Euclidean sets without interior which support Poincare inequalities for the Lebesgue measure.
Monday, May 2, 13:30-14:30, Burnside 920
Flavia Colonna (George Mason University)
THE TALK IS CANCELLED
Wednesday, May 11, 13:00-14:00, Burnside 920
A. Nabutovsky (Toronto)
Morse landscapes of Riemannian functionals and related problems

Wednesday, May 11, 14:30-15:30, Burnside 920
R. Rotman (Toronto)
Short geodesics between a pair of points on a closed Riemannian manifold
Monday, May 16, 13:30-14:30, Burnside 920
Alberto Enciso (Zurich)
Abstract: Given any possibly unbounded, locally finite link, we show that there exists a smooth diffeomorphism transforming this link into a set of stream (or vortex) lines of a vector field that solves the steady incompressible Euler equation in R^3. Furthermore, this diffeomorphism can be chosen arbitrarily close to identity in any C^r norm.
Wednesday, May 18, 13:30-14:30, Burnside 920
Nondegeneracy of the eigenvalues of the Hodge Laplacian for generic metrics on 3-manifolds
Abstract: From a qualitative point of view, one of the most attractive results in spectral geometry is K. Uhlenbecks proof of the fact that, for a "generic" set of C^r metrics, the eigenvalues of the Laplacian on a closed manifold are simple and its eigenfunctions are Morse (1972). In this talk we will show how Uhlenbeck's theorem can be extended to the case of differential forms on 3-manifolds using the Beltrami (or rotational) operator on co-exact 1-forms. This talk will be based on a joint work with Alberto Enciso (to appear in Trans. Amer. Math. Soc.)
Spectral Theory seminar
Thursday, May 19, McGill, 13:30-14:30, Burnside 920
Topological monsters in PDE
Abstract: Can the infinite jungle gym be the zero set of a harmonic function? Are there two harmonic functions whose joint zero set contains all knot and link types? In this talk we will show the existence of harmonic functions in R^3 exhibiting these and other topological monsters, keeping technicalities to the bare minimum. This talk will be based on joint work with Alberto Enciso.
Tuesday, May 24, 11:00-12:00, Concordia University LB-921-4
Piotr Oprocha (AGH University of Science and Technology, Krakow,Poland)
Abstract: The main aim of this talk is to relate notions of average shadowing property and asymptotic average shadowing property with other notions known from topological dynamics. We will focus mainly on relations of the above two notions with stronger forms of transitivity (e.g. topological mixing) and more standard versions of shadowing property (e.g. shadowing and limit shadowing).
Wednesday, May 25, 13:30-14:30, Burnside 920
Ivana Alexandrova (SUNY Albany)
Aharonov-Bohm Effect in Resonances of Magnetic Schrodinger Operators with Potentials with Supports at Large Separation
Abstract: Vector potentials are known to have a direct significance to quantum particles moving in the magnetic field. This is called the Aharonov--Bohm effect and is known as one of the most remarkable quantum phenomena. Here we study this quantum effect through the resonance problem. We consider the scattering system consisting of two scalar potentials and one magnetic field with supports at large separation in two dimensions. The system has trajectories oscillating between these supports. We give a sharp lower bound on the resonance widths as the distances between the three supports go to infinity. The bound is described in terms of the backward amplitude for scattering by each of the scalar potentials and by the magnetic field, and it also depends heavily on the magnetic flux of the field.

## ANALYSIS-RELATED TALKS ELSEWHERE, WINTER 2011

CRM-ISM colloquium Friday, March 4, UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420, 16:00-17:00
Dan Stroock (MIT)
Some random thoughts about Cauchy's functional equation
Abstract: pdf

## FALL 2010

Monday, August 2, 11:00, Room 920
Wednesday, August 4, 11:00, Room 920
Friday, August 6, 11:00, Room 920
Monday, August 9, 11:00, Room 920
Raphael Ponge (Tokyo)
Fefferman's program and Green functions of conformally invariant differential operators
Abstract: The following topics will be covered:
Fefferman's program in conformal geometry. Conformal invariants. Conformally invariant operators.
Construction of the conformal powers of the Laplacian (aka GJMS operators) via the ambient metric of Fefferman-Graham.
Singularities of Green functions and zeta functions.
Explicit computation of the logarithmic singularities of the Green functions of the conformal powers of the Laplacian.
The lectures are aimed at and should be accessible to graduate students.

Monday, August 2, 14:00, Room 920
Julian Edward (Florida International University)
An application of boundary control method to an inverse problem
Abstract: Suppose a circular membrane in R^2 has an unknown density which is radially dependent. Using the boundary control method, we show that this density can be recovered from certain boundary measurements.
Wednesday, September 8, 13:30-14:30, Burnside 920
Elijah Liflyand (Bar Ilan)
Two-sided weighted Fourier inequalities

Friday, September 24, 13:30-14:30, Burnside 920
Christoph Haberl (TU Vienna and Poly NYU)
Affine Sobolev inequalities
Abstract
Friday, October 8, 13:30-14:30, Burnside 920
Hans Christianson (North Carolina)
Local smoothing with a prescribed loss for the Schrodinger equation
Abstract: Local smoothing estimates express that, on average in time and locally in space, solutions to the Schrodinger equation are more regular than the initial data. It is known that the presence of trapped geodesics forces a loss in the local smoothing effect, but not too many examples have been studied. In this work (joint with J. Wunsch), we study some examples which fill in the gap between no loss and total loss in the smoothing effect.
Friday, October 15, 13:30-14:30, Burnside 920
N-body systems, quantum fields, many-body systems: a proof of the Mourre estimate
Abstract: We prove the Mourre estimate for systems with a variable number of of particles. The framework and the techniques are C*-algebraic.
Seminaire Analyse et sujets connexes
Tuesday, October 19, 9:30-10:30
Universite de Montreal, CRM, Salle 4336
Laurent Baratchart (INRIA, France)
Problemes extremaux de potentiel et approximation rationelle.
Friday, October 22, 13:30-14:30, Burnside 920
Discriminant separability, pencils of conics and Kowalevski integrability
Abstract: A new view on the Kowalevski top and Kowalevski integration procedure is presented. It is based on geometry of pencils of conics, a classical notion of Darboux coordinates, a modern concept of n-valued Buchstaber-Novikov groups and a new notion of discriminant separability. Unexpected relationship with the Great Poncelet Theorem for a triangle is established. Classification of strongly disriminatly separable polynomials of degree two in each of three variables is performed. Further connections between discriminant separability, geometry of pencils of quadrics and integrability are discussed.
Wednesday, November 3, 14:30-15:30 (cofee at 15:30)
Concordia univ, Library building, 1400 De Maisonneuve West, LB 921-4

Andrea Colesanti (University of Florence) Problems of Minkowski type
Abstract: The classical Minkowski problem requires to find a convex body (i.e. a compact convex set) with smooth boundary, given the Gauss curvature as a function of the outer unit normal. The problem admits a weak formulation in which the datum is the so called area measure of the convex body. Such measure can be also interpreted as the first variation of the volume with respect to the outer unit normal. The aim of the talk is to describe in some detail this point of view and to present similar problems where the volume is replaced by variational functionals like capacity, the first eigenvalue of the Laplace operator and the torsional rigidity.
Wednesday, November 3, 16:00-17:00 (cofee at 15:30)
Concordia univ, Library building, 1400 De Maisonneuve West, LB 921-4

Francis Clarke (Univ. Lyon)
A painless introduction to nonsmooth analysis and its applications
Abstrsct: An overview of the rudiments of Nonsmooth Analysis is provided. Then some applications to problems in optimization theory and control theory are outlined.
Friday, November 5, 13:30-14:30, Burnside 920
Der-Chen Chang (Georgetown)
Geometric analysis on the 3-D sphere
Abstract: The unit sphere S3 can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism. We also discuss some properties of the heat kernel for the sub-Laplacian.
Friday, November 12, 13:30-14:30, Burnside 920 (time and place to be confirmed)
S. Bezuglyi (Inst. for Low Temperature Physics, Academy of Sciences, Ukraine )
Homeomorphic measures on a Cantor set
Abstract: Two measures m and m' on a topological space X are called homeomorphic if there is a homeomorphism f of X such that m(f(A)) = m'(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history. The well-known result of Oxtoby and Ulam gives a criterion when a Borel measure on the n-dimensional cube [0, 1]^n is homeomorphic to the Lebesgue measure. The situation is more difficult for measures on a Cantor set. There is no complete answer to the above question even in the simplest case of Bernoulli trail measures. In my talk, I will discuss the recent results about classification of Borel probability measures which are ergodic and invariant with respect to aperiodic substitution dynamical systems. In other words, we consider the set M of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation. The properties of these measures related to the clopen values set S(m) are studied. It is shown that for every measure m from M there exists an additive subgroup G of real numbers such that S(m) + Z = G, i.e. in other words S(m) is group-like. A criterion of "goodness" is proved for such measures. Based on this results, we classify the good measures from M up to a homeomorphism. It is proved that for every good measure m from M there exist countably many measures m(i) from M such that the measures m and m(i) are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.
Friday, November 19, 13:30-14:30, Burnside 920
Brian Clarke (Stanford)
Geodesic distance on the manifold of Riemannian metrics
Abstract: On a fixed closed manifold, I will consider the manifold of all possible Riemannian metrics. This manifold is itself equipped with a canonical Riemannian metric, called the L^2 metric. I will give an explicit expression for the distance, with respect to the L^2 metric, between any two metrics on the base manifold. Additionally, the completion of the manifold of metrics can be described, and I will present an explicit expression for the unique minimal path between any two points in this completion. Time permitting, I will also discuss connections to other areas such as Teichmuller theory and the convergence of Riemannian manifolds.
Friday, November 26, 13:30-14:30, Burnside 920
Frederic Naud (Avignon)
The spectral gap of convex co-compact subgroups of arithmetic groups
Abstract: (Joint with Dmitry Jakobson). We investigate the spectral gap of the Laplace-Beltrami operator on certain infinite area Riemann surfaces and its relationship with various problems such as the hyperbolic lattice counting problem or the decay of waves.
Friday, December 10, 11:00-12:00, Burnside 920
On the near periodicity of eigenvalues of Toeplitz matrices
Abstract: Let $A$ be an infinite Toeplitz matrix with a real symbol $f$ defined on $[-\pi, \pi]$. It is well known (e.g. Szego) that the sequence of spectra of finite truncations $A_N$ of $A$ converges to the convex hull of the range of $f$. Some numerical experiments showed, a while ago, that for symbols $f$ with two discontinuities located at rational multiples of $\pi$, the eigenvalues of $A_N$ located in the gap of the range of $f$ asymptotically exhibit periodicity in $N$. Here, we quantify and prove the analog of this conjecture for the matrix $A^2$ in a particular case when $f$ is a piecewise constant function taking values $-1$ and $1$. This is a joint work with Alex Sobolev and Daphne Sobolev.
Friday, December 17, 13:30-14:30, Burnside 920
Junfang Li (Birmingham, Alabama)
A modified mean curvature type of flow and isoperimetric inequality
Abstract: We introduce a mean curvature type of flow and its fully nonlinear analogue. We use this new type of flow to prove isoperimetric inequality and Alexandrov-Fenchel inequalities.

## ANALYSIS-REALTED TALKS ELSEWHERE, FALL 2010

CRM-ISM Colloquium
Friday, October 22, 16:00-17:00
UdeM, Pav. A. Aisenstadt, 2920, ch. de la Tour, salle 6214.
Claude LeBris (ENPC)
Stochastic homogenization and related problems
Abstract: The talk will focus on homogenization theory in the non periodic context. It will be shown how some appropriately chosen deterministic generalizations of the periodic setting, and some "weakly random" generalizations can lead to theories that are both practically relevant and computationally efficient. The material presented in the talk covers joint work with X. Blanc, PL. Lions, F. Legoll, A. Anantharaman, R. Costaouec, F. Thomines.
CRM-ISM Colloquium
Friday, October 29, 16:00-17:00
UdeM, Pav. A. Aisenstadt, 2920, ch. de la Tour, salle 6214.
Mathieu Lewin (Cergy-Pontoise)
The Thermodynamic Limit of Coulomb Quantum Systems
Abstract: In this talk I will review the methods for studying the limit of infinitely many quantum particles interacting through the Coulomb potential, like electrons and nuclei in ordinary matter. I will in particular present a new approach which generalizes previous results of Fefferman, Lieb and Lebowitz. This is joint work with Christian Hainzl (Birmingham, Alabama) and Jan Philip Solovej (Copenhagen, Denmark).
Nonlinear analysis and dynamical systems seminar
Thursday, November 4, 14:00
Salle 4336, pav. Andre-Aisenstadt, Univ. de Montreal
Oxana Diaconescu
Lie algebras and invariant integrals for multi-dimensional polynomial differential systems
Abstract: The talk is devoted to application of Lie algebras of operators and of the theory of algebraic invariants to ordinary polynomial differential systems of first order. Lie theorem on integrating factor is generalized for multi-dimensional polynomial differential systems. Lie algebras of operators were constructed for n-dimensional systems of the Darboux type of degree m. With the help of these algebras the explicit forms of invariant particular and first GL (n, R)-integrals were obtained. Recurrent formulas of some invariant integrals for Darboux type systems were constructed.

2005/2006 Analysis Seminar