## 2008/2009 Analysis Seminar

Seminars are usually held on Mondays or Fridays at Concordia or at McGill
For suggestions, questions etc. please contact Galia Dafni (gdafni@mathstat.concordia.ca), Dmitry Jakobson (jakobson@math.mcgill.ca), Ivo Klemes (klemes@math.mcgill.ca) or Alexander Shnirelman (shnirel@mathstat.concordia.ca)

## WINTER 2009

Joint Seminar with Applied Mathematics
Monday, January 12, 2009, 14:30, Burnside 1205
David Anderson (Wisconsin)
Deterministic and Stochastic Methods for Biochemical Reaction Systems
Abastract: The dynamics of biochemical reaction systems can be modeled either deterministically or stochastically. Typically, the equations governing the dynamics of these models are quite complex. Further, there is oftentimes little knowledge about the exact values of the different system parameters, and, worse still, these system parameter values may vary from cell to cell. However, the network structure of a given system induces the corresponding equations (up to parameter values) governing its dynamics. I will show in this talk how this fact may be exploited to infer qualitative properties of large classes of biochemical systems and, most importantly, to learn which properties are independent of the details of the system parameters. I will give results for both stochastically and deterministically modeled systems. I will also discuss some recent work on numerical methods for the simulation of sample paths for stochastically modeled systems. The use of such methods is quickly increasing throughout the biology and biochemistry communities and therefore warrants more careful study.

Joint Seminar with Applied Mathematics
Monday, January 19, 2009, 14:30, Burnside 1205
Irina Mitrea (Virginia)
Boundary value problems for higher order elliptic operators
Abstract
Joint Seminar with Applied Mathematics
Monday, January 26, 2009, 14:30, Burnside 1205
Vera Mikyoung Hur (MIT)
Dispersive properties of the surface water waves
Abstract: I will speak on the dispersive character of waves on the interface between vacuum and water under the influence of gravity and surface tension. I will begin by giving a precise account of the formulation of the surface water-wave problem and discussion of its distinct features. They include the dispersion relation, its severe nonlinearity, traveling waves and the Hamiltonian structure. I will describe the recent work of Hans Christianson, Gigliola Staffilani and myself on the local smoothing effect of 1/4 derivative for the fully nonlinear problem under surface tension with some detail of the proof. If time permits, I will explore some oen questions regarding long-time behavior and stability.
Joint Seminar with Applied Mathematics
Monday, February 9, 2009, 14:30, Burnside 1205
George Haller (Morgan Stanley and MIT)
Aerodynamic Separation and Invariant Manifolds: Recent Progress on a Century-old Problem
Abstract: Flow separation - the detachment of fluid from a boundary - is a major cause of performance loss in engineering devices such as diffusers, airfoils and jet engines. In a landmark 1904 paper on boundary layers, Ludwig Prandtl derived a criterion for flow separation from no-slip boundaries in steady two-dimensional incompressible flows. Despite widespread effort, however, no unsteady or three-dimensional extension of Prandtl's criterion has emerged in the fluid dynamics literature. In this talk, I discuss recent success in extending Prandtl's criterion to unsteady three-dimensional compressible flows. This new separation theory relies on nonstandard dynamical systems concepts, such as nonhyperbolic invariant manifold theory and aperiodic averaging. Remarkably, these techniques render exact flow separation criteria that cannot be obtained from first principles. Beyond discussing the mathematics behind this new theory, I show numerical and experimental results condirming the new separation criteria. I also discuss applications to flow control and pollution tracking.
Joint Seminar with Applied Mathematics
Monday, February 16, 2009, 14:30, Burnside 1205
Rustum Choksi (Simon Fraser)
Mathematical Paradigms for Periodic Phase Separation and Self-Assembly of Diblock Copolymers
Abstract Energy-driven pattern formation induced by competing short and long-range interactions is ubiquitous in science, and provides a source of many challenging problems in nonlinear analysis. One example is self-assembly of diblock copolymers. Phase separation of the distinct but bonded chains in dibock copolymers gives rise to an amazingly rich class of nanostructures which allow for the synthesis of materials with tailor made mechanical, chemical and electrical properties. Thus one of the main challenges is to describe and predict the self-assembled nanostructure given a set of material parameters. A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations in 3D, I will attempt to characterize minimizers without any preassigned bias for their geometry. In particular, I will show how this simple model has given rise to some basic questions and answers in the modern calculus of variations.
Friday, February 20, 2009, Burnside 920, 14:30
Lp Boundedness of Riesz transform related to Schrodinger operators on a manifold
Abstract
Friday, March 6, 2009, 14:30, Burnside 920
Ivana Alexandrova (East Carolina)
The Structure of the Scattering Amplitude for Schrodinger Operators with a Strong Magnetic Field
Abstract: We study the microlocal structure of the semi-classical scattering amplitude for Schrodinger operators with a strong magnetic field at non-trapping energies. We prove that, up to any order, the scattering amplitude can be approximated by a semi-classical pseudodifferential-operator-valued Fourier integral operator.
Monday, March 9, 2009, 14:30, Burnside 920
S. Molchanov (North Carolina, Charlotte)
The statistics of the discrete spectrum for the Schrodinger operator with complex-valued potentials (with applications to optical waveguides).
Joint Seminar with Applied Mathematics
Wednesday, March 11, 2009, 14:30, Burnside 1205
Dmitry Pelinovsky (McMaster)
Global well-posedness and wave breaking in the short-pulse equation
Abastract: We prove global well-posedness of the short-pulse equation with small initial data in Sobolev space H^2. Our analysis relies on local well-posedness results of Schafer & Wayne (2004), the correspondence of the short-pulse equation to the sine-Gordon equation in characteristic coordinates, and a number of conserved quantities of the short-pulse equation. We also find sufficient conditions for the wave breaking to occur if the initial data have large H^2 norm. The analysis relies on the method of characteristics and it holds both on an infinite line and in a periodic domain. Numerical illustrations of the finite-time wave breaking are given for the periodic short-pulse equation.
Friday, March 13, 2009, 14:30, Burnside 920
Mike Wilson (Vermont)
How fast and in what sense(s) does the Calderon reproducing formula converge?
Abstract
Monday, March 16, 2009, 14:30, Burnside 920
A. Choffrut (Minnesota)
On steady-state solutions to Euler's equations
Abstract: The manifold of the steady-state solutions of 2d Euler's equation in a domain (and with suitable boundary conditions) is typically infinite-dimensional. The geometric interpretation of Euler's equations suggests a natural local parametrization of the manifold (under some non-degeneracy assumptions). This is established rigorously in some interesting situations. (Joint work with Vladimir Sverak).
Friday, March 20, 14:30, Burnside 920
Raphael Ponge (Toronto)
Pseudodifferential operators and Fefferman's program
Abstract: Motivated by the analysis of the Bergman kernel of a strictly pseudoconvex domain of C^n, Fefferman launched in the 70s the program of determining all the local invariants of a strictly pseudoconvex CR structure. Since then the program has evolved to include local and global invariants of other parabolic geometries (e.g. conformal geometry). In this talk we shall report on approach of Fefferman's program in terms of pseudodifferential and how this approach allows us to obtain new invariants in conformal and CR geometry.
Friday, March 27, 14:30, Burnside 920
G. Kolutsky (Moscow, visiting Toronto)
Geometrical Continued Fractions and Anosov Diffeomorphisms
Abstract: We show how an object from the combinatorially geometric version of the analytical number theory, namely geometrical continued fractions, appears in the classical smooth dynamics, namely in the problem on the topological classification of Anosov diffeomorphisms of tori.
Friday, April 3, 14:30, Burnside 920
Mar Gonzalez (IAS)
Half-Laplacian problems related to crystal dislocations
Abstract: Dislocations are line defects in crystals, and can be modeled using non-local operators. I will speak about a related evolution equation involving the half-Laplacian operator.
Thursday, April 16, 2009, 14:30, Burnside 1205 (please note day and room change)
Alain Pajor (Marne-la-Valee)
Compress sensing and geometry of polytopes
Abstract: The connection between "Compressed sensing" and "High dimensional Geometry and Convexity" is well known and studied in many papers in the recent years. In this talk we will give new examples of (random) highly neighborly polytopes which in compressed sensing theory means new examples of "sensing" matrices for the exact reconstruction of sparse vectors.
Friday, May 1, 2009, 14:30, Burnside 920
R. Martin (UC Berkeley)
Symmetric operators and reproducing kernel Hilbert spaces
Abstract
Tuesday, May 26, 14:30-15:30, Burnside 920
Leonid Berlyand (Penn State)
Homogenization of elasticity equations without scale separation
Abstract
Wednesday, June 3, 14:30-15:30, Burnside 1214
Paolo Salani (Florence)
Concavity properties and Brunn-Minkowski inequalities in free boundary problems
Abstract: In a recent paper with C. Bianchini, we prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant, giving a partial answer to a conjecture of Flucher and Rumpf. Moreover, we study the behaviour of the free boundary with respect to the given boundary data and we prove a uniqueness result regarding the interior problem.
Friday, June 26, 14:30-15:30, Burnside 920
Junfang Li (Alabama)
Li-Yau type Differential Harnack inequalities on Riemannian manifolds : linear heat equation
Abstract: We present new Li-Yau type gradient estimates for positive solutions of heat equation on Riemmannian manifolds with $Ricci(M)\ge -k$, $k\in \mathbb R$. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li-Yau-Hamilton differential Harnack inequality for heat kernels on manifolds with $Ricci(M)\ge -k$, which generalizes a result of L. Ni [NL1,NL4]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.

## ANALYSIS-REALTED TALKS ELSEWHERE, WINTER 2009

Applied Mathematics: Friday, January 16, 2009, 14:30-15:30, Burnside 1205
Margaret Beck
Nonlinear stability of time-periodic viscous shocks
Abstract: If a given solution of a PDE is stable, then, roughly speaking, any other solution that starts near it, stays near it for all time. This is an important concept in applications, because it is typically only the stable solutions that are observed in practice. I will outline two key mathematical difficulties that one can encounter when analyzing the stability of time-periodic solutions of dissipative PDEs on unbounded domains. Briefly, they are the presence of zero eigenvalues that are embedded in the continuous spectrum and the time- periodicity of the associated linear operator. In the context of viscous shocks in systems of conservation laws, I will show how these difficulties can be overcome. The method involves the development of a contour integral representation of the linear evolution, similar to that of a strongly continuous semigroup, and detailed pointwise estimates on the resultant Greens function, which are sufficient for proving nonlinear stability under the necessary assumption of spectral stability.
Dynamical Systems seminar: Friday, January 30, 2009, 10:00-11:00 Concordia, Library Building LB 921-4
Edson Vargas (Univ. of Sao Paolo)
Decay of geometry for Fibonacci critical covering maps of the circle
Abstract: We study the growth of Dfn(f(c)) when f is a Fibonacci critical covering map of the circle with negative Schwarzian derivative, degree d >= 2 and critical point c of order l > 1. As an application we prove that f exhibits exponential decay of geometry if and only if l <= 2, and in this case it has an absolutely continuous invariant probability measure, although not satisfying the so-called Collet-Eckmann condition

## FALL 2008

Friday, September 5, 2008, 14:30-15:30, Burnside 920
C.S. Lin (National Taiwan University)
Green function, elliptic functions and mead field equations on torus
Abstract: A mean field equations on torus is a second order nonlinear ellitpic equations with an exponential nonlinearity. The blowing up analysis of mean field equation has a close relations to the critical points of Green function on torus. In this talk, we will study the question of the number of critical points of a Green function via the mean field equations. We prove that the Green function has at most five critical points via studying the mean curvature equation.

Mini-course by David Ruelle (IHES)
Nonequilibrium statistical mechanics and smooth dynamical systems
Friday 13:00-14:30, starting September 12; Burnside 920
Description (pdf)
Friday, September 19, 2008, 14:30-15:30, Burnside 920
D. Grieser (Oldenburg)
Spectral approximation for fat graphs
Abstract: A fat graph is a family of Riemannian manifolds $M_\epsilon$, $\epsilon>0$, modelled on a finite metric graph $G$, in a way similar to an $\epsilon$-neighborhood of a straight-edge embedding of $G$ in some Euclidean space. The behavior of the spectrum and of spectral invariants of various geometric differential operators on $M_\epsilon$ as $\epsilon$ tends to zero has been studied by many authors in different contexts. We focus on a question arising in mathematical physics which has attracted much attention in the quantum graphs community recently: Which operator on the singular limit $G$ describes the asymptotic behavior of the eigenvalues of the Laplacian on $M_\epsilon$ appropriately? While for Neumann boundary conditions (or closed manifolds) the answer has been known for some time and can be obtained by relatively elementary methods, the case of Dirichlet conditions is harder and was solved only recently. This will be explained in the talk.
Joint Seminar with Applied Mathematics
Friday, October 3, 2008, 14:30, Burnside 1205
D. Pelinovsky (McMaster)
Spectrum of an advection-diffusion operator with sign-varying diffision coefficient

Monday, October 6, 2008, 14:30, Burnside 920
V. Zagrebnov (CPT-Luminy, Marseille)
Bose-Condensation in External Potentials
Abstract (pdf)
Wednesday, October 15, 2008, 13:30-14:30, Room 920
L. Nirenberg (Courant)
Remarks on fully nonlinear elliptic partial differential equations

Friday, October 24, 2008, 14:30-15:30, Burnside 920
S. Janson (Uppsala University)
Schatten norm identities for Hankel operators
Wednesday, October 29 (note a change of date), 13:30-14:30, Room 920
Robert Seiringer (Princeton)
Vortices and Spontaneous Symmetry Breaking in Rotating Bose Gases
Abstract We present a rigorous proof of the appearance of quantized vortices in dilute trapped Bose gases with repulsive two-body interactions subject to rotation, which was obtained recently in joint work with Elliott Lieb. Starting from the many-body Schroedinger equation, we show that the ground state of such gases is, in a suitable limit, well described by the nonlinear Gross-Pitaevskii equation. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.
Joint Seminar with Group Theory
Friday, November 7, 2008, 14:30, Burnside 920
A. Furman (Univ. of Illinois at Chicago)
Actions of Product Groups on Manifolds
Abstract This is a joint work with Nicolas Monod. We analyze volume-preserving actions of products of Kazhdan groups on Riemannian manifolds. Under a natural irreducibility assumption we obtain lower bounds on the dimension of the manifold in terms of the number of factors in the acting group, and strong restrictions for actions of non-linear groups. We prove our results by means of a new cocycle superrigidity theorem of independent interest, in analogy to Zimmer's programme.

Joint Seminar with Vermont
Monday, November 10, 2008, 14:30, Burnside 920
C. Perez (University of Seville)
Weighted estimates Singular Integral Operators and Sobolev inequalities
Abstract (pdf)
Friday, November 21, 2008, 14:30, Burnside 920
A. Komech (Texas A & M)
Global attraction to solitary waves in nonlinear models based on the Klein-Gordon equation
Abstract: We discuss recent results on global attraction to solitary waves in several U(1)-invariant models built upon the Klein-Gordon equation, such as the Klein-Gordon field interacting with finitely many nonlinear oscillators and with the mean field interaction. The main analytical tools of the approach are the Paley-Wiener arguments and the Titchmarsh Convolution Theorem, which allow to restrict the time-spectrum of the omega-limit trajectory to a single point. Physically, the global attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. This is a joint work with Alexander Komech, University of Vienna
Monday, December 1, 2008, 14:30, Burnside 920
W. Craig (McMaster)
The Navier - Stokes equations: an analysis overview
Monday, December 15, 2008, 12:30-13:30, Burnside 920
E. Kritchevski (UBC)
One dimensional Anderson model with non-homogeneous disorder
Abstract: We consider the random discrete Schrodinger operator H=L+V on the one dimensional integer lattice Z. The operator L is the discrete laplacian, (LF)(x)=f(x-1)+f(x+1), and V is a potantial, Vf(x)=v(x)f(x), where v(x) is a family of independent random variables. We will discuss a new method to establish localization, i.e. that generically the eigenfunctions of H decay exponentially. The method is robust enough to allow v(x) to have different probability distributions for different lattice points x. Moreover, the method allows to obtain lower bounds for the rate of decay of the eigenfunctions. The talk will be iven in the language of finite dimensional matrices and basic probability theory.

## ANALYSIS-REALTED TALKS ELSEWHERE, FALL 2008

Thursday, September 4, 2008, 13:00; McGill, Rutherford 326
G. Ben-Shach (McGill)
The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
CRM-ISM colloquium
Friday, September 26, 4:00-5:00pm
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
PDE aspects of the Navier-Stokes Equations
Abstract: We will explain the main difficulties arising in the mathematical analysis of the Navier-Stokes equations and we mention some recent results which are related to these problems.
CRM-ISM colloquium
Friday, October 3, 4:00-5:00pm
UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour, salle 6214
Elliott Lieb (Princeton University)
Some Calculus of Variations Problems in Quantum Mechanics
Abstract: Three examples are given, in order of historical development, of minimization problems in quantum mechanics arising from attempts to model the N-body Schroedinger equation by simpler energy functionals involving only densities. These simpler models are Thomas-Fermi theory, Hartree-Fock theory and the Mueller density matrix functional theory. This talk is for non-specialists: no knowledge of quantum mechanics is needed.
CRM-ISM colloquium
Friday, October 10, 4:00-5:00pm
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
Leonid Bunimovich (Georgia tech)
Visual Chaos: dispersing, defocusing, absolute focusing and astigmatism
Abstract: The mechanisms generating chaotic (hyperbolic) behavior in billiards will be discussed. It turned out that all focusing components of the boundary of chaotic billiards should be absolutely focusing. Absolute focusing seems to be a new notion in the geometric optics. The astigmatism comes into play in dimensions greater than two which forces to reduce the sizes of focusing components of chaotic billiard tables. We conclude with the simple visual examples of billiards with divided phase space where any number of chaotic ergodic components coexist with any number of integrable islands.
October 17, 20, 22, 23, 4:00-5:00pm
UdeM, Pav. Andre-Aisenstadt, 2920, ch. de la Tour, salle 1360 (Oct. 17); salle 6214 (Oct. 20, 22, 23)
Svante Janson (Uppsala)
Random Graphs
Abstract
CRM-ISM colloquium
Friday, October 24, 4:00-5:00pm
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
David Ruelle (IHES)
Nonequilibrium Statistical Mechanics and Smooth Dynamical Systems
Abstract: One idealization of nonequilibrium statistical mechanics simply gives general smooth dynamics on a compact manifold, with new interpretations and new questions, which we shall review. In particular we shall introduce the natural physical (SRB) measure associated with a diffeomorphism f, and ask if it depends smoothly on f. We shall also introduce an analytic susceptibility function and study its singularities.
CRM-ISM colloquium
Friday, October 31, 4:00-5:00pm
Robert Seiringer (Princeton)
Dilute Quantum Gases
Abstract: We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The discussion includes, for instance, results on the free energy in the thermodynamic limit, and on Bose-Einstein condensation, Superfluidity and quantized vortices in trapped gases. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a brief description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schroedinger equation.
Statistics Journal Club , Burnside 1214
Friday, November 28, 12:00-13:00
Igor Wigman (CRM)
Nodal lines and the distribution of the zeros of random trigonometric polynomials
Abstract: I will show a video and some pictures of nodal lines, mysterious lines occuring on musical instruments. I will also explain a recent related result concerning the distribution of the zeros of random trigonometric polynomials due to A. Granville-IW.
Workshop at CRM
Hilbert Spaces of Analytic Functions
December 8-12, 2008
Organizers: J. Mashreghi, T. Ransford (Laval), K. Seip (NTNU)

2005/2006 Analysis Seminar