## 2007/08 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 2008

CRM-ISM Colloquium, January 4, 4:00-5:00pm
UdeM, Pav. Andre-Aisenstadt, 2920 Chemin de la Tour, Room 6214
M. Jakobson (Univ. of Maryland)
Attractors and Invariant measures in low-dimensional dynamical systems
Abstract. We consider typical behavior of trajectories in dynamical systems from topological and measure theoretical prospectives. They coincide for uniformly hyperbolic systems, but can be different for non-uniformly hyperbolic ones. Sinai-Ruelle-Bowen (SRB) measures are crucial for understanding ergodic properties of dynamical systems. In the case when a system has an absolutely continuous SRB measure its typical trajectories exhibit random behavior and at the same time their statistical properties can be described quantitatively. We discuss several types of dynamics which arise in one-parameter families of low-dimensional dynamical systems.
Monday, January 14, 2:30pm (DATE AND TIME CHANGED!)
Burnside 920
Z. Yan (McGill)
On a Birkhoff mormal form theorem for 1-d Gross-Pitaevskii equations
Abstract. We consider a 1-dimensional Gross-Pitaevskii (GP) equation and want to determine if there exits any Birkhoff normal form theorem for it. In doing so, the following two factors plays important roles: one is to introduce suitable function spaces; and the other one is to obtain estimates on integrals of products of Hermite functions, which represent coefficients of mode coupling. The resulting system is a perturbed system of a completely resonant system. Then we made an analysis of the impact of the perturbation on the principal part of the GP system.
Friday, February 8, 2:30pm
Burnside 920
X. Xu (McGill)
$L^\infty$ and gradient estimates of spectral clusters on compact manifolds with boundary
Abstract. In this talk, I will discuss $L^\infty$ and gradient estimates of spectral clusters of both Dirichlet Laplacian and Neumann Laplacian on compact manifolds with boundary. For interior $L^\infty$ estimates, two approaches are discussed: one follows from the estimates of the remainder of Wey's Law (for smooth manifolds), and the other follows Smith's recent work (for $C^{1,\alpha}$ manifolds). The boundary $L^\infty$ estimates and gradient estimates are proved via the maximum principle arguments for Poisson equations.
Friday, February 15, 2:30pm
Burnside 920
A. Girouard (Univ. de Montreal)
Extremal problems for low eigenvalues on planar domains
Abstract. A classical result of G. Szego states that among all simply-connected planar domains of fixed area, the first nonzero Neumann eigenvalue is maximized by a disk. In this talk, I will discuss the maximization problem for the next eigenvalue. This is joint work with I. Polterovich and N. Nadirashvili. Along the way, I will survey some classical and recent results on extremization of small eigenvalues.
Thursday, February 28, 12noon
Burnside 920
M. Nasri (IMPA)
Two iterative methods for equilibrium problem
Abstract. The main purpose of this talk is the study of finite dimensional equilibrium problems. First we guarantee existence of solutions for equilibrium problems under reasonable assumptions. We then introduce a proximal point method for finding solutions of equilibrium problems, using a regularization technique. We also discuss an augmented Lagrangian method for solving this kind of problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization.
March 7, 2:00pm
Burnside 934
B. Colbois (Univ. Neuchatel)
Upper bounds on the spectrum of the Laplacian
Abstract. In this talk, I will present a geometric approach of getting upper bounds on the spectrum of the Laplacian (with Neumann boundary conditions) on domains of a Riemannian manifold with Ricci curvature bounded below (a joint work with D. Maerten) and for submanifolds in R^n (a work in progress with E. Dryden and A. El Soufi).
Applied Mathematics/Analysis Seminar, March 11, 3:30pm
Burnside 1205
Alex Barnett (Dartmouth)
Eigenmodes and quantum chaos: Lost on the frequency axis? Check your Dirichlet-to-Neumann map!
Abstract: The Dirichlet eigenmode (or drum') problem describes vibrations of an elastic membrane, acoustic cavity, or quantum particle, and is a paradigm for more complicated applications to electromagnetic and optical resonators. When the wavelength is much shorter than the cavity size this becomes a challenging multiscale problem, and boundary methods are essential. I will explain an accelerated cousin of the method of particular solutions (MPS, a global basis approximation method) which allows O(k) modes to be calculated in the effort usually required for a single mode, k being the wavenumber. It removes the need for expensive root-searches along the wavenumber (ie frequency) axis. At very high frequencies and many cavity shapes, it is the fastest method known, 10^3 times faster than either MPS or boundary integral methods. This has enabled large-scale numerical study of the asymptotic properties of planar eigenmodes. I will present recent data on dynamical tunneling' in Bunimovich's mushroom cavity, which has both chaotic and integrable motion. If time I will mention recent work with T. Betcke on the Helmholtz BVP using fundamental solutions bases, which in analytic shapes give spectral accuracy approaching only 2 degrees of freedom per wavelength on the boundary.
March 14, 14:30
Burnside 920
L. Berlyand (Penn. State)
Solutions with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation
Abstract. We study solutions of the Ginzburg-Landau (GL) equation for a complex valued order parameter $u$
-\Delta u+(1/\varepsilon^2) u(|u|^2-1)=0, \; x \in A \subset R^2.
This equation is of principal importance in the Ginzburg-Landau theory of superconductivity and superfluidity. For a 2D domain $A$ with holes we consider the so-called "semi-stiff" boundary conditions: the the Dirichlet condition for the modulus $|u|=1$, and the homogeneous Neumann condition for the phase $\arg(u)$. The principal result of this work is that there are stable solutions with vortices of this boundary value problem. The vortices are of a novel type: they approach the boundary and have bounded energy in the limit of small $\varepsilon$. By contrast, in the well-studied Dirichlet problem for the GL PDE, the vortices are distant from boundary and their energy blows up as $\varepsilon \to 0$. On the other hand there is no stable solutions with vortices to the homogeneous Neumann ("soft") problem. In this work we develop a variational method that allows us to construct local minimizers of the GL energy functional which corresponds to the GL PDE. We introduce the notion of the approximate bulk degree which is the key ingredient of our method. We show that, unlike the standard degree over a curve, the approximate bulk degree is preserved in the weak $H^1$-limit. This is a joint work with V. Rybalko.
Monday, March 17, 14:30 (date changed!)
Burnside 920
Y. Pautrat (Paris-Sud, visiting McGill)
A non-commutative Levy-Cramer theorem
Abstract. In classical probability theory, the Levy-Cramer theorem is the basic tool for proving the convergence in distribution of sequences of random variables. In particular, it shows that the convergence of (joint) characteristic functions determines the (joint) law of the limiting random variables. In non-commutative probability theory, the joint law of random variables does not exist in general, so joint characteristic functions may seem useless. In a joint work with Jaksic and Pillet, we proved, however, that convergence of "quasi characteristic functions" formally identical to the standard characteristic functions has interesting consequences for many functionals of the limiting random variables. We will give a short review of the principles of non-commutative probability theory, and discuss our result, which is stated in surprisingly simple and general terms.
March 28, 14:30
Burnside 920
Nir Lev (Tel Aviv)
Span of translates on the real line, and zeros of Fourier transform.
Abstract. A function $f \in L^p(R)$ is called a "generator" if the set of all translates of $f$ spans the whole space $L^p(R)$. How to decide whether a given function is a generator or not? We shall discuss this problem including recent joint work with A. Olevskii.
April 4, 14:00
Burnside 934
T. Shaposhnikova (Ohio State University and Linkoeping University)
Multipliers in spaces of differentiable functions with applications to PDEs
Abstract. New results on pointwise multipliers between Besov spaces will be presented. Various applications to differential and integral operators will be discussed. This is a joint work with V. Maz'ya.
CRM-ISM colloquium, April 4, 16:00
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
V. Mazya (Ohio State, University of Liverpool and Linkoeping University)
Unsolved mysteries of solutions to PDEs near the boundary
Abstract. Throughout its long history, specialists in the theory of partial differential equations gained a deep insight into the boundary behavior of solutions.Yet despite the apparent progress in this area achieved during the last century, there are fundamental unsolved problems and surprising paradoxes related to solvability, spectral, and asymptotic properties of boundary value problems in domains with irregular boundaries. I shall formulate some challenging questions arising naturally when one deals with unrestricted, polyhedral, Lipschitz graph, fractal and convex domains.
April 7-11
CRM
Workshop on Spectrum and Dynamics
Website
May 2, 10:30-11:30
Ignacio Uriarte-Tuero (U. Missouri and Fields Institute)
Removability problems for bounded, BMO and Hoelder continuous quasiregular mappings in the plane.
Abstract. A classical problem in complex analysis is to characterize the removable sets for various classes of analytic functions: H\"{o}lder, Lipschitz, BMO, bounded (this last case gives rise to the analytic capacity and the Painlev\'{e} problem which has been recently solved by Tolsa.) One can ask the same questions in the setting of K-quasiregular maps (since they are a K-quasiconformal map followed by an analytic map.) Most of the bounded case was dealt with in a joint paper with K. Astala, A. Clop, J.Mateu and J.Orobitg, [ACMOUT]. The BMO case was dealt with in [ACMOUT] except for a gap at the critical dimension (Question 4.2 in [ACMOUT].) I answered the question filling the gap in [UT]. The Lipschitz case was dealt with by A. Clop, as well as most of the H\"{o}lder case, where again a gap at the critical dimension was left. In a joint paper with A. Clop [CUT] we closed the gap. I will summarize the results and give some ideas of the proofs in the above papers. The talk will be self-contained.
References:
[ACMOUT] Kari Astala, Albert Clop, Joan Mateu, Joan Orobitg and Ignacio Uriarte-Tuero. Distortion of Hausdorff measures and improved Painlev\'{e} removability for bounded quasiregular mappings. Duke Math J., to appear.
[CUT] Albert Clop and Ignacio Uriarte-Tuero. Sharp Nonremovability Examples for H\"{o}lder continuous quasiregular mappings in the plane, submitted.
[UT] Ignacio Uriarte-Tuero. Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability. IMRN, to appear.

Thursday, May 29, 14:00-15:00
McGill, Room 920
Bernard Helffer (Paris-Sud)
Semi-classical analysis for Schroedinger operators with magnetic fields
Slides
June 2-7
CRM
Workshop "Mathematical Aspects of Quantum Chaos"
Website

## FALL 2007

Friday, September 7, 14:00-15:00
Burnside 920
Andrea Malchiodi (ISAS, Trieste)
Concentration phenomena for singularly perturbed elliptic PDEs
Abstract: We consider a class of nonlinear elliptic equations with a singular perturbation parameter $\epsilon$, which arise from physical or biologial models, like the nonlinear Schroedinger equation or the Gierer-Meinhardt system. In the last twenty years, a lot of attention has been devoted to study the asymptotics of solutions, and more recently some results about concentration at sets of positive dimension have been derived. We discuss the recent progress in this direction and some open problems.

Friday, September 14, 14:00-15:00
Burnside 920
Dmitry Jakobson (McGill)
On nodal sets of eigenfunctions
Abstract. We describe results about the topology of nodal sets on S^2 and R^2 (joint work with A. Eremenko and N. Nadirashvili). We next discuss recent results about approximation by nodal sets on real-analytic Riemannian manifolds (joint work with D. Mangoubi). A movie will be shown!
Friday, September 21, 14:00-15:00
Burnside 920
Jan Derezinski (Warsaw)
On the excitation spectrum of homogeneous Bose gas
Abstract. I will speak about the joint energy-momentum spectrum of the Bose gas in thermodynamic limit. Using rigorous mathematical language I will formulate some conjectures about its infimum and describe their relevance for physics of cold gases. I will describe a number of beautiful, although mostly heuristic, arguments going back to Landau and Bogoliubov, supporting a very interesting physical and mathematical picture of such systems.
Friday, September 28, 14:00-15:00
Burnside 920
Vitali Milman (Tel Aviv)
Asymptotic Geometric Analysis; Geometrization of Probability
Abstract. We study the asymptotic behavior of finite- (but very high-)dimensional normed spaces and convex bodies when dimension tends to infinity. Contrary to common intuition, which anticipates enormous diversity and chaotic behavior, we observe a uniform behavior for the whole family of finite- (but high-)dimensional spaces. In the first part of our talk we will discuss different and unexpected phenomena accompanying high dimension. In the second, the main part of the talk we will explain how the geometric theory of convexity is extended to a larger category of log-concave measures which bring inside this class of (probability) measures geometric vision and approach. This brings inside the theory functional versions for many geometric inequalities, and also leads to solutions of some central problems of the theory. The talk will be understandable to any graduate student in Mathematics.
Friday, October 19, 14:00-15:00
Burnside 920
Jun-Fang Li (CRM and McGill)
Monotonicity formulas under Ricci flow and Normalized Ricci flow
Abstract. We study the steady state of Ricci flow (RF) and Normalized Ricci flow(NRF). We are interested in finding integral quantities which have monotonicity properties under RF or NRF, and the first variation of these functionals vanishes if and only if at a steady state of RF or NRF. The ideas stem from G. Perelman's groundbreaking work in the study of Ricci flow. As applications we classified the steady states of RF and NRF(compact expanding and steady cases). A byproduct is our results improve some previous work on eigenvalues monotonicity properties under RF.
Joint Applied Mathematics/Analysis seminar
Monday, October 22, 14:30-15:30
Burnside 1205
George C. Hsiao (Univ. of Delaware)
Boundary Element Methods for the Last 30 Years
Abstract. Variational methods for boundary integral equations deal with weak formulations of the equations. Boundary element methods are numerical schemes for seeking approximate weak solutions of the corresponding boundary variational equations in finite-dimensional subspaces of the Sobolev spaces with special basis functions, the so-called boundary elements. This lecture gives an overview of the method from both theoretical and numerical point of view. It summarizes the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.

Friday, October 26, 14:00-15:00
Burnside 920
Thierry Daude (CRM and McGill)
Recovering the mass and the charge of a Reissner-Nordstrom black hole by an inverse scattering experiment
Abstract. In this talk, we shall study inverse scattering of massless Dirac fields that propagate in the exterior region of a Reissner-Nordstrom black hole. Using a stationary approach we shall determine precisely the leading terms of the high energy asymptotic expansion of the associated scattering matrix which, in turn, will permit us to recover uniquely the mass and the charge of the black hole.
Friday, November 2, 14:00-15:00
Burnside 920
Cristina Pereyra (Univ. of New Mexico)
Weighted inequalities and Bellman functions
Abstract. We present a modern perspective to the classical problem of weighted inequalities for singular integrals and dyadic model operators. The new ingredient is the use of Bellman functions techniques to track optimal dependence of the operator bounds on weighted Lebesgue spaces in terms of the $A_p$ or $RH_p$ characteristic of the weights. This is not just a mathematical exercise, some people do use these optimal estimates in areas such as quasiconformal mmapping theory and elliptic differential equations.
Friday, November 9, 14:00-15:00
Burnside 920
Philip Gressman (Yale)
Uniform estimates for cubic oscillatory integrals
Abstract. I will discuss the problem of proving uniform, optimal asymptotic estimates for scalar oscillatory integrals with a phase function which satisfies an appropriate third-order nondegeneracy condition. The proof relies on the construction of a nontrivial symmetric space structure adapted to the geometry of the phase.
Friday, November 16, 14:00-15:00
LB 655 (at Concordia)
Hong Yue (Oulu Univ, Finland)
The John-Nirenberg Inequality for $Q_\alpha(R^n)$ and the Related Fractal Function
Abstract. The John-Nirenberg inequality characterizes functions in the space BMO in terms of the decay of the distribution function of their oscillations over a cube. We prove separate necessary and sufficient John-Nirenberg type inequalities for functions in the space $Q_\alpha(R^n)$, introduced by Essen, Janson, Peng and Xiao, who conjectured a version of this inequality. Our results are a modified version of their conjecture, and we give two counterexamplesto show the necessity for this modification. The counterexamples provide us with a borderline case function for $Q_\alpha(R^n)$ which can be expressed in terms of a Haar wavelet decomposition. We vary the parameter in this decomposition and discuss the relation between the function and the spaces. In addition, we study the fractal properties of the function and determine the fractal dimensions of its graph. The properties and dimensions illustrate some form of regularity for functions in $Q_\alpha(R^n)$.
Friday, December 7, 14:00-15:00
Burnside 920
Stephan De Bievre (Univ. Lille)
The Unruh effect revisited
Abstract. This is joint work with M. Merkli. According to Unruh (1976), a detector accelerated through a relativistic quantum field in its vacuum state will respond as if bathed in a black body radiation at a temperature proportional to its acceleration. I will give a precise mathematical meaning to this statement and show how it can be understood as a problem of return to equilibrium'' in quantum statistical mechanics.

## ANALYSIS TALKS ELSEWHERE

CRM-ISM colloquium
Friday, October 26, 4:00-5:00pm
UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420
B. Khesin (Toronto)
Pseudo-Riemannian geodesics and billiards
Abstract. In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. In the talk I will describe the geometry of these structures, define pseudo-Euclidean billiards and discuss their properties. In particular, I will outline complete integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in pseudo-Euclidean space; these results are pseudo-Euclidean counterparts to the classical theorems of Euclidean geometry that go back to Jacobi and Chasles.

2005/2006 Analysis Seminar