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)

UdeM, Pav. Andre-Aisenstadt, 2920 Chemin de la Tour, Room 6214

Attractors and Invariant measures in low-dimensional dynamical systems

Burnside 920

On a Birkhoff mormal form theorem for 1-d Gross-Pitaevskii equations

Burnside 920

$L^\infty$ and gradient estimates of spectral clusters on compact manifolds with boundary

Burnside 920

Extremal problems for low eigenvalues on planar domains

Burnside 920

Two iterative methods for equilibrium problem

Burnside 934

Upper bounds on the spectrum of the Laplacian

Burnside 1205

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.

Burnside 920

Solutions with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation

-\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.

Burnside 920

A non-commutative Levy-Cramer theorem

Burnside 920

Span of translates on the real line, and zeros of Fourier transform.

Burnside 934

Multipliers in spaces of differentiable functions with applications to PDEs

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

Unsolved mysteries of solutions to PDEs near the boundary

CRM

Website

CRM, Pav. Andre-Aisenstadt Room 5340

Removability problems for bounded, BMO and Hoelder continuous quasiregular mappings in the plane.

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.

McGill, Room 920

Semi-classical analysis for Schroedinger operators with magnetic fields

Slides

CRM

Website

Burnside 920

Concentration phenomena for singularly perturbed elliptic PDEs

Burnside 920

On nodal sets of eigenfunctions

Burnside 920

On the excitation spectrum of homogeneous Bose gas

Burnside 920

Asymptotic Geometric Analysis; Geometrization of Probability

Burnside 920

Monotonicity formulas under Ricci flow and Normalized Ricci flow

Burnside 1205

Boundary Element Methods for the Last 30 Years

Burnside 920

Recovering the mass and the charge of a Reissner-Nordstrom black hole by an inverse scattering experiment

Burnside 920

Weighted inequalities and Bellman functions

Burnside 920

Uniform estimates for cubic oscillatory integrals

LB 655 (at Concordia)

The John-Nirenberg Inequality for $Q_\alpha(R^n)$ and the Related Fractal Function

Burnside 920

The Unruh effect revisited

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

Pseudo-Riemannian geodesics and billiards

2005/2006 Analysis Seminar

2004/2005 Seminar in Nonlinear Analysis and Dynamical Systems

2003/2004 Working Seminar in Mathematical Physics