Mathematical Physics Seminar

Winter Term 2017

Abstracts

Andrew Comech (Texas A&M, IITP Moscow): Stability of solitary waves in the nonlinear Dirac equation

We consider the point spectrum of non-selfadjoint Dirac operators which arise as linearizations at solitary wave solutions to the nonlinear Dirac equation. We prove that in the nonrelativistic limit (\omega\lesssim m) the solitary waves in the Dirac equation with scalar-type self-interaction ("Soler model") with "NLS-subcritical" nonlinearity are spectrally stable. The results are obtained together with Nabile Boussaid, University of Bourgogne -- Franche-Comte and are partially based on articles "On spectral stability of the nonlinear Dirac equation" (with Nabile Boussaid), JFA-2016, http://arxiv.org/abs/1211.3336 "Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity" (with Nabile Boussaid), http://arxiv.org/abs/1606.07308




Nicolas Delfosse (Caltech): A homological upper bound on critical probabilities for hyperbolic percolation

Percolation is one of most simple models which exhibits a phase transition. The determination of the threshold for this phase transition is usually quite difficult. It is the central problem of percolation theory. The relation between percolation and quantum error correction has been exploited in quantum information theory. In the present work, we propose to go in the other direction. We derive new bounds on the percolation threshold of hyperbolic graphs based on results of quantum information. This leads to the most precise upper bound on the percolation threshold of these graphs, improving bounds of Benjamini and Schramm.




Messoud Efendiyev (Helmholtz Zentrum Munich): Symmetry and Attractor

We study the asymptotic behaviour of solutions of nonlinear elliptic equations in asymptotically symmetric unbounded domains. We are interested in how the asymptotic symmetry of domain inherited to the symmetry of appropriate global attractor. To this end we use dynamical systems approach and new Liouville type results which are of independent interest as well.




Yoram Last (Hebrew University Jerusalem): Absolutely continuous spectrum and the spectra of periodic approximants

We discuss relations between absolutely continuous spectrum of discrete one-dimensional Schroedinger operators and the spectra of periodic approximants made by cutting and repeating finite pieces of the potential.




Jonathan Breuer (Hebrew University Jerusalem): A Transfer Matrix Approach to Scaled Limits of Christoffel-Darboux Kernels

We present an approach to computing the scaling limits of Christoff el-Darboux kernels using transfer matrix evolution.




Julien Sabin (U. Paris Orsay): Spectral cluster bounds for orthonormal functions

Sogge's L^p bounds are a way to measure the concentration of eigenfunctions of the Laplace-Beltrami operator on compact Riemannian manifolds associated to large eigenvalues. We generalize these bounds to systems of orthonormal functions, building a bridge between Sogge's result about concentration and the Weyl law, which in some sense is a manifestation of non-concentration. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators. Joint work with Rupert Frank.




Israel Michael Sigal (U. Toronto): The mean curvature flow

The mean curvature flow appears naturally in the motion of interfaces in material science, physics and biology. It also arises in geometry and has found its applications in topological classification of surfaces. In this talk I will discuss recent results on formation of singularities under this flow. In particular, I will describe the 'spectral' picture of singularity formation and sketch the proof the neck pinching results obtained jointly with Zhou Gang and Dan Knopf.




Rui Han (UC Irvine): Full Measure Reducibility and Localization for Quasi- Periodic Jacobi Operators: a Topological Criterion

We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasi-periodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region.