In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. Professor G. Zaslovski always expressed a special interest in the models of chaos containing strong fluctuations, e.g. L ́evy flights? We’ll consider several models of potentials constructed by the use of iid random variables which belong to the domain of attraction of the stable distribution with parameter α < 1. This is a report on joint work with S. Molchanov.

Support for travel expenses is available especially for students and postdocs.  Please register online and request support by January 15, 2015.

The mathematical problem of gradient estimates for solutions of divergence form elliptic systems with piece-wise smooth coefficients arises in studying composite materials in applied science.

We will start with ideas in joint works with Vogelius (2000) and Nirenberg (2003) about a decade ago, in particular an open problem in the paper with Nirenberg, then discuss recent progress in closely related topics, such as gradient estimates for solutions of the Lame system with partially infinite coefficients (Arch. Rational Mech. Anal. (2015), joint with JiGuang Bao and HaiGang Li).

This is an expository lecture accessible to first year graduate students.

Electrolyte and cell volume regulation is essential in physiological systems. After a brief introduction to cell volume control and electrophysiology, I will discuss the classical pump-leak model of electrolyte and cell volume control. I will then generalize this to a PDE model that allows for the modeling of tissue-level electrodiffusive, convective and osmotic phenomena. This model will then be applied to the study of cortical spreading depression, a wave of ionic homeostasis breakdown, that is the basis for migraine aura and other brain pathologies.

We describe a short, direct, alternative to the DeTurck trick to prove the
uniqueness of solutions to a large class of curvature flows of all orders,
including the Ricci flow, the L^2 curvature flow, and other flows related
to the ambient obstruction tensor. Our approach is based on the analysis
of simple energy quantities defined in terms of the actual solutions to the
equations, and allows one to avoid the step -- itself potentially
nontrivial in the noncompact setting -- of solving an auxiliary parabolic
equation (e.g., a k-harmonic-map heat-type flow) in order to overcome the
gauge-invariance-based degeneracy of the original flow. We also
demonstrate that, by the consideration of a certain energy
quotient/frequency-type quantity, one can give a short and quantitative
proof (avoiding Carleman inequalities) of the global backward uniqueness of
solutions to a large class of these equations.