Let $p>2$ be a prime number. The classical Hasse invariant is a modular form modulo p that vanishes on the supersingular points of a modular curve. Its non-zero locus is called the ordinary locus. While the Hasse invariant generalizes easily to moduli spaces of abelian varieties with additional structures, it happens often that the generalized ordinary locus is empty, and therefore the Hasse invariant is then tautologically trivial. We present an elementary and natural generalization of the Hasse invariant, which is always non-trivial, and which enjoys essentially all the same properties as the classical Hasse invariant. In particular, the usual applications generalize nicely, and we shall highlight the state-of-the-art in our talk.

Epidemiological models and immunological models have been studied largely independently. However, the two processes (within- and between-host interactions) occur jointly and models that couple the two processes may generate new biological insights. Particularly, the threshold conditions for disease control may be dramatically different when compared with those generated from the epidemiological or immunological models separately. We developed and analyzed an ODE model, which links an SI epidemiological model and an immunological model for pathogen-cell dynamics. When the two sub-systems are considered in isolation, the dynamics are standard and simple. That is, either the infection-free equilibrium is stable or a unique positive equilibrium is stable depending on the relevant reproduction number being less or greater than 1. However, when the two sub-systems are explicitly coupled, the full system exhibits more complex dynamics including backward bifurcations; that is, multiple positive equilibria exist with one of which being stable even if the reproduction number is less than 1. The biological implications of such bifurcations are illustrated using an example concerning the spread and control of toxoplasmosis. 

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.