For the last several years, we had been working on projects related to 
charge transport in solutions and proteins (ion channels). One of the key ingredients 
in these studies is the understanding of diffusion and its relations to other effects, 
such as hydrodynamics, electrostatics and other particle-particle interactions. 
Due to the non-ideal situations in almost all biological environments, such as 
the high concentration of charge densities, those conventional theories have to be modified or re-derived. 

In the talk, I will employ the general framework of energetic variational approaches, 
especially Onsager's Maximum Dissipation Principles to the problems of generalized diffusion. We will discuss the roles of different stochastic integrations, 
and the procedures of optimal transport in the context of general 
linear response theory in statistical physics.

A subset $X$ in $\{0,1\}^n$ is a called an $\epsilon$-biased set if for any nonempty subset $T\subseteq [n]$ the following condition holds. Randomly choosing an element $x\in X$, the parity of its T-coordinates sum has bias at most $\epsilon$. This concept is essentially equivalent or close to expanders, pseudorandom generators and linear codes of certain parameters. For instance, viewing $X$ as a generator matrix, an $\epsilon$-biased set is equivalent to an $[|X|, n]_2$-linear code of relative distance at least $1/2-\epsilon$. For fixed $n$ and $\epsilon$, it's a challenging problem to construct smallest $\epsilon$-biased sets. In this talk we will first introduce several known constructions of $\epsilon$-bias sets with the methods from number theory and geometrical coding theory. Then we will present a construction with a conjecture, which is closely related to the subset sum problem over prime fields.

What is the probability that a random abelian variety over F_q is ordinary? Using (semi)linear algebra, we will answer an analogue of this question, and explain how our method can be used to answer similar statistical questions about p-rank and a-number. The answers are perhaps surprising, and deviate from what one might expect via naive reasoning. Using these computations and numerical evidence, we formulate several ``Cohen-Lenstra" heuristics for the structure of the p-torsion on the Jacobian of a random hyperelliptic curve over F_q. These heuristics are the "l=p " analogue of Cohen-Lenstra in the function field setting. This is joint work with Jordan Ellenberg and David Zureick-Brown.

I will discuss a general approach for constructing SRB measures for diffeomorphisms possessing chaotic attractors (i.e., attractors with nonzero Lyapunov exponents). I introduce a certain recurrence condition on the iterates of Lebesgue measure called “effective hyperbolicity” and I will show that if the asymptotic rate of effective hyperbolicity is exponential on a set of positive Lebesgue measure, then the system has an SRB measure. Along the way a new notion of hyperbolicity -- "effective hyperbolicity'' will be introduced and a new example of a chaotic attractor will be presented. This is a joint work with V. Climenhaga and D. Dolgopyat.