In this talk I will introduce some basic ideas from probability theory
such as random walk, Markov chain and Brownian motion. Then I will
discuss how they play a role in analyzing some "real world" models of
physical phenomena such as polymer behavior, spread of pollutants and
solar magnetic fields.

Stochastic effects can be important for the behavior of processes involving small population numbers, so the study of stochastic models has become an important topic in the burgeoning field of computational systems biology. However analysis techniques for stochastic models have tended to lag behind their deterministic cousins due to the heavier computational demands of the statistical approaches for fitting the models to experimental data. There is a continuing need for more effective and efficient algorithms. In this talk I will focus on the parameter inference problem for stochastic kinetic models of biochemical reactions given discrete time-course observations of either some or all of the molecular species.

I will describe an algorithm for inferring kinetic rate parameters based upon maximum likelihood using stochastic gradient descent (SGD). A general formula will be derived for calculating the gradient of the likelihood function given discrete time-course observations. The formula applies to any explicit functional form of the kinetic rate laws such as mass-action, Michaelis-Menten, etc. Our algorithm estimates the gradient of the likelihood function by reversible jump Markov chain Monte Carlo sampling (RJMCMC), and then gradient descent method is employed to obtain the maximum likelihood estimation of parameter values. Furthermore, we utilize flux balance analysis and show how to automatically construct reversible jump samplers for arbitrary biochemical reaction models. We provide RJMCMC sampling algorithms for both fully observed and partially observed time-course observation data. I will illustrate the utility of the method with two examples: a birth-death model and an auto-regulatory gene network.

We extend the theory of Patterson-Sullivan measure to any regular
covering of a compact manifold using the Busemann compactification
and derive an integral formula for the volume entropy. As applications
we prove some rigidity theorems for the volume entropy.
This is a joint work with Francois Ledrappier.

Since 2005, UCLA has run an internal NSF-funded summer REU program in applied mathematics for talented UCLA students and, more recently, students from other local colleges. The REU program has been very successful and is continuing to evolve into a better program. The unique feature of this program is that the undergraduate research projects are intrinscially tied into ongoing research carried out by the faculty and graduate students. I will discuss my involvement with the REU program for the last 3 years and present some of the projects I have mentored.The goal is to suggest a possible template for other schools to develop their own REU program in mathematics.