I will give an overview of my work in mathematical cell biology.  First I will discuss topics related to polarity, specifically in the context of cell movement.  This and numerous other cell functions require identification of a “front” and “back” (e.g. polarity).   In some cases this can form spontaneously and in others sufficiently large stimuli are required.  I will discuss a mechanistic theory for how cells might transition between these behaviors by modulating their sensitivity to external stimuli.  In order to address this and analyze the systems being presented, I will describe a new non-linear bifurcation technique, the Local Perturbation Analysis, for analyzing complex, spatial biochemical networks.  This methodology fills a void between simple (but limited) stability techniques and more thorough (but in many cases impractical) non-linear PDE analysis techniques.  Additionally, I will discuss work related to early development of the mammalian embryo.  A vital first step in this process is the formation of an early placenta prior to implantation.  I will discuss a multi-scale stochastic model of this spatial patterning event and show that genetic expression noise is both necessary and sufficient for this event to occur robustly. 

We prove a smooth version of the classical Schwarz reflection principle for CR  mappings between an abstract CR manifold and a generic CR manifold embedded in euclidean complex space. Our results settle a conjecture of  X. Huang. The talk is based on a recent joint work with Ming Xiao.

We continue with the presentation of two different ways to generically add a club subset of a successor cardinal, under some GCH.  The first one is designed to destroy a given stationary set, and we show that it also forces diamond.  The second adds a club with "small" conditions and destroys saturated ideals.  We will discuss the open problem of whether this can be done without any cardinal arithmetic assumptions.

We will present two different ways to generically add a club subset of a successor cardinal, under some GCH.  The first one is designed to destroy a given stationary set, and we show that it also forces diamond.  The second adds a club with "small" conditions and destroys saturated ideals.  We will discuss the open problem of whether this can be done without any cardinal arithmetic assumptions.