We prove that the Dry Ten Martini Problem, i.e., all possible spectral gaps are open, holds for almost Mathieu operator with noncritical coupling and any irrational frequency. It is joint work with Artur Avila and Jiangong You. 

In this talk, we will talk about two phase transiton results for quasi-Periodic Schr\"odinger Operators. 

For continuous Sch\"odinger operators with large analytic quasi-periodic 
potentials of two frequencies, we obtain the exact power-law for phase transition in energy.

For the almost Mathieu operator with any fixed frequency, we locate 

the point where phase transition from  singular continuous spectrum to pure point spectrum takes place, 

which solves Jitomirskaya's conjecture

The key to set up Anderson Localization is to estimate the Green function. In this talk, I will introduce two ways to estimate the Green function.

One is the way of Harmonic analysis(based on Bourgain's book," Green's function  estimates for lattice Schrodinger operators and application"). The other way is by Jitomirskaya (based on two papers in the Annals of Math( 1999 and 2009). In the end, I will give an

extended result on Anderson Localization (this is a joint work with Xiaoping Yuan).

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.