In this talk we will consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy roughly N^alpha vertices, for some 0 < alpha < 1. The value of alpha is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. These results are in sharp contrast with the picture in the lattice case.
This is a joint work with Yael Dekel, Elchanan Mossel and Yuval Peres.

In Taubes' proof of the Weinstein conjecture, a main ingredient is the estimate on the spectral flow of a family of Dirac operators, which he used to obtain the energy bound. When the perturbation is a contact form, much evidence suggests that the asymptotic behavior of the spectral flow function is nicer. In this talk, we will explain how to improve the spectral flow estimate for some classes of contact forms.

A key phenomenon in the study of cell-to-cell communication and
protein regulation is the all-or-none, ultrasensitive dose response, which
transforms a continuous input into a digital output. Multisite systems are
often used in conjunction with allosteric effects to create such a behavior.
In this talk I describe a non-allosteric mechanism for a multisite system to
present strongly ultrasensitive behavior. Applications are given to protein
activation through multisite phosphorylation, clusters of receptors and DNA
regulation through histone modifications.

In this lecture, we will talk about a recent joint
work of Gordon Heier and myself about curvature characterizations
of uniruledness and rational connectivity of projective manifolds. A
result on projective manifolds with zero total scalar curvature will
also be discussed.

Our research focus on the mathematical modeling and numerical simulations of vascular tumor growth and chemotherapy. We have developed a model by coupling a discrete angiogenesis model and a continuous tumor growth model accounting for vascular and interstitial fluid dynamics (vascular flow, vascular fluid extravasation, interstitial fluid flow and lymphatic drainage), which affects the delivery of nutrients and therapeutical agents during tumor growth/treatment. The talk will discuss over (1) overview of vascular tumor growth, (2) the tumor vascular/interstitial pressure/flow and the related physiological factors, (3) traditional drug transportation, chemotherapy and the physical barrier therein (4) the novel treatment strategy.