We complete the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we explain the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.

Cancer often arises through a sequence of genetic alterations.  Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion.  To model this process we consider a generalization of the biased voter process on a lattice which incorporates successive mutations modulating individual fitness.  We will study the rate of mutant spread and accumulation of oncogenic mutations in this process.  We then investigate the geometry and extent of premalignant fields surrounding primary tumors, and evaluate how the risk of secondary tumors arising from these fields may depend on the cancer progression pathway and tissue type.  (joint work w/K. Leder, R. Durrett, and M. Ryser).

Advisor: Prof. Qing Nie
Committee Members: Prof. Frederic Wan, Prof. Long Chen

We discuss the existence of foliations that are invariant under the dynamics for systems that are isotopic to Anosov diffeomorphisms. Specifically, we examine partially hyperbolic diffeomorphisms with one dimensional center that are isotopic to a hyperbolic toral automorphism and contained in a connected component. We show in this case there is a center foliation. We will also discuss more general cases where there is a weak form of hyperbolicity called a dominated splitting. This is joint work with Jerome Buzzi, Rafael Potrie, and Martin Sambarino.