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.

When a holomorphic modular form is a newform, its L-function has nice analytic properties and associates a cuspidal automorphic representation, which is a restricted product of local representations. To recover the newform from the representation, Casselman considered the fixed line of the congruence subgroups of GL(2) at the conductor level on the local representations. A vector on this line shall encode the conductor, the L-function and the \epsilon-factor of the representation. This is called the theory of newforms for GL(2). Similar theory has been established for some groups of small ranks as well as GL(n). In this talk I will introduce one for SO(2n+1).

We continue 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.

We study the effect of defects in the periodic homogenization of
Hamilton-Jacobi equations with non convex Hamiltonians. More precisely, we
handle the question about existence of sublinear solutions of the cell
problems.