We will discuss the common submanifolds of two Hermitian symmetric spaces. In particular, we proved that the Euclidean space and a bounded symmetric domain cannot share a common submanifold. This is based on the joint work with Professor X. Huang.

The problem of understanding the local geometry of CR embedded submanifolds in CR manifolds arises naturally in several complex variables analysis, e.g., in the study of isolated singularities of analytic varieties (via their links). Significant work has been done on this in connection with rigidity phenomena and the classification of proper holomorphic mappings between balls. We develop from scratch a CR invariant local theory based on the CR tractor calculus associated to the Chern-Moser-Cartan connection. This produces the tools for constructing local invariants and invariant operators in a way parallel to the classical Gauss-Codazzi-Ricci calculus for Riemannian submanifolds.

Turan-type problems in graph theory ask how many edges a graph may have if a certain subgraph is forbidden. One can think of this as an optimization problem, as one is maximizing the global condition of number of edges subject to the local constraint that there is no forbidden subgraph. Problems in combinatorial number theory ask one to deduce properties of a set of (for example) integers while knowing only how large the set is. We study the connection between these two seemingly disjoint areas. Graphs coming from finite projective planes are intimately related to both areas.

I will survey some aspects of the theory of noncommutative completely positive kernels, which are the generalization of usual positive kernels to the setting of free noncommutative function theory. I will then use the language of noncommutative completely positive kernels to discuss the noncommutative generalizations of the interpolation and realization theorems for the Schur-Agler class. This is a joint work with J. Ball and G. Marx.