In this talk, we introduce the Laplacian eigenvalue problem and briefly go over its history.  Then we will present a recent result which gives a sharp lower bound of the fundamental gap for convex domain of spheres motivated by the modulus of continuity approach introduced by Andrews-Clutterbuck.  This is joint work with Lili Wang and Guofang Wei.

Let f: C' -> C be a cyclic cover of smooth projective curves. Its Prym variety is by definition the complement of the pullback of the Jacobian of C in the Jacobian of C'. It is an abelian variety with a polarization depending on the genus of C, the degree of f and the ramification type of the covering f. This gives a map from the moduli space of coverings of this type into the moduli space of abelian varieties of the corresponding type with endomorphism structure induced by the automorphism given by f, called Prym map. In many cases the Prym map is generically injective. Particularly interesting are the cases where the Prym map is finite and dominant. In this talk these cases will be worked out for covers of degree a prime number and twice an odd prime. In some cases the degree of the Prym map is determined. This is joint work with Angela Ortega.

I will describe a natural sequence of generalizations going from Turing style computational complexity theory and the P vs NP problem to the complexity theory of algebraic varieties. I will then explain how to use universal circuits to make an NP-complete sequence of projective varieties.

Ever since the 1970's, holomorphic twistor spaces have been used to study the geometry and analysis of their base manifolds. In this talk, we will introduce integrable complex structures on twistor spaces fibered over complex manifolds that are equipped with certain geometrical data. The importance of these spaces will be shown to lie, for instance, in their applications to bihermitian geometry, also known as generalized Kahler geometry. (This is part of the generalized geometry program initiated by Nigel Hitchin.) By analyzing their twistor spaces, we will develop a new approach to studying bihermitian manifolds. In fact, we will demonstrate that the twistor space of a bihermitian manifold is equipped with two complex structures and natural holomorphic sections as well. This will allow us to construct tools from the twistor space that will lead, in particular, to new insights into the real and holomorphic Poisson structures on the manifold.