The talk is based on some joint work with A.P.Rao and G.V.Ravindra. The relevant preprints are available at {xxx.arXiv.org/abs/math.AG/0507161} and {arXiv:math/0611620}. The first will appear in Commentari Math. Helv and the second in IMRN.

A vector bundle on a polarized projective variety (X,L) is called Arithmetically Cohen-Macaulay if all its middle cohomologies in all twists by powers of L vanish. A famous criterion of G.Horrocks states that a vector bundle on projective space is a direct sum of line bundles if and only if it is arithmetically Cohen-Macaulay (with respect to the usual polarization). It is well known that this criterion fails for other varieties, in particular for hypersurfaces in projective spaces. In my talk I will discuss the following results proved in the above articles. Any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in P^5 or on a general hypersurface of degree at least six in P^4 must be split.

Non abelian theta functions, which generalise the classical theta functions on the Jacobian of a curve X, are associated to a group G, an algebraic curve X and a level k. As the curve X moves in its moduli space, the spaces of non abelian theta functions carry a connection. If G->H is a map of groups, H-theta functions map to G-theta functions. I will discuss the problem of determining when these maps are flat, with applications to ``strange dualities''.

I shall discuss and prove a base change theorem for derived categories of smooth schemes, in the absence of flatness assumptions. As an application I shall present a Hochschild-Kostant-Rosenberg isomorphism for smooth, global quotient orbifolds.

This is a joint work with Jun Li.

In this talk I will describe the structures of the moduli spaces of genus-2 stable maps into P^n. This is done by first finding the structures of some natural sheaves over the moduli space. In the end, I will also discuss some applications such as desingularization.

We study systems of linear ordinary differential equations dy/dz=A(z)y, where A is a matrix-valued rational function of z. By definition, such equation is rigid if it is uniquely determined by the type of its singularities.

Our goal is to provide a classification of rigid equations. If all singularities of A are regular, this was done by N.Katz; we extend his results to arbitrary A.

In 1990, Candelas, de la Ossa, Green, and Parkes used the then-new technique of mirror symmetry to predict the number of rational curves of each fixed degree on a quintic threefold. The techniques used in the prediction were subsequently understood in Hodge-theoretic terms: the predictions are encoded in a power-series expansion of a quantity which describes the variation of Hodge structures, and in particular this power-series expansion is calculated from the periods of the holomorphic three-form on the quintic, which satisfy the Picard-Fuchs differential equation.

In 2006, Johannes Walcher made an analogous prediction for the number of holomorphic disks on the complexification of a real quintic threefold whose boundaries lie on the real quintic, in each fixed relative homology class. (The predictions were subsequently verified by Pandharipande, Solomon, and Walcher.) This talk will report on recent joint work of Walcher and the speaker which gives the Hodge-theoretic context for Walcher's predictions. The crucial physical quantity "domain wall tension" is interpreted as a Poincare normal function, that is, a holomorphic section of the bundle of Griffiths intermediate Jacobians. And the periods are generalized to period integrals of the holomorphic three-form over appropriate 3-chains (not necessarily closed), which leads to a generalization of the Picard--Fuchs equations.

Multigraded Hilbert schemes parametrize all homogeneous ideals with a fixed Hilbert function in a polynomial ring that is graded an abelian group. We will examine conditions which guarantee that a multigraded Hilbert scheme is smooth and irreducible. This talk is based on joint work with Diane Maclagan.