We obtain an inequality between the log canonical thresholds and multiplicities, which generalizes the classical inequality between arithmetic mean and geometric mean. We apply the result to study the the singularities of pairs under generic projection. As an application, we study the birational rigidity of hypersurfaces of degree n in Pⁿ.

We will discuss some relations between maximally degenerating variations of Hodge structure and families of Frobenius modules. A family of Frobenius modules is an abstraction based on the small quantum multiplication by elements of H^{1,1} on the cohomology of a Calabi-Yau manifold. In general, there is a correspondence between maximally degenerating polarized variations of Hodge structure and families of Frobenius modules.

Motivated by the passage from the small to the big quantum product we show how, under certain conditions, it is possible to unfold a family of Frobenius modules -- equivalently, a maximally degenerating variation of Hodge structure -- into a Frobenius manifold.

The talk presents a formula computing the number of curves of a given genus and degree through a collection of generic points in a toric surface (both over complex and real numbers). The computaion is done by means of the so-called tropical algebraic geometry. The answer is presented as a number of certain lattice paths in the corresponding convex polygon. The dimension of the curves in the count is equal to the length of the corresponding lattice paths.

Certain types of cohomological questions arising from vector bundles and their morphisms always seem to have special kinds of polynomials (such as Schur and Schubert polynomials) popping up in their answers. Moreover, when these special polynomials are expressed in terms of simpler special polynomials (such as monomials), the coefficients always seem to be positive integers that count combinatorial objects. I will explain how to view the cohomological questions in a universal geometric setting, and how positivity can then be explained geometrically.

I will give an account of Tom Bridgeland's new axioms for Douglas' notion of a stability condition on a triangulated category, relating it to bundles, (special) Lagrangians, and some curves in the plane.