An informal discussion on derived categories.

Let C be a smooth projective curve of genus g≥ 4 over the complex numbers and SU^s_C(r,d) be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of SU^s_C(r,d), there exists a distinguished hypersurface S_E consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety C_E consisting of vectors tangent to Hecke curves in SU^s_C(r,d) through E. Our main result establishes that the hypersurface S_E and the variety C_E are dual to each other. This gives, among others, a simple proof of the non-abelian Torelli theorem. This is a joint-work with S. Ramanan.

Quotients of a smooth variety by a group play an important role in algebraic geometry. In this talk, I will describe an interesting collection of quotient spaces (called toric stacks) defined by combinatorial data. As an application, I will relate the orbifold cohomology of a toric stack with a resolution of the underlying singular variety.

Arakelov theory has developed into an independent, fundamental discipline within algebraic geometry and arithmetic geometry which involves fascinating aspects of number theory as well as certain components of analysis, such as spectral invariants, automorphic forms, and heat kernel analysis. I will discuss results obtained in collaboration with Jurg Kramer (Humboldt University, Berlin) in which we investigate specific problems which can be viewed as questions in analysis, but, from our point of view, arise from issues in Arakelov theory. As time permits, I will present our results concerning integrals of star products of Green's currents, asymptotic bounds for special values of Selberg's zeta function, as well as our more recent work.

A famous result of Springer realizes representations of the Weyl group in the homology of certain subvarieties of the flag variety. Using a more recent construction of the group algebra of the Weyl group given by Lusztig, we will discuss a new approach to these representations in terms of constructible functions.

I'll begin with an elementary introduction to the Geometric Invariant Theory "master space" via the familiar plane Cremona transformation. This is fun, universally accessible and will involve plenty of pretty pictures. I'll then use GIT to construct the moduli spaces of G-constellations (generalizing Kronheimer's hyperkähler construction of resolutions of ADE singularities as described in Justin's course). Finally, I'll provide a disturbingly elementary construction of the relevant master space using initial (left) ideals in a noncommutative ring, namely the skew group ring.

Continuing from last week, joint work with D. Maclagan and R. Thomas.

For any smooth projective variety, we study a birational invariant, defined earlier on by Campana, depending on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers. We show how such invariant is related with other geometrical properties (as the fundamental group) and we describe new bounds for any projective variety of non-negative Kodaira dimension.

A celebrated Theorem of Del Pezzo and Bertini classifies the nondegenerate irreducible projective varieties of minimal degree (=1+ codimension). I will discuss extensions of these theorems to the general reducible case from various geometric, algebraic, cohomological and graph theoretic/combinatorial perspectives, based on recent joint work with David Eisenbud, Mark Green, Klaus Hulek.