Consider a packing of spheres in R^3 at close to maximum density. Do such packings tend to be perturbations of face-centered cubic (fcc) lattices, perturbations of hexagonal-close-packed (hcp) lattices, perturbations of other lattices, or something altogether different? I'll present the current evidence, including recent computer experiments by Hans Koch, Charles Radin and myself.

It is conjectured that the Riemann-Hilbert monodromy map between certain quiver varieties and character varieties of a punctured Riemann sphere induces a map on cohomology which preserves mixed Hodge structures and induces an isomorphism on the pure parts. Doing arithmetics and representation theory on the varieties I will exhibit some striking evidence in favor of this conjecture, with a more recent technique applying Letellier's calculation of Fourier transforms of invariant functions on a finite Lie algebra. This gives a powerful method to calculate the Poincare polynomial of Nakajima's starshaped quiver varieties and together with the conjecture hints at a deep connection between the character tables of finite Lie groups and algebras.

Given an open smooth variety M there is (conjecturally) a tautological compactification with a nice boundary, the so called log canonical compactification. I'll recall the (entirely elementary) construction, point out examples of known compactifications of moduli spaces that turn out to be of this sort, and then mention an example, moduli of hyperplane arrangments, where the compactification is not know, and indicate why finding it is of compelling interest.

A conic bundle is a 3-fold fibred over a surface with general fibre a smooth rational curve. These varieties are important in the classification of 3-folds as described by Mori's minimal model program. A conic bundle corresponds to an algebra over the surface with general fibre 2x2 matrices. We use a version of the MMP for this noncommutative surface to construct simple birational models of conic bundles. The noncommutative singularities which arise have many nice properties, for example, they are simultaneously Clifford algebras and quotients of matrix algebras. This is joint work with Daniel Chan and Colin Ingalls

We'll first review some classical geometry of affine spheres and Tzitzeica equation. After formulating the structure equations as an integrable system associated to twisted loop algebra of sl(3), we were able to find simple rational elements in this loop algebra whose dressing actions give exactly the classical Tzitzeica transformations. (They are then parallel to the example of pseudospheres, sine-Gordon equation and classical Backlund transformation.)

This is the first of two talks in conformal geometry. We will explain first the classical Yamabe problem (construct a constant scalar curvature metric in the same conformal class as a given metric), for a compact manifold without boundary. Then we will consider the same problem on a complete, non-compact manifold. PDE's play an important role in both, however, the talk will be Geometry oriented.

This is the continuation of the Gadget talk last week. Here we will consider the curvatures of a manifold, a generalization of the scalar curvature. They are defined as the symmetric functions of the eigenvalues of the Schouten tensor and give rise to interesting problems in conformal geometry. The end of the talk will try to relate to the material from the previous week.