I will describe a flat connection D on the Cartan subalgebra of a simple Lie algebra g with values in any g-module V and logarithmic singularities on the root hyperplanes obtained jointly with J. Millson. Its monodromy gives a one-parameter family of representations of the generalised braid group Bg of type g which deforms the action of (a finite extension of) the Weyl group of g on V. I will also relate, in analogy with the Kohno-Drinfeld theorem, this monodromy to the quantum Weyl group representations of Bg obtained by Lusztig via the quantum group U_qg.

The group of holomorphic automorphisms of an affine homogeneous space G/H is infinite dimensional, and has not been well-understood until recently. I will show how twisted automorphism arising from the one-parameter subgroups of G can be used to approximate any holomorphic automorphisms. It follows from this that the Abhyankar-Moh property does not hold for holomorphic maps. This is joint work with Varolin.

Negative cosmological constant gravity in 2+1 dimensions is known to have black hole solutions. These black holes can be of non-trivial topology with, for example, handles behind the horizon. We describe an analytic continuation procedure that sends a black hole spacetime into a hyperbolic 3-manifold having the topology of a handlebody. Physical (thermodynamical) properties of the black hole are encoded in the conformal geometry of the boundary Riemann surface.

In a joint work with I. Frenkel, we relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore can be viewed as identical. In particular, we are able to give an alternative and much simpler geometric proof of a result of E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado on the construction of bases of affine Lie algebra representations. At the same time, we give a simple parametrization of the irreducible components of Nakajima quiver varieties associated to infinite and cyclic quivers. We also define new varieties whose irreducible components are in one-to-one correspondence with highest weight representations of \widehat\mathfrakgl_n+1.

Cohomological invariants such as the orbifold Euler number and orbifold Hodge numbers have been defined and studied and have been shown to agree with the classical invariants of a nice (crepant) resolution of singularities. An orbifold cohomology and Chow ring structure compatible with those invariants has recently been introduced by Chen-Ruan, and by Abramovich-Graber-Vistoli. We associate to a simplicial toric variety a Deligne-Mumford stack (a stacky toric variety) and compute its orbifold Chow ring. This is joint work with Lev Borisov and Greg Smith.

In joint work with Gang Tian we identify the difference between the CM polarisation and the Chow polarisation on the ``Hilbert scheme''. As a consequence CM stability is shown to be equivalent with a weight inequality coming from Mumfords' G.I.T.

The Dynkin-Kostant classification of nilpotent orbits in a simple Lie algebra \mathfrak g proceeds by studying the equivalent problem of finding all embeddings of the simple three-dimensional Lie algebra, \mathfraksl₂, into \mathfrak g. This equivalent formulation produces a semisimple conjugacy class for each nilpotent orbit and this class turns out to be important for the unitary representation theory of the corresponding complex Lie group. In this talk, we present a new characterization of these semisimple classes which is (almost) purely combinatorial and explain some of the connections to representation theory. This is joint work with Paul Gunnells.

We will show how Configurations Spaces,taken from various points of view,play an important role in the understanding of the noncompactness in Conformal as well as Contact Form Geometry.

We will study the limit mixed Hodge structure of near by fundamental group for Mumford Tate degeneration. Periods can be written using multiple zeta values.

We will first recall the notion of mixed twistor structure introduced by Simpson. The principle behind this is his "Meta Theorem", which says that theorems for mixed Hodge structures should be generalized to theorems for mixed twistor structures. As some evidences, we explain our study on the asymptotic behaviour of harmonic bundles. A harmonic bundle is a generalization of a variation of Hodge structures, and it can be regarded as a variation of pure twistor structures. We can generalize some of the classical results of Cattani-Kaplan-Schmid and Kashiwara-Kawai. In the last part, we would like to mention our attempt toward the theory of twistor modules, which should be a generalization of Hodge modules. A decomposition theorem will be stated.

We count curves of genus 2 over finite fields and obtain detailed information about the cohomology of local systems. This determines the cohomology of the moduli spaces \cal M_2,n of curves with marked points. This is joint work with Gerard van der Geer.

We will talk about how to construct compact minimal submanifolds of compact Kähler manifolds and to relate the classical theory of deformations of holomorphic submanifolds due to Kodaira-Spencer and Bloch, to the one for minimal submanifolds. The main source of examples of volume minimizers in such manifolds is given by calibrated submanifolds by a well known extension of Wirtinger's inequality due to Harvey-Lawson. A classical problem is to understand under which conditions the converse holds. The more positive the curvature becomes, the more likely a positive solution can be expected. This is highlighted by the fact that the converse is indeed true: 1) for the complex projective space, by a result of Lawson and Simons; 2) for compact Kähler manifodls of positive bisectional curvature; 3) for any Kähler-Einstein surface with positive Ricci carvature, by a result of Wolfson (with the condition that the minimal surface be symplectic), etc. We will show already in the case of Kähler surfaces the existence of stable minimal surfaces wich are not J-holomorphic w.r.t. any complex structure compatible with the metric, if we relapse the Kähler-Einstein condition (but mantain constant scalar curvature); and in fact that we get such examples also for Kähler-Einstein manifolds of postive curvature (and non-negative bisectional curvature), in complex dimension at least 3.