Spectral curves and Langlands duality
Abstract: The moduli space of principal Higgs bundles with group G on a compact Riemann surface has the structure of an integrable system, where the generic fibre is an abelian variety. The SYZ approach to mirror symmetry suggests that dualizing the fibres leads to the moduli space for the Langlands dual group. In the talk we shall discuss the known cases of SL(n) and its dual PGL(n), Sp(m) and its dual SO(2m+1) and more interestingly that of G2. In the latter case we give a new, more concrete, approach to the description of the abelian variety and show that duality involves also an involution on the base of the integrable system.
On motives in finite characteristic
Abstract: Langlands correspondence gives the description of the set of irreducible motivic local systems of a given rank on a curve defined over a finite field F. Namely, it identifies it with the joint spectrum of a commuting family of matrices with integer coefficients (Hecke operators). If we pass to all finite extensions of F, we obtain a bizzare infinite countable set M on which acts simultaneously the Galois groups of Q and of F. The question is how can one "understand" this. I will describe two proposals, the first one relates M with polynomial dynamics over rational numbers. The second says that M comes from a finite scheme in certain tensor category of motivic nature associated with F. Both proposals generalize to higher-dimensional case.
Compatified Moduli Spaces and String Topology of the Free Loop Space of a Manifold
Abstract: There are canonical minimal energy area preserving flows on
Riemann surfaces with input punctures where the fluid enters at
given rates and output punctures where the flow exits at given
rates.These flows are pictorially easy to analyze as the surface
developes nodes.There are also versions on surfaces with boundary...
The combinatorics of the orbits touching rest points of the
flows leads to two discussions.
The first is a cell decomposition of the open moduli space of
Riemann surfaces (discussed first by Giddings and Wolpert in a
physics oriented paper and independently by C.F. Bodigheimer in a
more precise mathematical treatment). The open moduli space cell
discussion extends naturally to the nodal compactification of
moduli space using the above mentioned pictures.
The second discussion makes use of the combinatorics, in
particular how the flow moves ,splits and reconnects the orthogonal
one manifolds to the flow, and regularized transversality in
families of poly curves in a manifold to construct operations in
the algebraic topology of the free loop space as well as in that of
path spaces with boundary conditions. The regularization is a
technical device which diffuses the objects acted upon using a
measure on a finite dimensional space of diffeomorphisms
constructed from a coordinate cover.
Manifolds with positive curvature operators are space forms
Abstract: We prove a long standing conjecture of R. Hamilton: On a compact manifold the normalized Ricci flow evolves a Riemannian metric with positive curvature operator to a metric of constant positive sectional curvature. This is joint work with B. Wilking.
Gromov-Witten invariants and Gauge Theory
Abstract: The study of holomorphic curves in Kaehler manifolds has been a subject of remarkable development, both in mathematics, where it is known as Gromov-Witten theory, and in physics, where these questions are studied under the name of topological string theory. Surprising connections have been found with mirror manifolds, integrable hierarchies and invariants of knots and three-manifolds. I will concentrate on a recent development: the relation with gauge theories of so-called Donaldson-Thomas type. Both the case of rank one and the limit of infinite rank are of interest. In physics this corresponds to the study of quantum black holes.
Revised 19.12.2006
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