Tamas Hausel

Mirror symmetry, Langlands duality, and the Hitchin system

Among the major mathematical approaches to mirror symmetry are those of Batyrev-Borisov and Strominger-Yau-Zaslow (SYZ). The first is explicit and amenable to computation but is not clearly related to the physical motivation; the second is the opposite. Furthermore, it is far from obvious that mirror partners in one sense will also be mirror partners in the other. This paper concerns a class of examples that can be shown to satisfy the requirements of SYZ, but whose Hodge numbers are also equal. This provides significant evidence in support of SYZ. Moreover, the examples are of great interest in their own right: they are spaces of flat SLr-connections on a smooth curve. The mirror is the corresponding space for the Langlands dual group PGLr. These examples therefore throw a bridge from mirror symmetry to the duality theory of Lie groups and, more broadly, to the geometric Langlands program.

Hodge cohomology of gravitational instantons

This article was published in the Duke Mathematical Journal [© Duke University Press] and is also available at: http://projecteuclid.org/euclid.dmj/1082665286

Arithmetic harmonic analysis on character and quiver varieties

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.

Generators For The Cohomology Ring of The Moduli Space of Rank 2 Higgs Bundles

The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and Atiyah and Bott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle. This paper studies the larger, non-compact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle

Betti numbers of holomorphic symplectic quotients via arithmetic Fourier transform

K

Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles

. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes. The spaces of Higgs bundles with values in

Geometric interpretation of Schwarzschild instantons

K(n)

Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve

for

Quaternionic Geometry of Matroids

n > 0

Geometric construction of new Yang–Mills instantons over Taub-NUT space

turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes.