# Hausel Group

## Geometry and its Interfaces

Tamas Hausel's research interests include combinatorial, differential and algebraic geometry and topology. The main tool used is representation theory, connecting our investigations to number theory and physics.

The more specific focus is the study of the geometry, topology and arithmetic of several moduli spaces appearing in supersymmetric quantum field theories including moduli spaces of Yang-Mills instantons in four-dimensions, the moduli space of magnetic monopoles in three dimensions and the moduli space of Higgs bundles in two dimensions. The questions we are concerned with have motivations in mathematical physics, such as string theory and topological quantum field theory as well as in number theory, in particular in the realm of the Langlands program.

The immediate question is to describe the topology of such spaces, starting with the structure of holes of various dimensions. Besides the traditional techniques of global analysis and Morse theory, we also employ arithmetic methods, which in turn will invariably lead to problems in representation theory of various objects in algebra. Examples include the representation theory of quivers, finite groups and algebras of Lie type and various Hecke algebras.

What results is a colorful palette of subjects in mathematics and theoretical physics and our focus is on the interconnectedness of these fields inside mathematics.

**Contact**

Tamas Hausel

E-mail: tamas.hausel@
ist.ac.at

»Mathematics at IST Austria website

**Assistant**

Caroline Petz

Phone: +43 (0)2243 9000-1209

E-mail: caroline.petz@
ist.ac.at

**Team**

- Jan Leon Friedmann (PhD Student)
- Iordan Ganev, Postdoc
- Quoc P. Ho, Postdoc
- Penghui Li, Postdoc
- Sasha Minets, Postdoc
- Mikhail Shkolnikov, Postdo
- Tanya Kaushal Srivastav, Postdoc
- Hongjie Yu (Postdoc)

**Selected Publications**

- Hausel, T. Letellier, E., Rodriguez-Villegas, F.: Positivity for Kac polynomials and DT-invariants of quivers,
*Annals of Mathematics*, 177 (2013) 1147-1168, Issue 3,

- de Cataldo, M, Hausel, T.,Migliorini, L: Topology of Hitchin systems and Hodge theory of character varieties: the case A_1,
*Annals of Mathematics,*Volume 175 (2012), Issue 3, 1329-1407, arXiv:1004.1420 - Hausel, T. Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties,
*Duke Mathematical Journal*, Volume 160, Number 2 (2011), 323-400, arXiv:0810.2076 - Hausel, T.: Kac conjecture from Nakajima quiver varieties,
*Inventiones Mathematicae*, Volume 181, Number 1, 2010, 21-37, arXiv:0811.1569 - Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties,
*Inventiones Mathematicae*, 174, no. 3, (2008), 555--624, arXiv:math.AG/0612668 - Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons,
*Duke Mathematical Journal*, 122 Issue 3, (2004) 485-548, arXiv: math.DG/0207169 - Hausel, T., Sturmfels, B.: Toric hyperkaehler varieties, Documenta Mathematica, 7 (2002), 495-534, arXiv: math.AG/0203096
- Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality and Hitchin systems,
*Inventiones Mathematicae*, 153, No. 1, 2003, 197-229 arXiv: math.AG/0205236

#### Career

As of 2016 Professor, IST Austria

2012-2016 Professor, Chair of Geometry, EPF Lausanne

2005-2012 Royal Society University Research Fellow, University of Oxford

2007-2012 Tutorial Fellow in Pure Mathematics, Wadham College

2007-2012 University Lecturer in Pure Mathematics, University of Oxford

2002-2010 Assistant and Associate Professor, University of Texas at Austin

1999-2002 Miller Research Fellow, University of California, Berkeley

1998-1999 School of Mathematics, Institute for Advanced Study, Princeton

1998 PhD Trinity College, University of Cambridge

#### Selected Distinctions

2013 ERC advanced grant

2008 Whitehead prize

2005 Sloan Research Fellow