# Dr. Jan Swoboda (Universität München): “ Limits and degenerations of Higgs bundle moduli spaces I und II”

Sep 30, 2015 from 04:15 PM to 05:15 PM

LMS 4 - Raum 424 - Kleiner Hörsaal

**Abstract:
**

Moduli spaces of Higgs bundles arise naturally in several rather
different contexts: as generalizations of the concept of stable bundles
in algebraic geometry, as solution spaces of certain elliptic equations
over a Riemann surface, as representation varieties of surface groups as
well as phase spaces of integrable systems. Being complete, noncompact
hyperkähler manifolds, these are also of some interest in Riemannian
geometry.

In the first part of this lecture series I shall give an introduction to
the basic theory of Higgs bundles over a Riemann surface. I will focus
on the construction of its moduli space as the set of gauge equivalence
classes of solutions to Hitchin's self-duality equation, discuss
concepts such as stability and the Hitchin fibration, and describe its
realization as a hyperkähler quotient.

In the second part, I plan to report about recent joint work with
Mazzeo, Weiß and Witt on the asymptotic behaviour of solutions in the
limit of large Higgs fields. In a rather different direction, I shall
focus on the behaviour of the moduli space under degeneration of the
underlying Riemann surface, which can be understood using gluing methods
from geometric analysis.