SR1

I will briefly discuss the construction of the moduli spaces of (semi)stable bundles on a given curve. The main aim of the talk will be to describe various features of the geometry and topology of these moduli spaces, with emphasis on methods as much as on results. Topics may include irreducibility, cohomology, Verlinde numbers, Torelli theorems.

We prove uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes. (This is joint work with Piotr Chrusciel. arXiv:0806.0016)

We introduce the notion of $G$-Higgs bundle from studying the representations of the fundamental group of a closed connected oriented surface $X$ in a Lie group $G$. If $G$ turns to be the isometry group of a Hermitian symmetric space, much more can be said about the moduli space of $G$-Higgs bundles, but this also implies dealing with exceptional cases. We will try to face all these subjects intuitively and historically, when possible!

In this talk I will discuss some elementary notions of deformation theory in algebraic geometry like Schlessinger's Criterion. I will describe obstructions and deformations of sheaves in detail and will point out relations to moduli spaces of sheaves.