Oxford

Extending work of Klyachko, we give a combinatorial description of pure equivariant sheaves on a nonsingular projective toric variety X and construct moduli spaces of such sheaves. These moduli spaces are explicit and combinatorial in nature. Subsequently, we consider the moduli space M of all Gieseker stable sheaves on X and describe its fixed point locus in terms of the moduli spaces of pure equivariant sheaves on X. Using torus localisation, one can then compute topological invariants of M. We consider the case X=S is a toric surface and compute generating functions of Euler characteristics of M. In case of torsion free sheaves, one can study wall-crossing phenomena and in case of pure dimension 1 sheaves one can verify, in examples, a conjecture of Katz relating Donaldson--Thomas invariants and Gopakumar--Vafa invariants.

We will present a self-contained introduction to gauge theory, self-duality and instanton moduli spaces. We will analyze in detail the situation of charge 1 instantons for the 4-sphere when the gauge group is SU(2). Time permitting, we will also mention the ADHM construction for k-instantons.

We describe John Stalling's method of studying finitely generated free groups via graphs and moves on graphs called folds. We will then discuss how the theory can be extended to study the automorphism group of a finitely generated free group.

I will survey the theory of quasiHamiltonian spaces, a.k.a. group valued moment maps. In rough correspondence with historical development, I will first show how they emerge from the study of loop group representations, and then how they arise as a special case of "presymplectic realizations" in Dirac geometry.