Imperial

The Borel conjecture asserts that aspherical manifolds are topologically rigid, i.e., every homotopy equivalence between such manifolds is homotopic to a homeomorphism. This conjecture is strongly related to the Farrell-Jones conjectures in algebraic K- and L-theory. We will give an introduction to these conjectures and discuss the proof of the Borel conjecture for high-dimensional aspherical manifolds with word-hyperbolic fundamental groups.

I aim to give some review of how generalised geometries provide a natural

framework for describing supersymmetric string backgrounds. In particular I

will focus on a rewriting of type II supergravity in terms of generalised

structures. Hitchin functions appear naturally along with generalised

extensions of the Gukov-Vafa-Witten superpotential.