In the first part of the talk I will discuss the meaning of "geometry" in contemporary mathematics and some general developments. In the second part I will focus on a few particular problems which I know more about, such as questions involving topology and geometric structures on manifolds. The talk will in part be built around an imaginary conversation with Hilbert.

In the first part of the talk I will review differential geometric background involving associative submanifolds of 7 manifolds with G2 holonomy and an analogue of the Yang Mills instanton equation for connections on bundles over such a manifold. Then I will describe joint work with Ed Segal in which, assuming these objects have suitable formal properties, we define holomorphic bundles over moduli spaces of Calabi-Yau 3-folds which can be viewed as the complexification of Floer theory. In the last part of the talk I will consider in more detail the problem of establishing the foundations required for the theory. This involves pertubations of the equations and the question of how to count solutions at infinity, which are pairs consisting of a connection and an associative submanifold. The counting problem leads to a nonlinear generalisation of the spectral flow for eigenvalues of Dirac operators.

For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html

INSTITUTE FOR MATHEMATICAL SCIENCES LAUNCH WEEK

The talk will have two themes. The first is to give an account of recent work of Chen, LeBrun and Weber, proving the existence of a new Einstein metric on a certain 4-manifold. The second will describe a general class of numerical solutions to Calabi's extremal metric equation. The two themes are related by the fact that the metric of Chen, LeBrun and Weber is conformal to an extremal metric, so explicit geometric information can be obtained.