MSI Weekly Bulletin - Week starting Monday 8 October, 2007

We show that the Dirichlet to Neumann (DN) map for the Laplacian determines Einstein manifolds with boundary up to isometry. We shall also discuss the (non-compact) conformally compact Einstein case using an anologue of the DN map. This is a joint work with A. Sa Barreto (Purdue Univ.)

In algebraic geometry, singularities on curves or surfaces are understood as a special kind of local commutative ring. It turns out that there is a great deal of information about the singularity in acyclic complexes of projective modules defined over the local ring; I will explain the connection in a simple example, and discuss the global generalization.

Although the idea that a computer can verify mathematical proofs has been nurtured in Computer Science, it has its roots firmly in the schools of Mathematics and Logic. We will give some simple examples demonstrating that systems for checking proofs are now mature enough to express significant theorems. We then ask: Are these systems rigorous enough to be trusted? Are they useful tools for mathematicians? To start to answer these questions we will explain some of the principles behind systems for machine verification.

I give an introduction to Topological T-duality for principal circle bundles. I discuss the mathematical and physical aspects of the extension of this construction to $S^1-$spaces with fixed points