Oxford

The Landscape problem in String Theory is the fact that there are apparently a great many possible vacua; each leading to a very different four dimensional world. I will give a survey of the space of possibilities and then argue that we may, after all, live in a naturally defined tip of the distribution.

In the Examination Schools

Both Kazhdan and Haagerup properties turn out to be related to actions

of

groups on median spaces and on spaces with measured walls.

These relationships allows to study the connection between Kazhdan

property (T) and the fixed point property

for affine actions on $L^p$ spaces, on one hand.

On the other hand, they allow to discuss conjugacy classes of subgroups

with property (T) in Mapping Class Groups. The latter result

is due to the existence of a natural structure of measured walls

on the asymptotic cone of a Mapping Class Group.

The talk is on joint work with I. Chatterji and F. Haglund

(first part), and J. Behrstock and M. Sapir (second part).

Roughly speaking, a quasiregular map is a possibly-branched covering map with bounded distortion. The theory of such maps was developed in the 1970s to carry over to higher dimensions the more geometric aspects of the theory of complex analytic functions of the plane. In this talk I shall outline the proof of rigidity theorems describing the quasiregular self-maps of hyperbolic manifolds. These results rely on an extension of Sela's work concerning the stability of self-maps of hyperbolic groups, and on older topological ideas concerning discrete-open and light-open maps, particularly their effect on fundamental groups. I shall explain how these two sets of ideas also lead to topological rigidity theorems. This talk is based on a paper with a similar title by Bridson, Hinkkanen and Martin (to appear in Compositio shortly). http://www2.maths.ox.ac.uk/~bridson/papers/QRhyp/

Abstract: I will give an introduction to, and overview of, the AdS/CFT correspondence from a geometric perspective. As I hope to explain, the correspondence leads to some remarkable relationships between string theory, conformal field theory, algebraic geometry, differential geometry and combinatorics.