L3

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).

Recent joint work with Greg McShane has answered the following question: Which curves can be short in a given cell of the decomposition of Teichmueller space? The answer involves a new combinatorial structure called "screens on fatgraphs" as we shall describe. The techniques of proof involve Fock's path-ordered product expansion of holonomies, Ptolemy transformations, and the triangle inequalities. This is a main step in giving a combinatorial description of the Deligne-Mumford compactification of moduli space which we shall also discuss as time permits.

I will present a general formalism for understanding coloured operads of different flavours, such as cyclic operads, modular operads and topological field theories. The talk is based on arXiv:math/0701767.

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: In this talk, I will describe recent work in string phenomenology from the perspective of computational algebraic geometry. I will begin by reviewing some of the long-standing issues in heterotic model building and describe the difficult task of producing realistic particle physics from heterotic string theory. This goal can be approached by creating a large class of heterotic models which can be algorithmically scanned for physical suitability. I will outline a well-defined set of heterotic compactifications over complete intersection Calabi-Yau manifolds using the monad construction of vector bundles. Further, I will describe how a combination of analytic methods and computer algebra can provide efficient techniques for proving stability and calculating particle spectra.

Abstract: The deconfinement transition in gauge theories, in which the Polyakov loop acquires a non-zero expectation value, is described in AdS/CFT as the formation of a black hole in the dual graviational theory. We will explain how to compute the free energy diagram for the Polyakov loop by a constrained gravitational path integral, leading to a new class of black hole solutions.

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.

Abstract: Twistor-string theory is reformulated as a `half-twisted heterotic' theory with target $CP^3$. This in effect gives a Dolbeault formulation of a theory of holomorphic curves in twistor space and gives a clearer picture of the mathematical structures underlying the theory and how they arise from the original Witten and Berkovits models. It is also explained how space-time physics arises from the model. It intended that the lecture be, to a certain extent, pedagogical.