# TTT87: Leicester

University of Leicester.

Supported by the London Mathematical Society.

## Date

6 March 2013

## Speakers

### Matias del Hoyo (University of Utrecht)

Integrating two-term representations up to homotopy over Lie algebroids

Integrating two-term representations up to homotopy over Lie algebroids

Lie groupoids are a way to deal with singular smooth spaces. They constitute a unified framework that includes manifolds, Lie groups, actions, foliations and others. Lie algebroids are their infinitesimal counterpart and together they play a rich theory in development. While representations of groupoids and algebroids are too restrictive, representations up to homotopy are flexible enough so as to contain interesting examples.

Two-term representations up to homotopy can be regarded as vector bundles over groupoids and algebroids, in the spirit of the so-called Grothendieck construction for lax functors. In this talk I will provide an overview of the topic and discuss the derivation and integration of vector bundles, part of a joint work with H Bursztyn and A Cabrera.

### Ambrus Pal (Imperial College London)

Simplicial homotopy theory of algebraic varieties over real closed fields

Simplicial homotopy theory of algebraic varieties over real closed fields

I will talk about the homotopy type of the simplicial set of continuous definable simplexes of an algebraic variety defined over a real closed field, which I call the real homotopy type. There is an analogue of the theorem of Cox comparing the real homotopy type with the etale homotopy type, as well as an analogue of Sullivan's conjecture which together imply a homotopy version of Grothendieck’s section conjecture.

As an application I show that for geometrically rationally connected varieties over archimedean real closed fields the map from R-equivalence classes to homotopy fixed points is a bijection, but it is not a bijection in general. Moreover there is a version of Grothendieck's anabelian section conjecture for hyperbolic curves over real closed fields.

### Leila Schneps (Insitut de Mathematiques de Jussieu Paris)

Braids, Galois groups and Grothendieck-Teichmueller theory

Braids, Galois groups and Grothendieck-Teichmueller theory

Grothendieck first developed Grothendieck-Teichmueller theory as a new approach to understand the absolute Galois group of the rationals $Gal(\bar{\bf Q}/{\bf Q})$: namely by studying the action of this group on objects coming from topology (genus g surfaces with marked points) and from geometry (the moduli spaces classifying Riemann surfaces).

To date, the theory has yielded a remarkable group, the 'Grothendieck-Teichmueller group' GT with a very simple definition, which acts like $Gal(\bar{\bf Q}/{\bf Q})$ on fundamental groups of moduli spaces of curves, contains $Gal(\bar{\bf Q}/{\bf Q})$ as a subgroup, and is in fact conjecturally isomorphic to it.