# TTT82: Leicester

University of Leicester.

Supported by the London Mathematical Society.

## Date

2 December 2011

## Speakers

### Mark Grant (Nottingham)

Realising homology classes by immersions

Realising homology classes by immersions

In every codimension greater than or equal to two there exists a mod 2 homology class in a closed smooth manifold which cannot be realised by an immersion. This is joint work with András Szucs (Eotvos Lorand).

### Beatriz Gonzalez Rodriguez (ICMAT-CSIC Madrid)

Homotopical cocompleteness for relative categories closed by coproducts

Homotopical cocompleteness for relative categories closed by coproducts

In this talk we study homotopy colimits defined as left adjoints to the constant diagram functor in a suitable 2-category of relative categories. In case the relative category considered is closed by coproducts, we give a characterisation of homotopical cocompleteness based on the existence of 'homotopy coequalisers', interpreted as good homotopy colimits for diagrams of simplicial shape.

This characterisation might be understood as a homotopical version of the classical result stating that all colimits can be computed using coproducts and coequalisers.

### Simone Borghesi (Milano-Bicocca)

Brody's theorem for Deligne-Mumford stacks

Brody's theorem for Deligne-Mumford stacks

The classical Brody's theorem in complex geometry relates two notions of hyperbolicity of a complex space: it states that they are equivalent if the space is compact. In recent work with G Tomassini we generalised this theorem to Deligne-Mumford analytic stacks.

In this talk we will discuss the definitions of Kobayashi and Brody hyperbolicity for stacks, using an exotic homotopy theory, then I will mention about the key points in the proof of the theorem involving the use of complex variables in a framework dictated by the underlying homotopy theory.

### Burt Totaro (Cambridge)

New bounds for the cohomology of finite groups

New bounds for the cohomology of finite groups

Symonds showed that the cohomology ring of a finite group G with a faithful complex representation of dimension n is generated by elements of degree at most n^2. This was a remarkable advance, since no bound was known before.

Symonds's proof combined equivariant cohomology with commutative algebra (Castelnuovo-Mumford regularity). We give better bounds for the cohomology ring of a p-group. For some problems of this type, we give optimal bounds.