# TTT93: Leicester

School of Mathematics and Statistics, University of Leicester.

Supported by the London Mathematical Society.

## Date

12 November 2014

## Speakers

### James Griffin (Coventry)

Moduli spaces of labelled graphs

Moduli spaces of labelled graphs

Although Aut(Fr) has a finite simplicial set as a classifying space, its homology is extremely difficult to calculate and the problem just gets worse as the rank r increases. However by a result of Hatcher and Vogtmann the homology is known to be stable, and Galatius computed this stable homology to be that of the sphere spectrum.

More generally Hatcher and Wahl conjectured that automorphism groups Aut(H*G*...*G) of free products of groups are homologically stable. I'll prove this via a moduli space of labelled graphs and a little category theory.

### Sarah Whitehouse (Sheffield)

Binomial rings in topology

Binomial rings in topology

A binomial ring is a torsion-free commutative ring that is closed under the binomial operations r(r-1)...(r-n+1)/n! for all positive integers n. They were first introduced by Hall around 1969 and they are related to integer-valued polynomials, Witt vectors and lambda-rings. I will discuss properties of these rings and explain how some interesting examples arise in topology.

### Jeff Giansiracusa (Swansea)

Variations on a theme – ribbon graphs, topological conformal field theories and Hochschild homology

Variations on a theme – ribbon graphs, topological conformal field theories and Hochschild homology

Costello used a ribbon graph model of the moduli space of Riemann surfaces to show that open topological conformal field theories are equivalent to A-infinity categories, and that such a theory has a universal extension to an open-closed theory whose closed state space is the Hochschild chains on the open part.

I will explain how this picture generalises to unoriented surfaces, and surfaces equipped with principle G-bundles, among other structures. In the unoriented case, this is joint work with Ramses Fernandez Valencia; Mobius graphs and an involutive variant of Hochschild homology appear.