# TTT92: Sheffield

School of Mathematics and Statistics, the University of Sheffield.

Supported by the London Mathematical Society.

## Date

17 July 2014

## Speakers

### John Greenlees (Sheffield)

Ausoni-Bokstedt duality for topological Hochchsild homology

Ausoni-Bokstedt duality for topological Hochchsild homology

Calculations of Ausoni, Bokstedt and others show very striking Gorenstein duality in the topological Hochschild homology of local rings if we take coefficients in the residue field of characteristic p. The purpose of this talk is to give a non-calculational duality statement covering these and many other examples.

This uses the machinery of Gorenstein duality for ring spectra (Dwyer-Greenlees-Iyengar). The only calculational input is Bokstedt's calculation of THH(k) for a field k of characteristic p. Gorenstein ascent is then applied to a cofibration of commutative ring spectra proved by Dundas.

Previous examples of Gorenstein duality ultimately stem (ie, via Morita theory and ascent) from Poincare duality for manifolds or Frobenius duality for group rings, so this does seem to be a new class of examples.

### Patrick Orson (Edinburgh)

Double L-theory and localisation

Double L-theory and localisation

I will give an introduction to the chain complex methods of symmetric algebraic L-theory; a way of algebraically modelling the symmetric properties of topological manifolds, such as Poincare duality, and manifold cobordism. I will describe how I have applied these techniques to the study of the 'doubly-slice' problem in high-dimensional knot theory, by defining an algebraic 'double-cobordism' relation.

This new algebra admits a localisation exact sequence which I will describe and compare to the classical Witt group and L-group localisation exact sequences of Milnor-Husemoller and Vogel-Ranicki.

### Andy Baker (Glasgow)

Power operations in completed K-theory

Power operations in completed K-theory

I will describe the action of power operations on the p-completed K -theory cooperation algebra K^*(K). I'll also discuss some applications.