# TTT58: Sheffield

Department of Pure Mathematics, the University of Sheffield.

This was a working meeting with the usual time for participant discussion.

## Date

28 November 2006

## Speakers

Thomas Huettemann

Enumerating lattice points using line bundles

Rui Reis (Edinburgh)

KO-homology and type I D-branes

Andrew Stacey (Sheffield)

Algebra objects and algebraic topology

## Abstracts

### Thomas Huettemann

A subset S of R^n defines a formal Laurent series in n indeterminates, the lattice points of S corresponding to the exponent vectors of the monomials occurring in the series. In this way, a polytope P with integral vertices gives rise to a Laurent polynomial, but also to several Laurent series corresponding to certain cones associated to the vertices of P.

I will explain a surprising theorem of Brion relating these series to the original polynomial, and present a generalisation using the language of toric geometry which allows to treat the more general case of 'virtual polytopes' as well.

### Rui Reis

In the spirit of Paul Baum's and Ronald Douglas' geometric description of K-homology, I'll describe the analogous construction for KO-homology. Namely, an element in the KO-homology of a space X can be represented by a triple (M,E,f), where M is a closed spin manifold, E is a real vector bundle over M, and f is a continuous map from M to X. These triples can be interpreted as D-branes in type I string theory, and the element of the KO-homology group it represents can be interpreted as its charge.

The relevant feature of the construction is its relation to analytic KO-homology, namely proving that the two theories are isomorphic, which one achieves by considering, as in the complex case, an appropriate index theorem, the Cl_n index theorem in the case of KO-homology.

### Andrew Stacey

This talk will a progress report on joint work with Sarah Whitehouse aimed at understanding the structure of unstable operations of cohomology theories. I shall show how the theory as we understand it fits into a general picture of monoids in certain categories that arise naturally in general (universal) algebra.

The benefit of this approach is that by describing unstable operations in algebraic terms (ie as some sort of generalised algebra) one can start looking for nice generating sets and try to devise some simple descriptions. I shall start by outlining the simple general theory underlying this approach, show how cohomology theories fit into the picture, and end with some simple examples.

Note for Sheffield topologists: this is an update of the talk I gave at the Sheffield Algebra and Topology seminar; the central idea is still the same but we have a much better understanding of the general theory which makes the whole thing more straightforward.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.