# TTT62: OU, Manchester

The Open University, Sharston, Manchester.

## Date

4 December 2007

## Speakers

Richard Hepworth (Sheffield)

Orbifold Morse theory

Martin Crossley (Swansea)

Conjugation invariants and Reed-Muller codes

Frank Neumann (Leicester)

Moduli stacks of vector bundles over algebraic curves and Frobenii

## Abstracts

### Richard Hepworth: Orbifold Morse theory

Morse theory is a geometric way to understand the homology of manifolds. Orbifolds are spaces that locally look like the quotient of a manifold by a finite group. Can we generalise Morse theory to orbifolds? I will explain how this question relates to the 'crepant resolution conjecture' and give details of the answers I have obtained so far.

### Martin Crossley: Conjugation invariants and Reed-Muller codes

Some years ago Sarah Whitehouse and I tried to compute the subalgebra of the dual Steenrod algebra consisting of those elements fixed by the Hopf algebra conjugation. Recently we have been using the free Hopf algebra generated by the Steenrod squares to shed light on this problem. While we haven't yet solved the conjugation invariants problem, our work has revealed an interesting link with Reed-Muller codes.

### Frank Neumann: Moduli stacks of vector bundles over algebraic curves and Frobenii

After giving an introduction into moduli problems and moduli stacks, I will describe the l-adic cohomology ring of the moduli stack of vector bundles on an algebraic curve in positive characteristic and will explicitly describe the action of the various geometric and arithmetic Frobenius morphisms on the cohomology ring.

It turns out that using the language of algebraic stacks instead of geometric invariant theory this becomes surprisingly easy and topolgical in flavour. I will indicate how to prove the analogues of the Weil conjectures for the moduli stack. This is joint work in progress with Ulrich Stuhler (Goettingen).

## Further information

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