# TTT59: Manchester

School of Mathematics, University of Manchester.

## Date

13 February 2007

## Speakers

Andrew Baker (Glasgow and Oslo)

Andre-Quillen homology as a cellular theory

Based on joint work with Gilmour and Reinhard. Andre-Quillen homology (for simplicial rings) and the analogue for commutative S-algebras known as topological Andre-Quillen homology can be calculated for cellular S-algebras in terms of the cell structures.

I will explain how this works and then mention applications to the study of minimal atomic p-local commutative S-algebras and simplicial commutative algebras over a Noetherian local ring, generalising analogous results of Baker, May, Pereira for p-local spaces and spectra and Alshumrani for chain complexes over a Noetherian local ring.

Taras Panov (Moscow)

Algebraic torus actions, Kempf-Ness sets and real quadrics in C^m

In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical quotient by the quotient with respect to a maximal compact subgroup. We show that an appropriate notion of Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate subspace arrangement complements) arising in the 'geometric invariant theory' approach to toric varieties.

These 'toric' Kempf-Ness sets are known to toric topologists as moment-angle complexes. In the case of a projective toric variety, the Kempf-Ness set is the level surface for the appropriate moment map and can be written as a complete intersection of real quadrics in C^m. We proceed by studying the cohomology of these Kempf-Ness sets.

Victor M Buchstaber (Manchester and The Steklov Institute)

The universal equivariant genus for torus actions

Based on recent work with Panov and Ray. We consider the universal equivariant genus for stably complex 2n-dimensional manifolds equipped with an action of a k-dimensional torus, where k is less than or equal to n. In the case of quasitoric manifolds M we explicitly calculate this genus in terms of certain underlying combinatorial data. By way of application, we obtain a formula evaluating the complex cobordism class of M in terms of this data.

## Further information

The meeting is jointly supported by the London Mathematical Society and MIMS.