# TTT99: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

14 July 2016

## Speakers

### Neil Strickland (Sheffield)

An example in the geometry of surfaces

An example in the geometry of surfaces

### Yumi Boote (Manchester)

Symmetric squares of even spaces

Symmetric squares of even spaces

Abstract: The integral homology of the symmetric square of a CW complex has been known since 1960s, although the general answer is very complicated. However, the situation for the integral cohomology ring remains an open problem, except for a few special cases. One of the main difficulties is the computation of its multiplicative structure.

In this talk I shall outline the solution for even spaces; these have torsion free integral cohomology, concentrated in even dimensions. Examples include quaternionic projective spaces, the octonionic projective plane, and flag manifolds.

### Chris Braun (Lancaster)

Derived localisation

Derived localisation

Abstract: Localisation of commutative rings and modules is among the fundamental tools of commutative algebra and algebraic geometry. It has been well-understood and documented for a long time. On the other hand, localisation of noncommutative rings, or even categories, although more fundamental, is less well-understood and less well-behaved.

It appears in many contexts, indeed the homotopy category of spaces is the localisation with respect to weak equivalences and so in this sense noncommutative localisation is central in homotopy theory. Embracing a, by now well established, philosophy from homotopy theory and working with a derived version of noncommutative localisation allows us to obtain a better behaved theory of noncommutative localisation, employing methods from homotopical algebra.

In this theory, the localisation of a dg algebra, or more generally a dg category, can be seen to be, in a certain precise sense, equivalent to the Bousfield localisation of its category of dg modules. This general abstract result has a wide range of concrete applications to, among others, a general version of the group completion theorem, the K–theory localisation sequence as well as having consequences for the localisation of usual commutative rings.