# TTT103: Liverpool

University of Liverpool.

Supported by the London Mathematical Society.

## Date

30 June 2017

## Speakers

### Jon Woolf (Liverpool)

Stratified homotopy theory

Stratified homotopy theory

Abstract: Stratified spaces arise in many contexts within topology, geometry, and algebra. By fixing suitable stratifications of geometric simplices one can construct stratified versions of geometric realisation and of the total singular complex functor, giving an adjunction between simplicial sets and stratified spaces.

The Joyal model structure on simplicial sets can be cofibrantly transferred across this to obtain a model structure on the category of stratified spaces. The cofibrant-fibrant spaces and weak equivalences in this model structure are closely related to 'classical' notions in the theory of stratified spaces, however one also gains new insights, in particular a new notion of stratified fibration which has better properties than previous ones. This is joint work with Stephen Nand-Lal.

### Stephen Nand-Lal (Liverpool)

Notions of 'basepoint' for a stratified space

Notions of 'basepoint' for a stratified space

Abstract: There are a number of potential approaches to basing a stratified space. In this talk I will explain why some plausible approaches are unsatisfactory, and introduce the notion of basing for a stratified space which we believe to be correct. As evidence for this, there is an adjunction between stratified suspension and loop space functors, which allows us to construct, for suitably nice stratified spaces, an $\mathbb{N}$-indexed family of categories that behave analogously to the homotopy groups of a connected space. This is joint work with Jon Woolf.

### Alessio Cipriani (Liverpool)

Perverse sheaves as modules

Perverse sheaves as modules

Abstract: I will introduce the abelian category of perverse sheaves on a topologically stratified space together with some important properties and examples. I will then explain the construction of the projective cover of a simple perverse sheaf under the assumption that all strata have finite fundamental group.

This is the key ingredient needed to show that the category of perverse sheaves is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if, and only if, all strata have finite fundamental group.

### Thomas Eckl (Liverpool)

Topological classification of isolated holomorphic foliation singularities

Topological classification of isolated holomorphic foliation singularities

Abstract: Isolated singularities of holomorphic foliations can be topologically described by intersecting the leaves of the foliation with (small) spheres centred in the singularities. This works particularly well for holomorphic foliation singularities of Poincare type, but is also useful for other types.

After discussing results along these lines we give a complete classification of isolated singularities of plane holomorphic foliations in the Poincare domain and present partial results and conjectures in higher dimensions.