# TTT107: Liverpool

University of Liverpool.

Supported by the London Mathematical Society.

## Date

17 July 2018

## Speakers

### Anna Pratoussevitch (Liverpool)

Spaces of higher spin Klein surfaces

Spaces of higher spin Klein surfaces

We will discuss the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle.

We describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of R^d by a discrete group. The spaces of higher spin bundles on Klein surfaces have connections with singularity theory and real algebraic geometry.

We will discuss an application to real forms of Gorenstein quasi-homogeneous surface singularities, in particular to Brieskorn-Pham singularities. This is joint work with S Natanzon.

### Anwar Alameddin

Motivic measures through Motivic homotopy theory

Motivic measures through Motivic homotopy theory

I will recall an extension of the stable homotopy category, given by its motivic counterpart (over the complex numbers). Then, I will explain how some Euler-Poincaré characteristics factorise through a non-connected K-theory of the motivic stable homotopy category. I will also discuss how this can be applied to address some questions in algebraic geometry.

### Julian Holstein (Lancaster)

A tour of infinity local systems

A tour of infinity local systems

The category of local systems on a topological space has several well-known descriptions, as locally constant sheaves, representations of the fundamental group or vector bundles with flat connection. There is a natural derived analogue, the infinity category of infinity local systems, which has a multitude of equivalent definitions.

In this talk I will describe infinity local systems in several different ways and talk about relations between the different characterisations. In particular I will talk about recent work with Joe Chuang and Andrey Lazarev describing infinity local systems as certain categories of modules over dg algebras.