# TTT97: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

27 November 2015

## Speakers

### Fionntan Roukema (Sheffield)

A DIY exceptional pair construction kit

A DIY exceptional pair construction kit

The exterior of the minimally twisted four chain link contains two properly embedded annuli, a properly embedded torus, and the open three chain link is a trivial surgery on each component. Thus, it is fertile ground for growing 3-manifolds with properly embedded topological surfaces with non-negative Euler characteristic and small Seifert spaces.

We will visually see how the four chain link can be obtained by surgery on a hyperbolic chain link in multiple ways. Via some lovely pictures, and some boring GCSE algebra, we will develop a DIY kit for building hyperbolic manifolds with pairs/triples/quadruples/quintuples/sextuples of non-hyperbolic fillings with specified topological obstructions to hyperbolicity.

The simple construction we present will produce all (known) examples of certain exceptional pairs.

### André Henriques (Oxford and Utrecht)

Representation theory for fusion categories

Representation theory for fusion categories

Given a finite group G then, by definition, EG denotes a contractible G-space with free action.

Let me start by describing a question. The question is not well-defined but its answer is nevertheless interesting. If one restricts attention to linear representations of G, what is that best possible approximation to the notion 'EG'? We would like the answer to be: each irrep of G arises in the representation with infinite multiplicity. This is called a 'universal representation' of G, also known as a 'complete G-universe'.

If V is a universal representation then, in particular, the subset of vectors v in V whose orbit is free is contractible. So a universal representation does indeed look a little bit like EG. If one restricts attention to unitary representations on Hilbert spaces, then universal representations are unique up to contractible space of choice in the following strong sense: there is only one such representation up to isomorphism, and the automorphism group of the representation is contractible.

The goal of this talk is to explore analogous notions to the ones presented above when the group G is replaced by a fusion category (a semisimple tensor category with finitely many simple objects). A representation of a fusion category C consists of a ring R and a tensor functor C --> Bim(R) to the category of R-R-bimodules.

We will see that every fusion category admits a left regular representation (coming from the left action of C on itself). We will then describe what we believe to be a universal representation of C. We conjecture is that this representation is unique up to isomorphism, and also unique up to contractible space of choices. However, unlike in the case of finite group, that representation is not a direct sum of smaller representations.

### David Pauksztello (Manchester)

Silting pairs and stability conditions

Silting pairs and stability conditions

This will be a report on joint work with Nathan Broomhead and David Ploog. The notion of a silting object is a generalisation of tilting object, which turns up in the context of derived equivalences. Silting objects come equipped with a rich combinatorial structure, which is related to mutation in cluster theory.

In this talk, we shall discuss a CW complex arising from silting objects and their connection to the space of Bridgeland stability conditions for certain algebraic examples.