# TTT106: Leicester

University of Leicester.

Supported by the London Mathematical Society.

## Date

28 March 2018

## Speakers

### Jason Semeraro (Leicester and Heilbronn Institute for Mathematical Research)

Subgroup complexes of fusion systems

Subgroup complexes of fusion systems

Abstract: The poset of p-subgroups of a finite group has been of valuable importance in relating its cohomology, geometry, p-local structure and modular representation theory. By studying this poset, Luig Puig was led to the notion of a saturated fusion system – a category whose objects are the subgroups of a fixed p-group and whose morphisms are certain injective group homomorphisms between objects.

Every finite group determines a fusion system, but not conversely. Using the subgroup complex, we will present some new reformulations and generalisations to fusion systems of longstanding conjectures and results in modular representation theory. We will also prove that these conjectures hold for some fusion systems which do not arise from finite groups, including the smallest Benson-Solomon 2-fusion system.

### Elena Gal (Oxford)

Higher Hall algebras

Higher Hall algebras

Abstract: We recall the notion of a Hall algebra associated to a category. The example we have in mind is that of a quantum group arising in this way from the category of representations of an associated quiver. We explain how this construction can be done in a way that naturally includes a higher algebra structure, motivated by work of Toen and Dyckerhoff-Kapranov. We will then explain how this leads to new insights about the bi-algebra structure and categorification of quantum groups.

### Clark Barwick (Edinburgh)

Exodromy

Exodromy

Abstract: It is a truth universally acknowledged, that a local system on a connected topological manifold is completely determined by its attached monodromy representation of the fundamental group. Similarly, lisse ℓ-adic sheaves on a connected variety determine and are determined by representations of the profinite étale fundamental group. Now if one wants to classify constructible sheaves by representations in a similar manner, new invariants arise.

In the topological category, this is the exit path category of Robert MacPherson (and its elaborations by David Treumann and Jacob Lurie), and since these paths do not 'run around once' but 'run out', we coined the term exodromy representation.

In the algebraic category, we define a profinite ∞-category – the étale fundamental ∞-category – whose representations determine and are determined by constructible (étale) sheaves. We describe the étale fundamental ∞-category and its connection to ramification theory, and we summarise joint work with Saul Glasman and Peter Haine.