Titles and Abstracts

The schedule of the workshop can be downloaded here.

Scott Balchin (Warwick): Reconstructing tensor-triangulated categories.

Abstract: Tensor-triangulated categories naturally occur in various settings, for example, as the derived category of cochains on a compact Lie group G, or the stable module category of a Frobenius ring. To any such category, one can assign the so-called Balmer spectrum, which is a categorification of the Zariski spectrum of a ring. I will report on joint work with J.P.C. Greenlees which provides a machinery to build up the unit object of the category from smaller building blocks. These building blocks are formed by taking localised completions of the unit at primes of the Balmer spectrum. Once one has this reconstruction result for the unit object, various models can be considered which fracture the category in question. Not only does this result give an insight into the structure of the category, but it usually provides a convenient setting to do calculations in.

Carles Broto (UAB Barcelona): Fixed points of finite p-group actions on fusion systems.

Abstract: We discuss the notion of group action on fusion systems and associated localities or centric linking systems (joint work with Alex Gonzalez). Full Abstract.

Natalia Castellana (UAB Barcelona): Local Gorenstein duality.

Abstract: Benson and Carlson proved that if the mod p cohomology of a finite group is Cohen-Macaulay then it is also Gorenstein. Using the framework of Gorenstein ring spectra introduced by Dwyer-Greenlees-Iyengar, Benson-Carlson result is a reflection of the fact that the ring spectrum of geometric cochains on BG for a finite group is always Gorenstein and the Benson-Carlson theorem follows from the degeneracy of a local cohomology spectral sequence. We will introduce the notion of local Gorenstein duality for ring spectra and explain how it relates to the previous notion of Gorenstein ring spectra. We will also describe examples satisfying this duality coming from classifying spaces of fusion systems and loop spaces. This is joint work with T. Barthel, D. Heard and G. Valenzuela.

Jesper Grodal (Copenhagen) (Wednesday talk): "Spetses," through the looking glass

Jesper Grodal (Copenhagen) (Thursday talk): Local to global, viewed from homotopy theory

Gunter Malle (Kaiserslautern): Exotic weight conjectures.

Abstract: The weight conjectures propose that global data on the p-modular representation theory of a finite group are encoded in local information, namely the fusion system on a Sylow p-subgroup. For groups of Lie type, both can be described in terms of the Weyl group. In this talk we explain how to attach similar data to certain complex reflection groups and thus formulate and prove weight conjectures for objects baptized 'spetses', using exotic fusion systems arising from p-compact groups. This is joint work in progress with Jason Semeraro and Radha Kessar.

Nadia Mazza (Lancaster): Endotrivial modules and elementary abelian p-subgroups.

Abstract: The group of endotrivial modules of a finite group G is the group of units in the stable module category of G. It is a finitely generated abelian group, and its torsion free rank only depends on a certain subgroup poset and the p-fusion in G. After reviewing the needed concepts and results, we will present some recent results obtained in collaboration with Carlson, Grodal and Nakano.

Jean Michel (Paris 7) (Wednesday talk): Root systems for complex reflection groups.

Abstract: I will define root systems for complex reflection groups, which are defined over the ring of integers of a number field where they live, and explain the classification over the field of definition for irreducible groups. An application is the definition of the connection index, which is an algebraic number intrinsically attached to each well-defined complex reflection group, and the relationship with the bad prime numbers appearing in the theory of Spetses.

Jesper Møller (Copenhagen): The Euler characteristic of the orbit category meets theorems of Frobenius, Brown, and Steinberg.

Abstract: We consider two general combinatorial identities, valid for all finite groups. We use them to count the number of p-elements in finite groups of Lie type, reproving a theorem of Steinberg. We shall also see that a 1908 theorem of Frobenius and a 1975 theorem of Brown are, in some sense, equivalent.

Bob Oliver (Paris 13) (Tuesday talk): Simple fusion systems over finite p-groups with weakly closed abelian subgroup.

Abstract: Fix a prime p, an abelian p-group A, and a subgroup G\le Aut(A). In this situation, we look for all saturated fusion systems F over a p-group S such that A is normal in S and is weakly closed in F, C_S(A)=A, O^{p'}(Aut_F(A))=G, A is not normal in F, and O^{p'}(F)=F. We describe one procedure for doing this that works in many cases: by first showing that F=<N_F(A),C_F(Z(S))>, and then analyzing C_F(Z(S)). For example, when p=3 and (A,G) is isomorphic to one of the pairs (E_{3^4},A_6), (E_{3^5},M_{11}), or (E_{3^6},2M_{12}), then this procedure shows that F must be isomorphic to the 3-fusion system of one of the simple groups known to contain an extension of A by G. Full Abstract

Bob Oliver (Paris 13) (Wednesday talk): From complex reflection groups to fusion systems.

Abstract: We describe the procedure developed by many authors for starting with a complex reflection group realizable over the p-adics, and via p-compact groups, turning it into a fusion system over a discrete p-toral group and a family of fusion systems over finite p-groups. Full Abstract

Chris Parker (Birmingham): Computing fusion systems with Magma.

Albert Ruiz (UAB Barcelona): Classifying p-local compact groups over a fixed discrete p-toral group

Abstract: Classifying all saturated fusion systems over a fixed finite (or p-toral) p-group seems to be one way of getting "exotic'' examples of p-local finite (or compact) groups. In the finite case, the meaning of "exotic'' is very clear and has been used since p-local finite groups where defined by C. Broto, R. Levi and B. Oliver. In the infinite case, we must precise what does it mean to be "exotic''. In this talk we will see a classification of p-local compact groups over some special discrete p-toral groups (joint work with Bob Oliver) which include a family of p-local compact groups which are not p-compact groups (joint work with Alex Gonzalez and Toni Lozano).

Sergey Shpectorov (Birmingham): Fusion laws for algebras and representations.

Abstract: In a recent joint paper with De Medts, Peacock and Van Couvenberghe, we introduced the category of fusion laws. The initial motivation for this was its application to the theory of axial algebras, a new class of non-associative algebras inherently related to groups. However, during the work on the paper, we came across interesting connections with the representation theory of groups. In the talk, I will introduce the basics of fusion laws and discuss the connections with axial algebras and with representation theory.

Markus Szymik (NTNU Trondheim): What is the spectral sequence of a group extension?

Abstract: Group extensions are classified by cohomology and there are spectral sequences for the cohomology of extension groups. How many are there? How different are they? What are the most critical test cases? In this talk, I will present a topological context for these questions that leads to answers to these questions in joint work with Frank Neumann (in progress).

Antonio Viruel (Malaga): Acyclic 2-dimensional complexes and Quillen’s conjecture.

Abstract: Let G be a finite group and A_p(G) be the poset of nontrivial elementary abelian p-subgroups of G. Quillen conjectured that O_p(G) is nontrivial if A_p(G) is contractible. We prove that O_p(G) is not trivial for any group G admitting a G-invariant acyclic p-subgroup complex of dimension 2. In particular, it follows that Quillen's conjecture holds for groups of p-rank 3. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher--Smith. This is a joint work with Kevin Ivan Piterman and Ivan Sadofschi Costa.