University of Chicago
Department of Mathematics

Fusion systems are an abstraction of the p-local structure of finite groups: They model the inclusion of a Sylow p-subgroup in an ambient finite group, together with ambient conjugation (or "fusion") data. The work of Broto-Levi-Oliver has shown how these algebraic, almost combinatorial objects are closely related to the homotopy type of the p-completion of classifying spaces of finite groups. In this talk, I will examine how the fundamental notion of group actions on finite sets can be understood in the context of fusion systems. I will try to motivate the notion of a "fusion action system," as well as the basic properties they should satisfy. There is also a strong parallel with the work of BLO: Fusion action systems have a notion of "classifying space," which, in the presence of an actual ambient group G acting on a finite set X, reconstructs the Borel construction EG\times_G X up to p-completion. I shall finish with an overview of this classifying space story, and suggest some applications to fusion system theory.

In recent work of Baas-Dundas-Richter-Rognes, the authors prove that the classifying space of 2-vector bundles, K(Vect) is equivalent to the algebraic K-theory of the connective K-theory spectrum ku. In this talk we will show that K(Vect) is the group completion of the classifying space of the 2-category of 2-vector spaces, which is a symmetric monoidal 2-category. We will explain how to use the symmetric monoidal structure to produce a Gamma-2-category, which will give an infinite loop space structure on K(Vect). Then we will show that the equivalence of BDRR is a map of infinite loop spaces.

A celebrated result of Freed, Hopkins and Teleman relates the twisted equivariant K-groups of a compact Lie group G to the Verlinde algebras of G, making these K-groups highly interesting objects to study. In this talk, we explain how a twisted version of the Atiyah--Segal completion theorem connects the twisted equivariant K-theory of G to the Gruher--Salvatore string topology of BG, and discuss a conjecture linking these K-groups to another variant of string topology of BG defined by Chataur and Menichi.

This is joint work with Cristina Costoya. We construct a model structure in the category of orbifold groupoids and prove that its homotopy category is the category of orbifolds introduced by Moerdijk. To each orbifold groupoid we can associate a category fibered in groupoids (CFG) which is not necessarily a stack. We will say that an orbifold groupoid is a stack groupoid if the associated CFG is a stack. The fibrant objects of our model are the stack groupoids.

Abstract: In this talk I will present joint work with Radu Stancu, that gives a completely new way to look at fusion systems, via the double Burnside ring. More precisely every fusion system has a ``characteristic idempotent'' in the double Burnside ring, from which one can reconstruct the fusion system. Furthermore, characteristic idempotents are exactly those that satisfy a certain Frobenius reciprocity relation, and thus one obtains a surprising bijection between fusion systems and idempotents satisfying Frobenius reciprocity, giving rise to the new point of view on fusion systems. I will discuss the bijection, and then talk about how concepts such as fusion subsystems and normal extensions can be formulated in terms of characteristic idempotents. The bijection has interesting implications for the stable homotopy theory of classifying spaces, which will be discussed in the algebraic topology seminar on November 10th.

In recent work with Radu Stancu we proved a bijection between saturated fusion systems on a finite p-group and certain idempotents in its p-localized double Burnside ring. The Segal conjecture relates the double Burnside ring to stable maps between classifying spaces, and so we can regard saturated fusion systems as objects in stable homotopy theory. In the talk I will show how, adopting this point of view, we can answer a long-standing question on stable splittings of classifying spaces, and give a (best possible?) generalization of the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. Time permitting, I will present a new, simplified model for p-local finite groups.

A Pi-algebra is a graded group with additional structure that makes it look like the homotopy groups of a space. Given one such object A, one may ask if it can be realized topologically: Is there a space X such that pi_*(X) is isomorphic to A as a Pi-algebra, and if so, can we classify them? Work of Blanc-Dwyer-Goerss provided an obstruction theory to realizing a Pi-algebra, where the obstructions (to existence and uniqueness) live in certain Quillen cohomology groups of Pi-algebras. What do these groups look like, and can we compute them? We will tackle this question from the algebraic side, focusing on Quillen cohomology of truncated Pi-algebras. We will then use the obstruction theory to obtain results on the classification of certain 2-stage homotopy types, and compare them to what is known from other approaches.

Homotopy n-types are an important class of topological spaces: they amount to CW complexes whose homotopy groups vanish in dimension higher than n. The problem of modeling homotopy types is relevant both in higher category theory and homotopy theory and received contributions from both areas. There is a particularly simple model of homotopy types in the path connected case, consisting of n-fold categories internal to groups, also called catⁿ-groups. This model, however, has the disadvantage that is it does not have an algebraic description of the Postnikov decomposition nor it is easy to establish algebraically when a map of catⁿ-groups is a weak equivalence. In this talk we introduce a new model of connected n-types through a subcategory of catⁿ-groups, which we call weakly globular, for which the above issues are resolved in transparent way. We also describe other homotopical properties of this model, and discuss the relevance of these structures for higher category theory. In the second part of this talk we discuss the use of the notion of weak globularity to deal with the non-path connected case in low dimension (n=2) and we illustrate an application. The second part of this talk is joint work with David Blanc.