Draft Schedule

The programme is still evolving - offers of talks definitely welcomed!

Clark Barwick: Equivariant stable homotopy theory via parametrized higher category theory

This sequence of talks is an exposition of ajoint monograph project with Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah

Introductory talk: Transfers in equivariant stable homotopy theory

In this talk, I will explain how to model the seemingly very delicate topological act of stabilization with respect to representation spheres of groups with purely algebraic structures - Mackey functors - and to rewire the whole of equivariant homotopy theory accordingly. This has two benefits: (1) Stripping out these structures permits us to get extremely refined information about – and universal characterizations of – the basic constructions of equivariant stable homotopy theory. (2) At the same time, we are now able to untether equivariant stable homotopy theory from the world of groups; this opens the door to many more interesting structures and many more interactions with other areas.

Modes of equivariance

In this talk, I will dive deeper into the study of the core structures of equivariant homotopy theory. I will introduce orbital ∞-categories and the concomitant parametrized ∞-category theories. We will look at some universal properties of the parametrized ∞-categories of B-spaces and B-spectra for an orbital ∞-category B. We will find that there are two interesting sorts of functors between orbital ∞-categories – left and right orbital functors. We will describe how these account for the different sorts of "fixed point functors" one encounters. Throughout this talk, I'll weave in as many examples as I can think of.

Some technical aspects of parametrized higher category theory

It's one thing to say how parametrized ∞-categories should work, but how do you make everything precise in a way that it can be used? In this talk, I'll introduce the necessary language, I'll explain some of the tricky bits of the theory, and I'll show all of this working on examples. I won't assume any tremendously detailed knowledge of any particular model of ∞-categories, however.

Parametrized higher algebra

In this talk, I'll introduce parametrized ∞-operads and parametrized symmetric monoidal ∞-categories, and I'll explain how these concepts operate in the examples of primary interest.

Abstracts of talks

Stefan Schwede (Universitaet Bonn)

Equivariant bordism from the global perspective, I & II (pdf file of talk)

Global homotopy theory is, informally speaking, equivariant homotopy theory in which all compact Lie groups acts at once on a space or a spectrum, in a compatible way. In this talk I will advertise a rigorous and reasonably simple formalism to make this precise, using orthogonal specctra. I will then illustrate the formalism by a geometrically motivated example, namely equivariant bordism of smooth manifolds.

Jay Shah (MIT)

Stability and additivity in parametrized higher category theory

There are two equivalent ways to stabilize G-spaces to obtain G-spectra (for G a finite group). On the one hand, one inverts all of the representation spheres; on the other hand, one can form the usual stabilization and then enforce an equivariant additivity condition. We investigate extensions of these procedures to the more general context of cocartesian fibrations over certain base ∞-categories.

John Berman (University of Virginia)

Burnside categories encoding structure on symmetric monoidal categories

A number of closely related constructions have at times gone by the name `Burnside category'. We show that these different constructions arise as 2-Lawvere theories for various kinds of structured symmetric monoidal categories (cartesian monoidal or additive, for example). More generally, we describe a correspondence between smashing localizations and enriched Burnside categories. A major benefit of this correspondence is that we can directly introduce group actions to some variants of the Burnside category, immediately recovering Guillou-May and Barwick's G-spectra and Hill-Hopkins and Barwick et al.'s G-symmetric monoidal categories, and obtaining new notions of G-cartesian monoidal and G-additive categories.

Saul Glasman (Princeton IAS)

Calculus and equivariance

Functors from the category of spectra which are n-excisive in the sense of Goodwillie can be thought of as Mackey functors indexed by the category of finite sets of cardinality at most n and surjections. The proof of this result, which is still a work in progress, contains several interesting ideas, and I'll give a detailed sketch.

Assaf Libman (University of Aberdeen)

Stable decompositions of classifying spaces (of compact Lie groups and p-local finite groups).

[last updated 08/12/2015]