# Talbot 2016: Equivariant stable homotopy theory and the Kervaire invariant

## Mentored by Douglas Ravenel and Mike Hill

###
April 3-9, 2016

Herriman, UT

The list of talks is given below, and more detailed descriptions of each, along with references, can be found on this syllabus prepared by the mentors.

The canonical reference is, of course, Hill-Hopkins-Ravenel's paper. The authors have also written an Introduction to the Arf-Kervaire invariant problem.

Eva's notes from the workshop are here: (pdf, tex sources).

Doug Ravenel's notes from the workshop can be found here.

## Monday

**Talk 1.**Historical introduction (Doug Ravenel). (slides)

**Talk 2.**Overview of the proof of the Kervaire invariant theorem (Mike Hill).

**Talk 3.**The odd primary Arf invariant theorem (Foling Zou).

**Talk 4.**Review of category theory, including Kan extensions, enriched functors and the Day convolution. (Yexin Qu)

**Talk 5.**Introduction to equivariant homotopy theory I. (J.D. Quigley)

## Tuesday

**Talk 6.**Introduction to equivariant homotopy theory II. (Fei Xie)

**Talk 7.**Model Categories I. (Ugur Yigit)

**Talk 8.**Model Categories II. (Alex Yarosh)

**Talk 9.**The Mandell-May definition of $G$-spectra including Yoneda spectra (a.k.a. $S^{-V}$), the tautological presentation and smash product. (Renee Hoekzema)

## Wednesday

**Talk 10.**The homotopy of $G$-spectra, including long exact sequences of homotopy groups for fiber and cofiber sequences. (Allen Yuan)

**Talk 11.**The positive complete model structure and why we need it. (Hood Chatham)

## Thursday

**Talk 12.**The norm functor, multiplicative properties, and their relation to geometric fixed points. (Benjamin Boehme)

**Talk 13.**Slice filtration and slice spectral sequence. (Koen van Woerden)

**Talk 14.**Dugger's computation for real $K$-theory. (Agnes Beaudry)

**Talk 15.**The construction of $MU_R$ and its slice differentials. (Eva Belmont)

## Friday

**Talk 16.**The Slice, Reduction, and Gap theorems. (Akhil Mathew)

**Talk 17.**The periodicity theorem. (Mingcong Zeng)

**Talk 18.**The detection theorem. (Zhouli Xu)

**Talk 19.**Future directions (Doug Ravenel)