# 2015 Workshop

## Invariants of Structured Ring Spectra with Bjørn Dundas and Birgit Richter

28 June - 4 July, 2015

Klosters, Switzerland

### About

The first European Talbot workshop took place in Klosters, Switzerland from June 28 to July 4, 2015. The goal was to bring together a group of 25-30 graduate students and post-docs to work on a focused topic under the guidance of two senior mentors.

Most of the talks were given by the participants, with enough free time in the afternoon and evenings for further discussions and interaction. The character of the workshop was expository in nature, starting with the basic ideas and leading to a survey of the most recent developments in the field. Since all participants will stay together at a Chalet, jointly responsible for cooking and cleaning, we hope to have created an informal and inspiring atmosphere.

### Topic

The topic of the workshop is ``Invariants of structured ring spectra'' and will be mentored by Prof. Bjørn Ian Dundas and Prof. Birgit Richter.

Structured ring spectra are a homotopical enhancement of the algebraic theory of associative and commutative rings. They are of fundamental importance for applications both within algebraic topology and beyond. Based on a good symmetric monoidal category of spectra, it is possible to generalize concepts from ordinary algebra and algebraic geometry to the homotopical setting. For example, one can define Picard and Brauer groups of ring spectra, study their Galois theory, or compute (topological) Hochschild (co)homology.

Many algebraic and arithmetic properties of rings are encoded in their algebraic K-theory. Similarly, the algebraic K-theory of a ring spectrum contains a wealth of important information, and interacts in highly interesting ways with the chromatic perspective on stable homotopy theory. Moreover, trace methods have been employed very successfully in reducing computations to easier invariants like topological cyclic homology.

After a review of the some of the abstract aspects of the foundations for structured ring spectra, we will introduce and study several of the aforementioned invariants, with a particular emphasis on algebraic K-theory.

### Syllabus

Here is a preliminary version of the syllabus.

### Schedule

**Monday:**

10:30-Introductory talk (Birgit Richter/Bjørn Dundas)

12-Thom Spectra and Bousfield localization (Piotr Pstragowski)

15-Infinite loop space machines and K-theory (Alyson Bittner)

17-(DG and simplicial) rings and structured ring spectra (Magdalena Kedziorek)

**Tuesday:**

9:30-Galois extensions of structured ring spectra (Bregje Pauweis)

11-Picard and Brauer groups (Drew Heard)

14-Hochschild and topological Hochschild homology (Amalie Høgenhaven)

16-Higher and iterated THH (Bruno Stonek)

**Wednesday**

9:30-Topological André-Quillen (co)homology and Postnikov towers of commutative S-algebras (Markus Hausmann)

11-Obstruction theory (Bogdan Gheorghe)

14-Excursion

**Thursday:**

9:30-Algebraic K-theory of rings and ring spectra: introduction (Sarah Yeakel)

11-Algebraic K-theory of rings and ring spectra: big theorems (Gabe Angelini-Knoll)

14-The Dennis trace map (Cary Malkiewich)

16-Topological cyclic homology and the cyclotomic trace map (Valentin Krasontovitsch)

**Friday:**

9:30-Algebraic K-theory of ring spectra: geometric features of Algebraic K-theory (Mark Penney)

11-Algebraic K-theory of ring spectra: localization sequences and logarithmic structures (Calvin Woo)

14-Rognes' redshift conjecture (Bjørn Dundas)

16-Discussion: What are future directions? (led by Mentors)

### Application

The application deadline has now passed.

### Funding

The local expenses for all participants will be covered by the workshop, in particular the costs for accommodation and meals. Limited travel funds might become available, but we encourage participants to seek other sources of financial support for this.

### Contact

For questions or suggestions, please send an email to the organizers organisers@europeantalbot.org.

This workshop is organized by Tobias Barthel and Sean Tilson, and its concept has been inspired by the US Talbot workshop. We gratefully acknowledge funding and support from the Homotopy Theory Group of the EPFL and the Conférence Universitaire de la Suisse Occidentale, especially Kathryn Hess, the Danish National Research Foundation through the Centre for Symmetry and Deformation, University of Copenhagen, and the Research Council of Norway.