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# The Transatlantic Transchromatic Homotopy Theory Conference II

## Warm-up conference

**Unfortunately, due to the current coronavirus pandemic, we have had to postpone the conference until 2021. We are currently also organizing a one-day conference to be held via Zoom on Monday 3rd August 2020. **

## Registration for the warm-up conference

Registration for the warm-up conference is now closed. For late registrations, please contact Tobias Barthel with your name, institution, and email address

# Schedule for warm-up conference

**Date: August 3, 2020. **

Introduction: 15:55 CEST, 9:55 EDT.

Speaker 1 (16:00 CEST, 10:00 EDT): Georg Tamme, Regensburg University, * Some results about chromatic localizations of algebraic K-theory *

A classical result of Waldhausen states that algebraic K-theory preserves 1-connective rational equivalences between connective ring spectra. This has important consequences for the rational algebraic K-theory of rings which are Z-torsion. I will present a generalization of Waldhausen's result to chromatic localizations at higher height and discuss some applications, in particular to vanishing results for chromatically localized algebraic K-theory of ring spectra. This is joint work with Markus Land and Lennart Meier.

Speaker 2 (17:00 CEST, 11:00 EDT): Nicola Bellumat, University of Sheffield. *Iterated chromatic localization*

The work of Ravenel, Devinatz, Hopkins and Smith in the 80s provided the basis of chromatic homotopy theory: its protagonists are the Morava theories E(n) and K(n), whose associated Bousfield localizations give us optimal means to decompose the stable homotopy category. It comes naturally to wonder how the compositions of such localizations behave: there are classical results regarding the relationship of the Bousfield classes of wedges of the above spectra which lead us to expect some kind of regularity. In this talk I will present a joint work with N. Strickland which provides a positive result in this direction: we show that, fixed an upper bound n for the chromatic height, the compositions of localizations with respect to spectra which are wedges of K(i), for i less than or equal n, are only finitely many up to isomorphism.

Speaker 3 (17:40 CEST, 11:40 EDT): Eva Belmont, Northwestern. *ℝ-Motivic homotopy theory and the Mahowald invariant. *

The Mahowald invariant is a highly nontrivial map (with indeterminacy) from the homotopy groups of spheres to itself with deep connections to chromatic homotopy theory. In this talk I will discuss a variant of the Mahowald invariant that can be computed using knowledge of the R-motivic stable homotopy groups of spheres, and discuss its comparison to the classical Mahowald invariant. This is joint work with Dan Isaksen.

Speaker 4 (18:20 CEST, 12:20 EDT): William Balderrama, UIUC. *Approximating higher algebra by derived algebra.*

The higher algebraic structures arising in homotopy theory are difficult to access directly. However, a general heuristic tells us that by understanding the operations which act naturally on the homotopy groups of these objects, one can build obstruction theories and other tools for working with them. In this talk, I will describe a conceptual framework in which this heuristic is realized. I will focus on the particular example of K(h)-local commutative algebras over Lubin-Tate spectra, and give some applications to highly structured orientation theory.

Speaker 5 (19:00 CEST, 13:00 EDT): Paul VanKoughnett, Purdue. *The action of the Morava stabilizer group on the Devinatz-Hopkins ring*

The cohomology of the Morava stabilizer group acting on the Lubin-Tate ring is a difficult input to computations in the homotopy of the K(n)-local sphere. Devinatz and Hopkins introduced an approximation to the Lubin-Tate ring, isomorphic to it after divided-power completion, but on which the action of the Morava stabilizer group becomes linear. We present some calculations of the cohomology of this action, and relate them to known results about the Lubin-Tate ring.

Speaker 6 (19:40 CEST, 13:40 EDT): Vesna Stojanoska, UIUC *Some K(n)-local Spanier-Whitehead duals*

This is a report on work joint with Beaudry, Goerss, and Hopkins. Fix an odd prime p, and let n=p-1. The Morava stabilizer group then has maximal finite subgroups G whose p-Sylow subgroups have order p. Its action on the Lubin-Tate spectrum E defines chromatically interesting but manageable spectra of homotopy fixed points E^{hG}. In this talk, I will explain how to determine that E^{hG} is K(n)-local Spanier-Whitehead self-dual with a shift of -n^2(2p+1). Some crucial theoretical input, like the so-called Linearization Hypothesis (proved by Clausen in general, or as part of this collaboration in certain cases) will be assumed but not dwelled on. I will focus on the concrete applications, featuring good old characteristic classes.

Post-conference: Discussion rooms.

## Aims and Scope

Transchromatic phenomena appear in a variety of contexts. These include stable and unstable homotopy theory, higher category theory, topological field theories, and arithmetic geometry. The aim of the conference is to bring together people from all of these areas in order to understand the relationship between each others work and to further demystify the appearance of transchromatic patterns in such disparate areas.

## Invited lectures

- Agnes Beaudry (Colorado)

- Mark Behrens (Notre Dame)

- Dan Berwick-Evans (UIUC)

- Irina Bobkova (Texas A&M)

- Natalia Castellana (UAB Barcelona)

- Paul Goerss (Northwestern)

- John Greenlees (Warwick)

- David Hansen (MPIM Bonn)

- Hans-Werner Henn (Strasbourg)

- Nick Kuhn (Virginia)

- Jack Morava (JHU)

- Vesna Stojanoska (UIUC)

- Neil Strickland (Sheffield)

- Georg Tamme (Regensburg)

Titles and Abstracts available [here].

## Financial Support

Limited financial support for the conference and younger participants has been provided by the SFB 1085: Higher Invariants. Funding is also available through the NSF grant DMS-1955705 for early-career US based participants. The deadline for full consideration of
funding applications has been extended until **April 5th, 2020**.

## Practical Information

MAP: Points of Interest.

PUBLIC TRANSIT: Local bus system and route map.

There are many useful buses typically, but the 6 will suffice for getting to and from the city center and the conference room.

One can also walk to the conference room from the 'Universitat' bus stop on campus, which is better served by the bus system. The 6, 11, C6, and C11 buses all travel between the HBF (central station) and the Universitat bus stop. The 6, 11, C6, and C11 towards campus can all be caught here and one of them stops at this stop about every 5 minutes during working hours.

You can pay for a bus ride on entering the bus or purchase a strip of tickets from one of the ticket machines. A strip of tickets costs 9.00 euros and it allows five bus rides.

ACCOMMODATION:

Regensburg is a tourist destination and we encourage guests to book their rooms as early as possible. Here is a list of hotels in the area:

1. Hotel Kaiserhof am Dom (In the city center, must take a bus to the University.)

2. Hotel Muenchner Hof (In the city center, must take a bus to the University.)

3. Hotel Apollo (Near the conference, but limited eating options.)

4. Hotel Jakob (In the center, must take a bus to the University.)

5. Hotel Central (In the city center, must take a bus to the University.)

One can also check the standard alternatives:

1. Hotels.com

2. Airbnb

TRAVEL: One can reach the university by following the instructions here.

INTERNET: Access to **eduroam** is available throughout the mathematics building. For those without eduroam access we will obtain a temporary guest account through the university.

## Organizers

- Tobias Barthel (Bonn)
- Drew Heard (NTNU)
- Niko Naumann (Regensburg)
- Nathaniel Stapleton (Kentucky)