# Workshop on Homotopy Theory

## Date and Location

The workshop will take place on May 5th, 2018 at the University of Regensburg in the SFB Seminar room M311.

## Aims and Scope

The aim of this workshop is to bring together early career researchers working in topics related to homotopy theory to discuss recent advances, foster collaboration, and say farewell.

## Lectures

• Hoang-Kim Nguyen (University of Regensburg)

## Financial Support

Limited financial support for the conference and younger participants has been provided by the SPP 1786: Homotopy theory and algebraic geometry and SFB 1085: Higher Invariants.

## Practical Information

MAP: Points of Interest.

PUBLIC TRANSIT: Local bus system. There are many useful buses typically, but the 6 will suffice for getting to and from the city center and the university (the nearest stop is the Universität stop). The work shop is on the third floor of the math building. The above maps may prove helpful.

ACCOMMODATION: We will reserve a block of hotel rooms at the [ www.hotel-central-regensburg.de Hotel Central].

One can also check the standard alternatives:

2. Airbnb

• TRAVEL: One can reach the university by following the instructions here.
• INTERNET: Access to eduroam is available throughout the university. If by some miracle, Eduroam works for you, then you can use that. Otherwise the Bavaria offers free wifi as well through BayernWLAN.

## Organizers

• Justin Noel (Regensburg)
• Georgios Raptis (Regensburg)

## Schedule

• All lectures will take place in the SFB Seminar Room in the math wing M311
• Registration and coffee breaks will be held nearby.
• On Saturday evening we will gather at Alte Linde.
Saturday
09:40 - 10:00 Registration and Coffee
10:00 - 10:55 Malkiewich
10:55 - 11:10 Coffee
11:10 - 12:05 Patchkoria
12:15 - 13:45 Lunch (Unikat)
13:45 - 14:40 Frankland
14:40 - 14:55 Coffee
14:55 - 15:50 Nguyen
15:50 - 16:00 Coffee
16:00 - 16:55 Land
16:55 - 17:05 Clean up
18:00 - 22:00 Dinner (Alte Linde)

## Registered Participants

• Peter Arndt (Düsseldorf)
• Uli Bunke (Regensburg)
• Denis-Charles Cisinski (Regensburg)
• Martin Frankland (Osnabrück)
• Moritz Groth (Bonn)
• Cary Malkiewich (Binghamton / MPIM)
• Niko Naumann (Regensburg)
• Hoang-Kim Nguyen (Regensburg)
• Justin Noel (Regensburg)
• Irakli Patchkoria (Bonn)
• Matan Prasma (Regensburg)
• Georgios Raptis (Regensburg)
• Daniel Schaeppi (Regensburg)
• Florian Strunk (Regensburg)
• Sean Tilson (Wuppertal)
• Koen van Woerden (Regensburg)

## Titles and Abstracts

• Cary Malkiewich (Periodic orbits and topological restriction homology):

I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be unwound into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.

• Irakli Patchkoria (Computations in real topological cyclic homology):

In this talk I will present recent computations in real topological cyclic homology (TCR). We will see that for a broad class of commutative rings with involution, the ring of components of TCR can be described in terms of Witt vectors over the components of the real topological Hochschild homology. We will also discuss connections to polynomial maps and extra functoriality of Witt vectors. Finally, we will mention the recent progress on TCR(F_p). This is joint work with E. Dotto and K. Moi.

• Martin Frankland (Towards the dual motivic Steenrod algebra in positive characteristic): Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Ostvaer generalized those results to a base field of characteristic $p$, as long as the coefficients are mod $\ell$ with $\ell \neq p$. The case $\ell = p$ remains conjectural.

In joint work with Markus Spitzweck, we show that over a base field of characteristic $p$, the conjectured form of the mod $p$ dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins-Morel-Hoyois isomorphism, a statement about the algebraic cobordism spectrum MGL.

• Hoang-Kim Nguyen (Adjoint functor theorems in infinity-categories)
• Markus Land (Genuine L-theory):

In this talk I will give an overview of what this project is about and how it relates (hopefully) to higher Grothendieck-Witt theory and real algebraic K-theory.