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The Transatlantic Transchromatic Homotopy Theory Conference II - Andy Baker 70

Dates and Location

The conference will take place during the five days July 31 -- August 4, 2023 at the University of Regensburg.

Aside from Wednesday afternoon, the conference will take place in lecture hall H51. Registration and coffee breaks will be held nearby.

If the speakers grant permission, we will stream the talks of the conference via Zoom.

Join Zoom Meeting https://uni-regensburg.zoom.us/j/66198197999?pwd=dytKRHJCUWhNaDNySENlNlZwNDJXUT09

Meeting ID: 661 9819 7999 Passcode: 324323

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. We will also celebrate Andy Baker's 70th birthday at the conference.

The conference will consist of 15 invited talks and a number of contributed talks.

Schedule

Time Monday Time Tuesday Time Wednesday Time Thursday Time Friday
9:30-10:00 Registration 9:30-10:20 Greenlees 9:30-10:20 Kuhn 9:30-10:20 Carmeli 9:30-10:20 Ravenel
10:00-10:50 Richter 10:20-10:40 Coffee break 10:20-10:40 Coffee break 10:20-10:40 Coffee break 10:20-10:40 Coffee break
10:50-11:10 Coffee break 10:40-11:30 Castellana 10:40-11:30 Behrens 10:40-11:30 Bobkova 10:40-11:30 Stojanoska
11:10-12:00 Strickland 11:40-12:10 Davies 11:30-13:00 Lunch 11:40-12:10 Deaton 11:40-12:30 Berwick-Evans
12:10-12:40 Subramanian 12:10-14:00 Lunch 12:30-14:00 Lunch Farewell
12:40-14:00 Lunch 14:00-14:50 Kong from 15:00 Biergarten Alte Linde 14:00-14:50 Henn
14:00-14:50 Burklund 15:00-15:30 G. Li 15:00-15:30 Balderrama
15.00-15.30 Levy

Invited lectures

Title and Abstracts:

Douglas Ravenel -- What is an infinity category?

This is an expository talk about infinity categories. I will give the definition (after reviewing simplicial sets) and describe the homotopy coherent nerve of the ordinary category of topological spaces in explicit detail. I will also give an example illustrating that ordinary colimits are the same as homotopy colimits. If time permits, I will also talk about the infinity category analog of Bousfield localization and define the infinity category of spectra.

Birgit Richter -- Loday Constructions of Tambara functors

Brun showed that pi_0 of every genuine commutative G ring spectrum is a G-Tambara functor. We define a Loday construction for G-Tambara functors for any finite group G. This definition builds on the Hill-Hopkins notion of a G-symmetric monoidal category and the work of Mazur, Hill-Mazur and Hoyer who prove that for any finite group and any G-Tambara functor R there is a compatible definition of tensoring a finite G-set X with R. We extend this to a tensor product of a G-Tambara functor with a finite simplicial G-set, defining the Loday construction this way. We investigate some of its properties and describe it in examples. This is joint work with Ayelet Lindenstrauss and Foling Zou.

Jack Davies -- Homotopical uniqueness of the topological q-expansion map

The work of Lurie in his series on elliptic cohomology has provided us with a spectral algebro-geometric interpretation of periodic topological modular forms TMF. What we still lack is such an interpretation for the dualisable Tmf and connective tmf forms. In this talk, we discuss the homotopical uniqueness of the transchromatic topological q-expansion map and some of its applications. In particular, we will see how this uniqueness, together with Lurie's construction of TMF, leads to operations on Tmf and tmf, as well as connective models for Behrens' Q(N) spectra away from the prime 2.

Robert Burklund -- Algebraic K theory, redshift and the telescope conjecture

I will discuss joint work with Jeremy Hahn, Ishan Levy and Tomer Schlank wherein we show that the algebraic K-theory of the K(1)-local sphere is a counterexample to the height 2 telescope conjecture.

William Balderrama -- The equivariant J-homomorphism and RO(G)-graded periodic phenomena

I will describe how the G-equivariant J-homomorphism can be "desuspended" in a way that produces nontrivial RO(G)-graded periodicities in equivariant stable homotopy theory, such as in the G-equivariant stable stems. When G = C_2, these are essentially versions of James periodicity, and I will explain how this recovers and unifies theorems of Bredon, Araki and Iriye, and Behrens and Shah.

Shachar Carmeli -- Higher Descent for Chromatically Localized Algebraic K-theory

I will describe work in preparation, joint with Ben-Moshe, Schlank, and Yanovski, proving descent for T(n+1)-local algebraic K-theory with respect to p-local π-finite group actions on T(n)-local categories, generalizing results of Thomason for height 0 and Clausen, Mathew, Naumann, and Noel for actions of discrete p-groups in arbitrary height. I will then discuss the compatibility of K-theory with the chromatic cyclotomic extensions from a previous work with Schlank and Yanovski, and how it gives a non-trivial example of hyperdescent for K(n+1)-local K-theory. I will also discuss the compatibility of K-theory with other constructions related to ambidexterity, such as the chromatic Fourier transform and the higher Kummer theory from a joint work with Barthel, Schlank, and Yanovski.

Nicholas Kuhn -- The equivalence of chromatic Smith and Floyd theorems, with applications

When G is a finite p-group, understanding the topology of the Balmer spectrum of the G-equivariant stable homotopy category amounts to determining the validity of chromatic Smith theorems, for each H<G, and pair (m,n). This says that whenever X is an appropriately finite G-complex, if X^G is K(n)* acyclic then X^H is K(m)* acyclic. If one fixes G, H, and n, one is trying to calculate the blue shift number (m-n), where m is smallest such that the chromatic Smith theorem holds. With Chris Lloyd, I've shown these imply stronger looking chromatic Floyd theorems which give comparisons between the dimensions of K(n)*(X^G) and K(m)*(X^H). This implication has interesting application when X = Gr_d(V), a Grassmanian associated to a representation V of G. Using the contrapositive formulation, examples of the form Gr_1(V) allow us to compute blue shift numbers for some infinite families of nonabelian groups, by getting better lower bounds than were previously known. Using the implication directly, together with known valid chromatic Smith theorems for cyclic groups, we can deduce nonequivariant results: lower bounds for dim K(n)*(Gr_d(R^m)), which seem to be exact.

Guchuan Li -- "Algebraic" approaches of computing homotopy groups of topological modular forms

The computation of homotopy groups of topological modular forms usually needs nontrivial topology information. In this talk, we present two new approaches of the 2-primary computation based on equivariant and motivic techniques respectively. These new approaches use more algebraic input and provide new information. In particular, the equivariant approach avoids the use of Toda brackets. The motivic approach settles a sign in the multiplicative structure, which is the last unresolved detail about the multiplicative structure in Bruner and Rognes' book. This talk is based on joint projects with Zhipeng Duan, Dan Isaksen, Hana Jia Kong, Yunze Lu, Yangyang Ruan, Guozhen Wang, and Heyi Zhu.

Vignesh Subramanian -- Fixed points via Tilting

Given an E_infty-algebra A over F_p, just as in classical algebra, there exists a homotopical coherent version of Frobenius on A, called the Tate valued Frobenius A -> A^{tC_p}. In this talk, we recall the notion Frobenius perfect F_p-algebra and construct a certain version of perfection A^flat, we call this construction Tilting and give an explicit formula for the computation of homotopy groups of the tilt via power operations. As an application, given X a finite G-CW complex where G is an elementary abelian group, we offer a recipe to recover the p-local homotopy type of the genuine fixed point X^{G} from the Borel equivariant cohomology of X. This application can be considered a categorification of Smith theory, which plays a significant role in the ideas surrounding proof of the Sullivan conjecture. This is joint work with Robert Burklund

Mark Behrens -- tmf resolutions at the prime 2

Around 20 years ago, Goerss, Henn, Mahowald, and Rezk started an industry of studying K(2)-local homotopy at bad primes using finite TMF resolutions. I will discuss how these resolutions detect a swath of elements at the prime 2 in the Isaksen-Wang-Xu range. This discussion involves a synthesis of work with/by many folks over the years, including Beaudy, Bhattacharya, Bobkova, Culver, Goerss, Henn, Hill, Hopkins, Mahowald, Ormsby, Petersen, Quigley, Stapleton, Stojanoska, and Xu.

John Greenlees -- Torsion models for Noetherian tensor triangulated categories

For suitably enhanced tensor triangulated categories with Noetherian Balmer spectrum one may construct adelic models by assembling data from information at each Balmer prime p. The data is morally of the form of complete modules over the p-localized p-completed unit object (though it may be more convenient to localize and complete the category itself): this would assemble abelian groups from p-complete modules over the p-adic integers and from rational vector spaces. An alternative is to use p-torsion modules as the information at p: this would assemble abelian groups from p-torsion modules over the p-adic integers and from rational vector spaces. The adelic models have better monoidal properties, but objects in torsion models may be more accessible. For example in the case of rational spectra equivariant for an r-torus there is an abelian adelic model of injective dimension r but the torsion abelian model is of injective dimension between r+1 and 2r (conjecturally 2r). (joint work with Scott Balchin, Luca Pol, Jordan Williamson).

Ishan Levy -- Some consequences of the failure of the telescope conjecture

I will describe work joint with Robert Burklund, Shachar Carmeli, Jeremy Hahn, Tomer Schlank, and Lior Yanovski about some consequences of the failure of the telescope conjecture. In particular I will explain that the average p-ranks of the stable stems are unbounded and that the Poincare duality class in the K(2) local homotopy of a finite complex lifts to a T(2)-local class.

Natalia Castellana Vila -- Descent in tensor triangular geometry

Given a tensor triangulated category (tt-category), one way to study it is by classifying its thick, smashing and localizing tensor ideals. Descent methods apply when one can reduce these problems to another tt-category via a tt-functor with good properties, e.g. base change with repect to a descendable commutative algebra. In this work we describe equalizer diagrams relating lattices of localizing and smashing ideals through base change, which yield to a coequalizer diagram for Balmer spectra. We apply these results to faithful Galois extensions. This is joint work with T. Barthel, D. Heard, N. Naumann, L. Pol and B. Sanders.

Daniel Berwick-Evans -- Quantum field theories and formal group laws

Hans-Werner Henn -- On the Brown Comenetz dual of the $K(2)$-local sphere at the prime 2 (joint work in progress with Paul Goerss)

The Brown Comenetz dual I of the sphere represents the functor which on a spectrum X is given by the Pontryagin dual of the 0-th homotopy group of X. For a prime p and a chromatic level n there is a K(n$-local version I_n of I. In particular, for a type n-complex X the homotopy groups of the function spectrum F(X,I_n) are given by the Pontryagin-dual of the homotopy groups of the K(n)-localization of X. By work of Hopkins and Gross the homotopy type of the spectra I_n for a prime p is determined by its Morava module if p is sufficiently large with respect to n and this Morava module is explicitly known. For small primes the result of Hopkins and Gross determines I_n modulo an “error term”. This talk is a report on work in progress with Paul Goerss on the case n=p=2. The “error term” is given by an element in the exotic Picard group which in this case is an explicitly known abelian group of order 2^9. Ideally we would like to nail down the homotopy type of the error term. This is what has been done successfully in the case n=1 and p=2. In the case n=2 and p=3 we had been able to characterize the error term in the exotic Picard group which in this case was isomorphic to (Z/3)^2. In the case under consideration we first use chromatic splitting in order to narrow down the choices for the error term and then propose to use specific information about the homotopy groups of a particular finite type 2-complex to completely determine the error term within the exotic Picard group.

Millie Rose Deaton -- Bialgebras, partitions, and class functions

I will introduce partition functors as a generalization of bialgebras and discuss a close relationship to global functors from equivariant homotopy theory. In this language, I will describe a kind of “class functions” for partition functors along with a universal property and rational isomorphism. When applied to the representation theory of symmetric groups, this recovers the usual integer valued class functions and a universal property for them.

Vesna Stojanoska -- Toward the Brauer group of topological modular forms

The Brauer group of an E_\infty ring spectrum R is the group of R-Azumaya algebras modulo Morita equivalence. It was introduced by Baker, Richter, and Szymik over 10 years ago, and it encodes interesting arithmetic information about R. In the last decade, a multitude of systematic approaches to its study have been developed (cf. work of Antieau, Gepner, Hopkins, Lawson, Lurie). Still, a lot of mysteries remain when it comes to the Brauer group of periodic ring spectra such as TMF. In joint work with Antieau and Meier, we study the subgroup of Br(TMF) consisting of those Azumaya algebras which are étale locally trivial. Among other things, we find that there are infinitely many 2-torsion elements in Br(TMF). I will explain where this calculation comes from, including the importance of understanding the Picard sheaf, rather than just the Picard group, in order to get a good handle on the Brauer group.

Conferece Picture

Transchromatic.jpg

Conference Videos

Financial Support

Limited financial support for the conference and younger participants has been provided by the SFB 1085: Higher Invariantsand by the NSF under grant no. DMS-1955705.


Conference Poster

Poster

Practical Information

MAP: Points of Interest.

Map of the University.

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 (nearest stops are either 'An der Kreuzbreite' or 'Neuprull').

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 and X4 buses all travel between the HBF (central station) and the Universitat bus stop. The 6, 11, X4 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 10.50 euros and it allows five bus rides.

ACCOMMODATION: We have reserved a block of hotel rooms at the Hotel Apollo. Participants will need to book and pay for their own accommodation, although the Hotel Apollo should be able to arrange shared rooms for participants with limited financial support (all pre-PhD participants requesting financial support should request a shared room). Please refer to PROMOTION CODE "SFB 1085" when making a reservation!

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 and the SFB building. For those without eduroam access we will obtain a temporary guest account through the university.


Organizers

List of participants

Tobias Barthel, Drew Heard, Niko Naumann, Luca Pol, Nat Stapleton, Mark Behrens, Nick Kuhn, Vesna Stojanoska, Natalia Castellana, Hans-Werner Henn, Irina Bobkova, Dan Berwick-Evans, John Greenlees, Doug Ravenel, Birgit Richter, Shachar Carmeli, Robert Burklund, Hana Jia Kong, Leonard Mushunje, Alexander Pacun, Marwa Mosallam, Millie Deaton, Guchuan Li, William Balderrama, Leonard Tokic, Tzu-Yi Yang, Vignesh Subramanian, Bhavna Joshi, Andy Baker, Jack Davies, Himanshu Yadav, Elie Alhajjar, Maxwell Johnson, Sabri Khadidja, Malthe Sporring, Jacob Lebovic, Christian Nassau, Lucas Piessevaux, Felix Nass, Neil Strickland, Nicholas Kuhn, Zachary Halladay, Samuel Hsu, Kartik Tandon, Marco Varisco, Mingyuan Hu, Thomas Blom, Laurent Smits, Shai Keidar, Shay Ben Moshe, Langwen Hui, Qi Zhu, Connor Grady, Naruki Masuda, Jonas Linssen, Elizabeth Tatum, Scott Balchin, Aras Ergus, Willow Bevington, Pier Federico Pacchiarotti, Jonathan Mann, Xu Jun, Prasit Bhattacharya, Ishan Levy, Sebastian Chenery, Yuli Rudyak, Julie Rasmusen, Viktor Burghardt, Foling Zou, Pavel Sechin, Venkata Sai Narayana Bavisetty, Javier Aguilar Martin, Piotr Pstragowski, Sil Linskens, Jana Nickel, Benjamin Dünzinger, Mark Backhaus, Cheikh Khoule, Jordan Williamson, Gabrielle Yangqing Li.

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