Homotopy theory in honor of Paul Goerss
20-24 March 2023 @ Northwestern
Confirmed speakers
Agnes Beaudry (Colorado)
Mark Behrens (Notre Dame)
John Greenlees (Warwick)
Hans–Werner Henn (Strasbourg)
Kathryn Hess (EPFL)
Mike Hill (UCLA)
Mike Hopkins (Harvard)
Marc Hoyois (Regensburg)
Dan Isaksen (Wayne State)
Rick Jardine (Western Ontario)
Tyler Lawson (Minnesota)
Jacob Lurie (IAS)
Doug Ravenel (Rochester)
Charles Rezk (Illinois)
Tomer Schlank (MIT/Hebrew)
Nat Stapleton (Kentucky)
Vesna Stojanoska (Illinois)
Zhouli Xu (San Diego)
Organizers
Benjamin Antieau and John Francis.
Registration
Please register here. Funding is no longer available.
Warning: there is spam email going out to some people about conference hotel booking. We will announce details from our @northwestern.edu addresses when the time comes.
Funding
Limited support is available for reimbursing participants’ lodging and travel expenses. This support is primarily from students and postdocs based in the United States. Decisions will be made by 10 February 2023.
Lodging
All speakers and participants are asked to book their own hotel, whether or not they will be reimbursed. Evanston has several hotels within walking distance of Northwestern, including Hyatt House Evanston, Hilton Garden Inn Evanston, and Hilton Orrington. The nicest is probably Hyatt House.
Travel
Northwestern is accessible via Amtrak through Chicago Union Station and via plane through Chicago O’Hare or Midway airports. One can take a cab from any of these stations or take public transportation on the Red or Purple ‘L’ Lines or on the UP-N Line of the Metra.
Parking
Parking is available at either the South or North Campus Parking Garage for $9/day. The South Garage is a 5-10 minute walk from the location of the conference, the North Garage is a 15-20 minute walk. More options are available, but for these a pass must be purchased at the parking office at the South Campus garage. For more details, see here.
Room
Talks will take place in Harris 107 [Google Maps].
Reception
The reception will take place from 1700 to 1900 Monday evening in the Atrium Dining Room at the Allen Center [Google Maps] and is open to everyone attending the conference.
Tentative schedule
M | Tu | W | Th | F |
---|---|---|---|---|
0900 Coffee | 0900 Coffee | 0900 Coffee | 0900 Coffee | 0900 Coffee |
0930 Rezk | 0930 Behrens | 0930 Lurie | 0930 Ravenel | 0930 Stojanoska |
1030 Coffee | 1030 Coffee | 1030 Coffee | 1030 Coffee | 1030 Coffee |
1130 Hill | 1130 Hopkins | 1130 Hoyois | 1130 Xu | 1100 Hess |
1230 Lunch | 1230 Lunch | Free afternoon | 1230 Lunch | 1200 Break |
1415 Isaksen | 1430 Henn | 1430 Stapleton | 1215 Greenlees | |
1515 Coffee | 1530 Coffee | 1530 Coffee | 1315 Conference Ends | |
1545 Lawson | 1630 Beaudry | 1630 Schlank | ||
1700 Reception |
Titles and abstracts
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 collaboration with many folks over the years, including Beaudy, Bhattacharya, Bobkova, Culver, Hill, Hopkins, Mahowald, Ormsby, Petersen, Quigley, Stapleton, Stojanoska, Xu.
Beaudry, The big computation. In the last ten years, our understanding of $K(2)$-local homotopy theory at the prime $2$ has increased by leaps and bounds. Many advancements were made possible by the duality resolution of Bobkova–Goerss–Henn–Mahowald–Rezk. Resolutionary techniques have opened the door to computations of the cohomology of the Morava stabilizer group $\mathbb{G}_2$ with coefficients in the homotopy groups of Morava E-theory $E_2$. They have also given rise to low-stem computations of the homotopy groups of the $K(2)$-local sphere. In turn, these computations have given us a better understanding of chromatic splitting, of the $K(2)$-local Picard group, and of duality in the $K(2)$-local category.
In this talk, I will give a revisionist history of a small part of this now long story, with a focus on how the duality resolution and chromatic splitting have shaped the politics of this 2-primary saga. The work presented is based on collaborations with Irina Bobkova, Paul Goerss, Hans-Werner Henn, Viet-Cuong Pham and Vesna Stojanoska. The talk also borrows from various works of Bohmann, Goscinny, Hopkins, Mahowald, Miller, Ravenel and Uderzo.
Greenlees, Rational equivariant cohomology theories for compact Lie groups. Several structural questions have emerged at least twice in topology: once in chromatic homotopy theory and once in equivariant topology (completions and localization, fracture squares, Balmer spectra, support, telescope conjecture, sheaves, filtrations, ….). In the chromatic world they arise in hard-core form, and in equivariant topology they reach a benign algebraic manifestation in the rational case. My talk is from this gentler world.
The overall project is to build an algebraic model for rational G-equivariant cohomology theories for all compact Lie groups G, and when G is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of G. The talk will focus on structural features of the expected model for general G such as those above, and feature recent joint work with Balchin and Barthel.
Henn, On the Brown Comenetz dual of the K(2)-local sphere at the prime 2 (joint work 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. 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. For small primes the result of Hopkins and Gross determines I_n modulo an “error term”. For n=1 every odd prime is sufficiently large and the case of the prime 2 has been understood for almost 30 years. For n>2 comparatively little is known if the prime is small. For n=2 every prime bigger then 3 is sufficiently large. The case p=3 had been settled in joint work with Paul Goerss several years ago. This talk is a report on work in progress with Paul Goerss on the case 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. We use chromatic splitting in order to narrow down the choices for the error term.
Hess, The universal Hochschild shadow: from bicategories to $(\infty, 2)$-categories, (joint with N. Rasekh and with K. Adamyk, T. Gerhardt, I. Klang, and H. Kong). The theory of shadows, first introduced by Ponto, is an axiomatic, bicategorical framework that generalizes (topological) Hochschild homology and satisfies analogous important properties, such as Morita invariance. I’ll explain how to generalize Berman’s extension of Hochschild homology to bicategories in order to prove that there is an equivalence between functors out of the Hochschild homology of a bicategory and shadows on that bicategory, from which it follows that Hochschild homology of bicategories actually provides a universal shadow on bicategories and which enables us to formulate Morita invariance functorially. I’ll then describe the infinity categorical generalization of this story, parts of which are still conjectural.
Hill, Equivariant approaches to chromatic homotopy. I’ll talk about how developments in equivariant homotopy theory over the last decade have resulted in new ways to study and understand Goerssian questions in chromatic homotopy. This is a preliminary report on work with various subsets of Beaudry, Bobkova, Lawson, Shi, Stojanoska, & Zeng.
Hopkins, Cohomology of motivic Eilenberg-MacLane spaces. I will describe joint work with Tom Bachmann on the cohomology of motivic Eilenberg-MacLane spaces, and give several applications.
Hoyois, Non-A^1-homotopy theory. Recently, there have been several attempts at developing extensions of motivic homotopy theory that include non-A^1-invariant phenomena, such as the algebraic K-theory of singular schemes or de Rham cohomology in positive characteristic. These are usually based on extensions of the category of schemes itself, such as schemes with modulus or log structures. In joint work with Toni Annala and Ryomei Iwasa, we consider a very naive extension of stable motivic homotopy theory, in which we simply remove the A^1-invariance axiom. We show that many basic results in A^1-homotopy theory can be proved in this context, using the invertibility of P^1 in a clever way. We construct in particular a non-A^1-invariant refinement of algebraic cobordism, which is related to algebraic K-theory by a Conner-Floyd isomorphism.
Isaksen, Bimotivic homotopy theory. I will discuss a possible new approach to computing the stable homotopy groups. The basic idea is to combine the Adams and Adams-Novikov spectral sequences into one computable object.
Lawson, Filtrations and their descendants. Our understanding of the homotopy theory of CW-complexes used to be heavily based on the cellular approximation theorem. In this talk we’ll take inspiration from this to define an operation taking a filtered space X to a “descendant” filtered space DX. We’ll discuss how, when we have categories of filtered objects, applying D to the function spaces has a “page-turning” effect on the homotopy theory, giving a low-tech way to turn a category of filtered objects into an E_2-category. Unstably, we can recover spaces of functions between CW-complexes; stably, we can construct a near relative of Pstragowski’s synthetic spectra from the category of modules over a spectral Rees ring.
Lurie, Lie algebras in stable homotopy theory. In this talk, I’ll review the classical theory of Lie algebras and discuss an “equation-free” approach which can be extended to the setting of homotopy-coherent mathematics. If time permits, I’ll discuss some connections with the calculus of functors.
Ravenel, Hiking in the Alps: C_p-fixed points of Lubin-Tate spectra. This is joint work with Mike Hill and Mike Hopkins. We resume our long lost study of the homotopy fixed point spectrum of C_p acting on the Lubin-Tate spectrum E_n when n is divisible by p-1 for a prime p.
Rezk, Reflections on free colimit completions. In this talk I’ll describe some basic but elegant observations about free adjoining classes of colimits to infinity-categories. Then I’ll give an interpretation of these results in terms of a ‘global’ analogue of presheaves.
Schlank, Lubin Tate spectra as algebraically closed fields. The Goerss-Hopkins-Miller theorem gives an $\bE_\infty$-structure on Lubin-Tate spectra. In this talk, I shall discuss a defining property of Lubin-Tate spectra attached to algebraically closed fields among all $K(n)$ (or $T(n)$) local $\bE_\infty$-rings. We show that they are characterized exactly as those that satisfy an analog of Hilbert’s Nullstellensatz. This suggests that one can consider such Lubin-Tate spectra as ``chromatic `algebraically closed fields’. In addition, we show that every non-zero $T(n)$-local $\bE_{\infty}$-ring $R$ admits an $\bE_\infty$-ring map to such a Lubin–Tate theory. If time permits, I shall present some consequences, for example, constructing $\bE_{\infty}$ complex orientations of algebraically closed Lubin–Tate theories and proving redshift for the algebraic $\mathrm{K}$-theory of arbitrary $\bE_{\infty}$-rings. This is a joint work with R. Burklund and A. Yuan.
Stapleton, A universal relation between multiplicative and additive power operations. This talk is in the realm of global equivariant algebra. We will give a notion of “integer valued class functions on the symmetric group” that makes sense for an arbitrary global Green functor. If that global Green functor is equipped with the structure of a global power functor, we will show that this notion can be used to provide a natural home for an exponential relation between multiplicative and additive power operations. Finally, we will show that under an extra hypothesis, this relation is universal. This is joint work in progress with the Kentucky Bourbon Seminar.
Stojanoska, Toward the Brauer group of topological modular forms. The Brauer group of an E∞ 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.
Xu, The Adams differentials on the classes h_j^3. In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes h_j, resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel proved that the classes h_j^2 support non-trivial differentials for j \geq 7, resolving the celebrated Kervaire invariant one problem. The precise differentials on the classes h_j^2 for j \geq 7 and the fate of h_6^2 remains unknown.
I will talk about joint work with Robert Burklund: In Adams filtration 3, we prove an infinite family of non-trivial d_4-differentials on the classes h_j^3 for j \geq 6, confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory – C-motivic stable homotopy theory and F_2-synthetic homotopy theory – both in an essential way. Along the way, we also show that h_j^2 survives to the Adams E_5-page and that h_6^2 survives to the Adams E_9-page.
Acknowledgments
The organizers thank the Northwestern Department of Mathematics for funding. The drawing above is due to Fomenko and is reproduced from Mathematical Impressions, by Fomenko.
Conference mug and shirt
Mug on Zazzle and Shirt on Zazzle.