JMM 2025 Homotopy theory Special Session
This year, the JMM will happen in Seattle on January 8-11, with a special session on homotopy theory on January 10-11. The official AMS website can be found here. In addition, there will be a satellite session at the University of Washington on Sunday, January 12.
We hope to see you there!
Friday morning
(617, Seattle Convention Center)- 8:00am Equivariant Weiss Calculus
Yang Hu - The classical Weiss calculus is an important tool for studying functors from vector spaces to topological spaces, of which notable examples include V -> BO(V), V -> BU(V), V -> BTop(V), etc. Each such functor has an associated tower of fibrations indexed by the natrual numbers, called the Taylor tower, whose layers are infinite loop spaces (i.e., cohomology theories). Weiss calculus is therefore regarded as a framework of studying unstable problems using tools from stable homotopy theory. In a joint work with Prasit Bhattacharya, we present a theory of G-equivariant Weiss calculus, where G can be any finite group. In our framework, the Taylor towers are indexed by G-representations and the layers are classified by genuine equivariant cohomology theories. Time permitting, we will discuss potential applications.
- 8:30am A synthetic approach for differentials in equivariant slice spectral sequences
Yuchen Wu - The theory of synthetic spectra has led to significant breakthroughs in the computation of Adams-type spectral sequences. The formalism underlying these synthetic techniques is also applicable to the study of other spectral sequences arising from filtered spectra. In this talk, I will employ synthetic methods to compute some differentials in equivariant slice spectral sequences for the Hill–Hopkins–Ravenel theories.
- 9:00am Algebraic geometry of Tambara functors
Danika Van Niel - Algebraic geometry and algebraic topology have long standing connections which have birthed important fields, techniques, and results. For example, equivariant tools have been used to explore questions in enumerative geometry, and motivic homotopy theory combines the study of algebraic geometry and homotopy theory. In this talk, we will introduce Tambara functors, an equivariant analogue to rings, and discuss equivariant analogues to classical definitions which are part of the field of algebraic geometry. For example, we will discuss the spectrum of a Tambara functor as defined by Nakaoka, and the definition of a ghost map of a Tambara functor as defined by Thevenaz. We will then discuss results which define an equivariant analogue of the affine line, and allude to our results about spectra of various Tambara functors, joint with: David Chan, David Mehrle, J.D. Quigley, and Ben Spitz.
- 9:30am Applications of Equivariant Homotopy Theory to the Mapping Class Groups of Some 4-Manifolds
Scott Tilton - The (equivariant) Bauer-Furuta invariant is a 4-manifold invariant living in (equivariant) stable homotopy groups. It is a refinement of the Seiberg-Witten invariant, and the families version has been used to distinguish families of 4-manifolds. This talk expands on uses of the Pin(2)-equivariant families Bauer-Furuta invariant to some important 4-manifolds.
- 10:00am Classifying fields in C_p^n-equivariant algebra
Noah Wisdom - For a finite group, G-Tambara functors are basic objects in equivariant algebra and appear as the structure adherent to the homotopy groups of a G-E_infty ring spectrum. Nakaoka defined G-Tambara fields and explained how to check whether a given G-Tambara functor is a field. For G = Z/p, we provide a recipe to construct an arbitrary Tambara field, discovering that in characteristic q =/ p a Tambara field is completely determined by a choice of group action on a field. In characteristic p, the Frobenius endomorphism and Artin-Schreier theory give rise to interesting examples. Our methods are very general and provide a partial classification of G-Tambara fields for finite groups G.
- 10:30am Computations with motivic spectral sequence for THH
David Lee - I will talk about how we can use the motivic spectral sequence to compute the integral homotopy groups of the topological Hochschild homologies of the integers, the Adams summand, and conjecturally, the truncated Brown-Peterson spectra.
- 11:00am Modeling Equivariant Simplicial Sets with Simplicial Coalgebras
Sofia Rose Martinez Alberga - Given a commutative ring R, a pi_1-R-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an R-homology equivalence between universal covers. When R is an algebraically closed field, Rivera and Raptis described a full and faithful model for the homotopy theory of spaces up to pi_1-R-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the cobar functor. In this article, we prove a G-equivariant analog of this statement through generalizations of a celebrated theorem of Elmendorf. We also prove a more general result about modeling G-simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras.
- 11:30am Equivariant Dyer-Lashof Operations
Alexander Waugh - For any G-ring spectrum R, there is an algebra of stable power operations which act on the underlying spectrum of any E_infty-algebra in R-modules. When R is HF_p, the algebra of such operations is well understood and is often called the Dyer-Lashof algebra. Having a full computation of this algebra of operations has lead to many interesting results, such as the Hopkins-Mahowald theorem. G-equivariantly, \underline{HF}_p is analogous to HF_p. Little is known about the associated Dyer-Lashof algebras for these rings, with the only complete computation being for G = C_2 and p=2. In this talk, we will recall a geometric construction of the classical power operations and generalize it to any finite group G. We will then examine how these operations behave under taking geometric fixed points and, time permitting, construct two infinite families of \underline{HF}_2-Dyer-Lashof operations for any finite group G.
Friday afternoon
(Ballroom 6E, Seattle Convention Center)- (AMS Invited Address)
2:10PM Arithmetic aspects of enumerative geometry
Kirsten Wickelgren - Enumerative geometry answers questions such as "How many lines meet four lines in space?" and "how many conics pass through 8 general points of the plane?" Fixed answers to such questions (here, 2 and 12) are obtained by working over an algebraically closed field like the complex numbers. Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. Homotopy theory on the other hand, studies continuous deformations of maps. In its modern form, it provides a framework to study shape in more algebraic and analytic contexts. This talk will introduce some interactions of homotopy theory with the arithmetic of solutions to enumerative problems in geometry. The study of such interactions was initiated in joint work with Jesse Kass and independently by Marc Hoyois and Marc Levine.
Saturday morning
(617, Seattle Convention Center)- 8:00am Corepresentational functor calculi
Kaya Arro - FI-calculus is an infty-categorification of representation stability. We describe an axiomatic framework for a family of functor calculi, which we call corepresentational functor calculi, modeled on FI-calculus. This framework encompasses a “dual” calculus to the Goodwillie-Weiss embedding calculus and, conjecturally, the Weiss orthogonal and unitary calculi, as well as a range of other examples we discuss. As one expects, the axioms give rise to Taylor towers in which homogeneous layers are classified by certain Taylor coefficients. Examples of corepresentational functor calculi are easily proliferated: given a cartesian fibration whose codomain is equipped with a corepresentational functor calculus, we show that the calculus lifts to a calculus on the domain of the fibration, allowing for the extension of established calculi to more structured settings. We conclude by extending a result from FI-calculus describing additional structure carried by the Taylor coefficients of a functor to any corepresentational functor calculus lifted from FI-calculus along a right fibration (and we provide examples of such calculi).
- 8:30am Equivariant formal group laws and Quillen theorem
Yunze Lu - Quillen proves that the universal ring for formal group laws is isomorphic to the homotopy of complex cobordism MU. I will discuss equivariant complex oriented cohomology theories and discuss an equivariant version of Quillen’s theorem for abelian compact Lie groups.
- 9:00am Equivariant monodromy and moduli
Sidhanth Raman - Topological and enumerative problems that arise in algebraic geometry are often best studied in families, i.e. via moduli spaces. When questions such as "what is the shape of..." or "how many..." are asked in the presence of a group action, additional complexity arises when zooming in on the stacky sublocus of the moduli space. We will discuss ongoing joint work with Thomas Brazelton that synthesizes homotopical and Hodge theoretic methods in various algebro-geometric problems. To stay grounded, we will focus on one such problem: the study of symmetric cubic surfaces. We will explain why this symmetric moduli space is a Shimura curve, and compute the monodromy group of the 27 lines cover restricted to the symmetric locus.
- 9:30am Homology of generalized Hurwitz spaces via Fox-Neuwirth cells
Anh Trong Nam Hoang - Homological stability of Hurwitz spaces, certain finite covers of unordered configuration spaces of the plane, and their generalizations plays a fundamental role in the recent flurry of remarkable work connecting topology and number theory, such as Ellenberg-Venkatesh-Westerland, Liu-Wood-Zureick-Brown, and Ellenberg-Landesman. Recently, Ellenberg-Tran-Westerland introduced a new approach to study the homology of Hurwitz spaces, starting with integrating the classical Fox-Neuwirth stratification of configuration spaces into the twisted setting. In this talk, we will explain how to expand this framework to study the homology of some generalized Hurwitz spaces, which has applications in number theory.
- 10:00am Involutions and the Brauer group in derived algebraic geometry
Lucy Yang - The Brauer group is an important invariant of rings and schemes; it measures both the arithmetic and geometric complexity of its input. Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) have established a connection between the existence of (anti-)involutions on the central simple algebras used to define the Brauer group and 2-torsion Brauer classes. Moreover, the presence of more general forms of involutions is related to the behavior of Brauer classes under corestriction along quadratic extensions. In this work, we introduce and study a generalization of these ideas to derived algebraic geometry. We investigate how the data of an involution on is reflected in additional structure on its category of modules. Using the theory of Poincaré infinity-categories by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, we introduce involutive versions of the Picard and Brauer group and relate them to their non-involutive counterparts.
- 10:30am Nil K groups and a question of Bass
Noah Riggenbach - Algebraic K-theory, as defined by Quillen in 1971, is a machine which to any ring R builds a topological space K(R) whose topology and in particular its homotopy groups contain deep information about R. Using algebraic K-theory we can then study certain problems in algebra using tools from algebraic topology. An example of such a question, which in some form is due to Serre, is when projective modules over R[x,y] are of the form P[x, y] where P is a projective R-module. In this talk I will discuss work, joint with Elden Elmanto, which uses algebraic K-theory to study this and closely related questions.
- 11:00am On the Computability of Immersions
Helen Epelbaum - Suppose someone hands you a pair of smooth manifolds, and a smooth map between them. Can you decide whether there is an immersion homotopic to this map? We will reduce this to a question about lifting a certain bundle, and then use rational homotopy theoretic techniques to provide an algorithm when the codimension of the manifolds is odd.
- 11:30am Algebraically Closed Tambara Functors
Ben Spitz - Tambara functors are "equivariant generalizations" of commutative rings – for each finite group G, there is a notion of G-Tambara functor, and when G is trivial group, this notion coincides exactly with that of a commutative ring. It is then interesting to ask: which objects play the role of algebraically closed fields in the category of G-Tambara functors? There is a reasonable definition to work with; Burklund-Schlank-Yuan introduced the notion of "Nullstellensatzian objects" in a category C, and noted that the Nullstellensatzian objects in the category of commutative rings are precisely the algebraically closed fields. We present a classification of the Nullstellensatzian objects in the category of G-Tambara functors, and as a corollary show that the K-theory of Nullstellensatzian Tambara functors reduces to the K-theory of algebraically closed fields.
Saturday afternoon
(617, Seattle Convention Center)- 1:00pm Parameterized proper equivariant incomplete stable homotopy theories and their operadic algebras
Bar Roytman - Advances in the study of multiplicative structures in equivariant stable homotopy theory is largely responsible for the recent surge of activity in the area. An outgrowth of this is renewed interest in equivariant stable homotopy categories that lie between the naive and genuine ones. Many fundamental equivariant homotopical structures, such as loop space structures and spaces associated with representation theory, are prohibitively difficult to express with current combinatorial tools. Consequently, it is valuable to rely on model categories based on topological spaces to represent them. We develop the model categorical foundations of parameterized proper equivariant stable and unstable homotopy theories of Lie groups, including their operad and operadic algebra categories. For spaces in the parameterized setting, we use Johnstone’s topological topos, which we show to be equivalent to the category of light condensed sets. The models of the stable theories we construct are variants of -modules and orthogonal spectra. Our stabilizations are with respect to collections of representation spheres for semidirect extensions of the group of equivariance. The equivariant stable homotopy categories we obtain can be Quillen equivalent yet have nonequivalent categories of commutative algebras when the corresponding linear isometry operads are nonequivalent. We explain this and the role of linear isometry operads for more general algebra categories. This paper is the first in a series dedicated to justifying an obstruction theory for highly structured equivariant orientations from Thom spectra and its applications to Fujii–Landweber Real bordism.
- 1:30pm The duals of some higher real K-theories at p=2
Juan C. Moreno - Higher real K-theories are central objects in chromatic homotopy theory, where they capture important information about the K(n)-local category. Work of Beaudry-Goerss-Hopkins-Stojanoska shows that taking the dual of one of these theories amounts to a twist by a certain representation sphere. Interestingly, in every case known to the speaker a higher real K-theory ends up being self-dual up to some suspension. In this talk we identify the representation sphere twist in some cases, working always at the prime 2. Then we make use of equivariant computational techniques developed by Hill-Hopkins-Ravenel in their solution to the Kervaire invariant problem to identify an integer suspension shift at some low heights.
- 2:00pm The generalized Tate diagram of the equivariant slice filtration
Guoqi Yan - The generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, I will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of Hill—Hopkins—Ravenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and allows us to extract information about isomorphism regions between these equivariant filtrations. As an application, we will begin by proving a stratification theorem for the negative cone of the slice spectral sequence. Building upon the work of Meier—Shi—Zeng, we will then utilize this stratification to establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines within the negative cone of the slice spectral sequences associated with periodic Hill—Hopkins—Ravenel and Lubin—Tate theories. Finally, we utilize our analysis to identify duality phenomenon of the differentials in the equivariant slice spectral sequence. This is joint work with Yutao Liu and XiaoLin Danny Shi.
- 2:30pm Toward Computations of C_3-Equivariant Stable Stems
Yueshi Hou - In this talk, I will first survey and compare various existing computations of the C_2-equivariant stable stems, including those by Bredon, Landweber, Araki-Iriye, Behrens-Shah, and Guillou-Isaksen. Then, I will present some ongoing computations of the C_3-equivariant stable stems, which are based on generalizing some of the aforementioned techniques from the C_2 case. This is joint work with Shangjie Zhang.
- 3:00pm Towards the K(2)-local homotopy groups of RP^2∧CP^2∧HP^2 at p=2
Sihao Ma - Let Y_H = Sigma^-7 RP^2∧CP^2∧HP^2 be a type 1 spectrum with 8 cells. In this talk, I will talk about the computation of the algebraic duality spectral sequence associated to Y_H and the topological duality spectral sequence converging to the homotopy groups of E_C^(hS^1_C) \wedge Y_H, a spectrum closely related to L_K(2) Y_H.
- 3:30pm Transfer systems for rank two elementary Abelian groups
Christy Hazel - Transfer systems are combinatorial objects of interest due to their connections to equivariant homotopy theory. Using transfer systems, we are able to translate questions about equivariant commutativity into questions about subposets on a given lattice. In this talk, we’ll define transfer systems, discuss their connections to equivariant homotopy theory, and then highlight some recent results about transfer systems for rank two elementary abelian groups. This joint work with L. Bao, T. Karkos, A. Kessler, A. Nicolas, K. Ormsby, J. Park, C. Schleff, and S. Tilton was part of the 2023 eCHT REU.
- 4:00pm Twisted Atiyah-Bott-Shapiro Maps
Cameron Krulewski - The Atiyah-Bott-Shapiro orientation MTSpin -> KO produces a KO-theory Thom class from a spin manifold. Freed-Hopkins developed twisted versions MTH(s) -> Sigma^(-s) KO of this map for the special case of suspensions of KO-theory, and we generalize their construction. We use the fermionic group formalism, which is an approach that leverages the compatibility between twists of KO-theory and twists of spin bordism. While the classical ABS map is 7-connected, the same is not true for these twisted maps. We develop methods for computing the kernel and cokernel of these maps in low dimensions using Smith long exact sequences, which are long exact sequences in bordism induced from zero-section maps. They can offer insight into manifold generators for bordism groups. Our work is motivated by condensed matter physics, where twisted ABS maps play a central role in so-called free-to-interacting maps between KO-theory and certain invertible TQFTs. We show that the free-to-interacting maps we construct commute with another physical process termed dimensional reduction.
- 4:30pm Weight Filtrations and Derived Motivic Measures
Anubhav Nanavaty - Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher K groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of K_0) to so-called "derived" ones, i.e. on the level of K_i for all i. I will describe some of this work, which shows that if one closely studies how the Gillet-Soulé weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher K groups of varieties, and relating them to other interesting algebro-geometric objects in the literature.
Sunday
(Room C-38, Padelford Building, Department of Mathematics, University of Washington)- 9:30am Real kq-resolutions
Jackson Morris - A fundamental computation in stable homotopy theory is the bo-based Adams spectral sequence. Mahowald’s computation of the bo-resolution gives the v_1-periodic stable stems and proves the height 1 telescope conjecture at the prime 2. In motivic homotopy theory, the very effective spectrum kq plays an analogous structural role, and Culver-Quigley computed the kq-resolution over C. We compute the kq-resolution over R, then show how to use a result of Bachmann-Ostvaer in a novel way to compute the -cooperation algebra over. This is work in progess.
- 10:00am Homotopy Incoherent Norm Maps - Generalizing E_k-Operads
Ben Szczesny - Transfer systems and N_infty-operads play crucial roles in equivariant homotopy theory, encoding information about norms and transfers. In this talk, we present a novel method that lifts transfer systems to N_infty-operads by taking slices of coinduced operads. Unlike existing approaches to realizing N_infty-operads from transfer system data, our method is concrete and offers broader applicability. Notably, it can be used to construct equivariant operads with specified fixed point data that extend beyond the class of N_infty-operads. We will then explore various applications of this approach, demonstrating how it lifts key properties of transfer systems—such as joins—to the corresponding operads. This work not only simplifies certain constructions in equivariant operad theory but also provides new tools for building "incomplete" equivariant structures.
- 10:30am Applications Of Exodromy In Topology
Peter Haine - The exodromy theorem in topology says that the infinity-category of constructible sheaves on a suitably nice stratified space (X, P) is equivalent to the infinity-category of functors out of the exit-path infinity-category of (X, P). In this talk, we’ll explain two quick applications of the exodromy theorem. The first, joint with Mauro Porta and Jean-Baptiste Teyssier, is on the representability of derived moduli spaces of constructible sheaves. These moduli spaces are generalizations of character varieties. The second, joint with Qingyuan Bai, is a splitting result of localizing invariants of infinity-categories of constructible sheaves with coefficients in any dualizable stable presentable infinity-category.
- 11:00am Topological Hochschild Homology With Coefficients
Logan Radcliffe Hyslop - Let R be a connective (suitably structured) ring spectrum equipped with a ring map pi_0(R) -> F_p, where F_p is a finite field of prime order. Suppose further that the ring pi_*(F_p otimes_R F_p) is a quotient of the dual Steenrod algebra (e.g., if is a truncated Brown-Peterson spectrum). In this situation, we will explain how the Brun spectral sequence allows for a straightforward computation of the topological Hochschild homology of R with coefficients in F_p, THH(R, F_p).