A peculiarity of the $\infty$-categories literature is that proofs are often written without reference to a concrete definition of an $\infty$-category, a practice that creates an impediment to formalization. We describe three broad strategies that would make $\infty$-category theory formalizable, which may be described as "analytic," "axiomatic," and "synthetic." We then highlight two parallel ongoing collaborative efforts to formalize $\infty$-category theory in two different proof assistants: the "axiomatic" theory in Lean and the "synthetic" theory in Rzk. We show some sample formalized proofs to highlight the advantages and drawbacks of each approach and explain how you could contribute to this effort. This involves joint work with Mario Carneiro, Nikolai Kudasov, Dominic Verity, Jonathan Weinberger, and many others.

To compute the cohomology of the Steenrod algebra or more general algebras, we can use the May(-type) spectral sequences or minimal resolutions to do the calculation. These two methods are both useful and give us different insights about the Ext groups. However, when you have both results from these two methods computing the same Ext group, it is very difficult to compare them and people usually rely on some ad hoc methods in low dimensions. In this talk, I will introduce an algorithm based on Gröbner basis to obtain the May spectral sequence from a minimal resolution.

It is well known that operads in a closed symmetric monoidal category may be viewed as monoids in symmetric sequences. In topology, it is often sensible to work with unital operads and their (reduced) monads. I will discuss a variant of symmetric sequences, which we call the $\Lambda$-sequences, and a monoidal product called the Kelly product, so that monoids in $\Lambda$-sequences give unital operads. We also introduce a method to embed any symmetric monoidal category in a category where finite colimits commute with the tensor product, thus generalizing the machinery to symmetric monoidal categories not necessarily being closed. This is based on joint works with Peter May, Ruoqi Zhang and Aowen Fan.

I will report on joint work with Michael Batanin and Florian De Leger studying the homotopy theory of the Grothendieck construction, given a base category $\mathcal B$ and a functor $F$ from ${\mathcal B}^{op}$ to $C\!AT$. When $\cal B$ is an appropriate category of operad-like objects, and $F(O)$ is the category of $O$-algebras, we produce a model structure on the Grothendieck construction. From such a model structure on the category of pairs $(O,A)$, we produce a "horizontal" (semi-)model structure on the base $\mathcal B$ and "vertical" (semi-)model structures on the fibers $F(O)$. This recovers in a unified framework all known (semi-)model structures on categories of operads and their algebras, and produces new model structures, e.g., on the category of twisted modular operads. Additionally, we study when these model structures are left proper, and when a weak equivalence in the base $\mathcal B$ gives rise to a Quillen equivalence of fibers. Applications include change of rings, rectification of operad-algebras, and strictification for categorical structures. The relevant arXiv papers are at arXiv:2311.07320 and arXiv:2311.07322.

The Grothendieck–Teichmüller group is a mysterious pro-algebraic group which arises as an automorphism group of rationalized little disks operads. It also has a description in terms of Kontsevich's graph complexes. I will describe a filtration of $QS^1$ whose associated graded is a model for the ring of functions on the Grothendieck–Teichmüller group. Based on joint work with Brown, Chan, and Payne.

In this talk about joint work with V. Burghardt, P. Hu and P. Somberg, I will discuss "perverse" $t$-structures on equivariant spectra coming from separate shifts on each isotropy, glued together using an analogue of the "6-functor formalism" of constructible sheaves. This connects with a program recently formulated by P. Scholze, leading, in the second part of my talk, to a discussion of ongoing joint work with B. Roytman on structures arising from the May–Guillou approach to equivariant spectra via infinity-Mackey functors, and other related topics.

Effective Cartier divisors underpin Katz and Mazur's approach to the arithmetic theory of the moduli spaces of elliptic curves, after Drinfeld and Deligne. In this talk, with motivation from chromatic homotopy theory and power operations, I'll report on joint work with Xuecai Ma in which we define relative effective Cartier divisors for a spectral Deligne–Mumford stack. Specifically, given a commutative algebra $R$ over the sphere spectrum $\mathbb S$, we prove that, as a functor from connective $R$-algebras to topological spaces, the space of such divisors is representable. We then solve various moduli problems of level structures on spectral abelian varieties. In particular, we obtain higher-homotopical refinement for finite levels of the Lubin–Tate tower as commutative $\mathbb S$-algebras, which generalize Morava, Hopkins, Miller, Goerss, and Lurie's spectral realization at the ground level. Moreover, passing to the infinite level and then descending along the equivariantly isomorphic Drinfeld tower, we obtain a Jacquet–Langlands dual to the Morava E-theory spectrum, along with homotopy fixed point spectral sequences dual to those studied by Devinatz and Hopkins. These serve as potential tools for computing higher-periodic homotopy types from pro-étale cohomology of $p$-adic general linear groups.

Representation theory in characteristic $0$ typically addresses different questions than algebraic topology. What the two fields do have in common is that they both can be approached by studying certain specific categorical structures. I will mostly talk about the method of interpolated algebraic geometry, which I discovered by generalizing P. Deligne's interpolated categories of representations of algebraic groups. In the main part of my talk, I will focus on a particular application to a complete computation of the conjectured Howe duality correspondence (theta correspondence) for orthogonal and symplectic groups over finite fields (previously studied by A. Aubert, J. Michel, R. Rouquier, S. Gurevich, R. Howe, S.-Y. Pan, Z. Yun, and others).

In the late 1930's (around the time Peter May was born), P.A. Smith began the study of the fixed points of actions of finite $p$-groups on finite complexes using mod $p$ homology. In particular, with $H$ a subgroup of $G$, whenever $X$ is a finite $G$-CW complex such that $X^H$ is $H{\mathbb Z}/p$-acyclic, then so is $X^G$. A decade later, E.E. Floyd proved the stronger result that for any finite $G$ complex $X$, $\dim H_*(X^H;{\mathbb Z}/p)$ will be at least as big as $\dim H_*(X^G;{\mathbb Z}/p)$.

Recently many people have been studying chromatic versions of these theorems with the Morava K-theories replacing mod $p$ homology in the above statements. One can ask for what $m$ and $n$ is it true that if $X^H$ is $K(m)$-acyclic then $X^G$ must be $K(n)$-acyclic, and one can similarly ask about analogues of Floyd's theorem.

This first question has a nice answer when $G$ is a cyclic group. Then we can show that a chromatic Smith theorem for $(G,H,n,m)$ implies the analogous chromatic Floyd theorem.

In our talk, we will quickly survey this, and then focus on applications:

1. a fixed point theorem for $C_2$ actions on $\rm SU(3)/SO(3)$.
2. lower bounds for $\dim K(n)_*({\rm Gr}_d({\mathbb R}^m))$, which seem to be exact.
3. lower bounds for 'blue shift numbers' which determine the topology of the Balmer spectrum of $G$-equivariant spectra for some $p$-groups $G$.

There are many open questions here, and if there is time, we will state a couple of these.

Some of this work is joint with Chris Lloyd, and some is joint with William Balderrama.

This talk will describe my on-going program with Mohammed Abouzaid to use modern homotopical algebra and equivariant stable homotopy theory to produce refined invariants of symplectic manifolds. The goal is to try to give a broad overview of what's going on and in particular emphasize where homotopy theory plays an important role.

The Kervaire invariant is the obstruction for a framed cobordism class to have a homotopy sphere. It is important to know if this invariant is non-trivial when classifying homotopy spheres in a given dimension. We will give an account on recent joint work with Lin and Xu solving the last case of the Kervaire invariant problem in dimension 126.

In this talk, I will give an overview of the proof that $h_6^2$ survives in the Adams spectral sequence. This is joint work with Weinan Lin and Guozhen Wang.

The computation of Equivariant bordism spectra remains largely open, especially in the non-abelian case. I will explain a surprising injectivity statement for the map induced in complex bordism by the procedure which equips to a $G$-manifold with the trivial $G$-action. Such a result holds both for geometric and homotopical bordism, but fails for unoriented and oriented bordism. The proof does not use sophisticated tools from homotopy theory, relying instead on refinements of Hironaka's resolution of singularities theorem.

In this talk, we will discuss several computational structures that are present in the equivariant slice spectral sequence. These structures are obtained by analyzing the equivariant slice filtration using the generalized Tate diagram of Greenlees and May. As an application, we will utilize these structures to investigate norms of Real bordism theories and their quotients. These quotients hold significant importance in Hill–Hopkins–Ravenel's resolution of the Kervaire invariant one problem, as well as in the study of fixed points of Lubin–Tate theories by finite subgroups of the Morava stabilizer group. Using these structural insights, we prove periodicity, vanishing lines, and transchromatic theorems. This talk contains joint work with Zhipeng Duan, Mike Hill, Guchuan Li, Yutao Liu, Lennart Meier, Guozhen Wang, Zhouli Xu, Guoqi Yan, and Mingcong Zeng.

The general topic is that of constructing algebraic models of rational $G$-spectra for a compact Lie group $G$. The first example of this (if you don't count Serre's model in the case of the trivial group) is in my work with Peter May (Appendix to the Tate volume, published 30 years ago in 1995). It states that when $G$ is finite, the category of rational $G$-spectra is a product of blocks, one for each conjugacy class of subgroups $H$. The factor corresponding to $H$ is ${\mathbb Q}[W_G(H)]$-modules, where $W_G(H)=N_G(H)/H$.

Last year, at IWOAT24 I described the structural features of the category of rational $G$-spectra for arbitrary compact Lie groups $G$. This year I want to emphasize that the purpose of the model is to give a concrete and practical method of calculation and construction. Accordingly I will focus on $G={\rm SU(3)}$ and describe the algebraic model: the group is small enough to let us be completely explicit but the methods apply rather generally.

[2501.(06914, 11200, 15584), 2502.(00959, 06017)]

Topological Hochschild homology, an invariant of ring spectra, is the realization of a cyclic object defined using Connes' cyclic category and it carries an action of the circle. Real topological Hochschild homology, an invariant of ring spectra with involution, is the realization of a dihedral object defined using the dihedral category and it carries an action of ${\rm O}(2)$. In this talk, we describe a simultaneous generalization of these constructions, a topological version of homology which takes as input rings with twisted group action, which generalize rings with involution. A new example of interest of this construction is quaternionic topological Hochschild homology, which carries a ${\rm Pin}(2)$-action. This is joint work with Gabriel Angelini-Knoll and Maximilien Péroux.

The classical stable homotopy groups display rich and intricate periodicity patterns. In the motivic setting, certain exotic periodicity classes arise that have no classical counterparts. In this talk, I will discuss one such phenomenon–$w_1$-periodicity–in the $\mathbb C$-motivic sphere, focusing on specific classes. This is joint work with Dan Isaksen, Guchuan Li, Yangyang Ruan, and Heyi Zhu.

Equivariant topological Hochschild homology of a commutative algebra in positive characteristic (over a primary cyclic group for the same prime) is a module over the constant Eilenberg–Mac Lane spectrum. Therefore, it has a model in the derived category of modules over the constant Green functor. In this talk, I will discuss how one can get a description of such a model for smooth algebras in positive characteristic using descent to semiperfect rings. This is joint work with V. Burghardt, I. Kriz and P. Somberg, expanding our previous work on the case of polynomial algebras.

Homotopical combinatorics is a recent offshoot of equivariant homotopy theory that studies transfer systems and related structures. These are combinatorial gadgets that correspond to homotopy types of $N_\infty$ operads, and a basic problem in the area is to compute the lattice $\mathbf{Tr}(G)$ of all $G$-transfer systems for a given finite group $G$. These lattices become complicated very quickly, and only a handful of general computations are known. In this talk, I will present a square of weak Grothendieck fibrations that systematizes previous work, and which will hopefully lead to new calculations of $\mathbf{Tr}(G)$. This is joint work with Mike Hill.