Motivic power operations acting on the mod-\( \ell \) motivic cohomology of smooth \( \mathbb{F}_p \)-schemes were constructed by Voevodsky and played a key role in his proof of the Milnor and Bloch–Kato conjectures. In this talk, I will describe upcoming joint work with Elden Elmanto, in which we extend Voevodsky's operations on mod-\( p \) motivic cohomology from characteristic \( 0 \) to characteristic \( p \), thereby obtaining the long-sought-after motivic power operations at the characteristic. These operations satisfy all expected properties (except possibly generating, together with the Bockstein, all the endomorphisms of \( H\mathbb{F}_p \)). If time permits, I will also discuss how to extend these operations to the motivic cohomology of singular \( \mathbb{F}_p \)-schemes, as recently defined by Elmanto–Morrow and Kelly–Saito. Additionally, I may mention other applications, such as defining obstructions to lifting motivic cohomology classes to algebraic cobordism classes, solving the motivic Steenrod problem for singular varieties at the characteristic, and identifying algebraic cycles that are not smoothable at the characteristic.

Synthetic homotopy theory is a general framework for constructing interesting contexts for doing homotopy theory: using the data of a spectral sequence in some category $\mathcal{C}$, one can construct another category which can be viewed as a deformation of $\mathcal{C}$. The motivating example of such a theory (due to Gheorghe, Wang, and Xu) is ($p$-complete, cellular) $\mathbb{C}$-motivic spectra, which is a deformation of $\mathcal{C}=\mathrm{Sp}$. Burklund, Hahn, and Senger showed that $\mathbb{R}$-motivic homotopy theory is a deformation of the category of $C_2$-equivariant spectra. I will discuss work in progress to construct deformations of $G$-equivariant homotopy theory for other groups $G$. This is joint with Gabriel Angelini-Knoll, Mark Behrens, Hana Jia Kong, and Maxwell Johnson.

About thirty years ago, Bökstedt–Madsen investigated and described the object named in the title, after completion at an odd prime number p, motivated by its relation to the p-complete algebraic K-theory of the p-adic integers. I will report on joint work with Sanath Devalapurkar, in which we describe a new way of analyzing this object, based on a description of the p-complete topological Hochschild homology of the integers in terms of the image of J spectrum.

Finite dimensional vector spaces over a field k admit a purely categorical description as the dualizable objects inside the symmetric monoidal category Vect of all k-vector spaces. Via the cobordism hypothesis, this provides a complete classification of one dimensional topological field theories with target Vect. The goal of this talk is to present an analogous picture in the context of categorified linear algebra and two dimensional topological field theory. More precisely, I will give a complete classification of the fully dualizable objects in a certain symmetric monoidal 2-category of categories linear over k. We will also discuss variants of this story in which k is allowed to be a commutative ring or even a connective commutative ring spectrum, where (derived) Brauer groups play a central role.

The category of elements (a discrete version of the Grothendieck construction) gives an equivalence between the categories of functors from a fixed category C to Set, and of discrete fibrations over C. It is intimately linked with the study of representable functors, as a well-known result shows that a functor is representable if and only if its category of elements has a terminal object. Hence, the category of elements gives us a way to characterize representable functors, and through them, universal properties, which are then used to understand key constructions such as adjunctions and (co)limits.

In this talk we will introduce a category of elements for enriched functors, and explain how this enjoys all of the desired (enriched) categorical properties. This is based on joint work with Lyne Moser and Paula Verdugo.

Waldhausen's A-theory of spaces — an extension of Quillen's higher algebraic K-theory of rings — is central to the study of moduli spaces of manifolds. In this talk, we will discuss a generalization of A-theory that takes as input an orbifold and which we expect to have rich geometric applications in analogy with the manifold setting. As an orbifold is locally described in terms of equivariant data, our discussion will involve an equivariant generalization of A-theory, due to Malkiewich–Merling, as well as related work in equivariant A-theory that is joint with David Chan, Anish Chedalavada, and Andres Mejia.

This talk will in some sense be a follow-up to the last talk I gave at UChicago (that being said, I will not assume any background from that talk). In this talk I will focus on some consequences of the existence of the category of synthetic cyclotomic spectra (constructed in joint work with Antieau), including: a new proof of the cohomological bounds on syntomic cohomology due to Antieau-Mathew-Morrow-Nikolaus; a filtered version of a fiber sequence originally due to Devalapurkar-Raksit; and a higher Traverso theorem.

In 1999, Mark Mahowald and Charles Rezk introduced a class of spectra which are particularly amenable to understanding using the classical Adams spectral sequence, called fp-spectra. As first described by Rognes, these play a pivotal role in generalizing Quillen-Lichtenbaum conjectures to the setting of ring spectra.

The Quillen-Lichtenbaum conjectures were proven for truncated Brown-Peterson spectra by Dylan Wilson and Jeremy Hahn in 2021, who in this way discovered the first highly non-obvious example of an fp-spectrum in the form of algebraic K-theory. This led them to ask about a general structure result for fp-spectra, and to conjecture that they can all be built out of particularly simple ones.

I will talk about recent joint work with David Lee where we prove a monochromatic analogue of the Hahn-Wilson conjecture, and deduce the original conjecture at height one.

In this talk I will discuss how to generalize the axioms of braided monoidal structure, and more generally En algebras, for higher categories. In particular, by repeated use of the Eckmann-Hilton argument, we will derive a ''minimal'' set of axioms for braided monoidal structures for general m-categories.

Drinfeld has defined a certain formal group closely related to the prismatization of G_m and to the theory of Chern classes in prismatic cohomology. I'll explain how to identify this formal group with one obtained from THH and chromatic homotopy theory. I'll also explain how this identification, together with a Lichtenbaum–Quillen property for syntomic cohomology, yields an "algebraization" of this formal group conjectured by Drinfeld.

Classically, the Algebraic $K$-theory of spaces ($A$-theory) is used to study manifold topology from a homotopical perspective. In the equivariant setting, Malkiewich and Merling constructed a genuine $G$-spectrum $A_G(X)$ together with an assembly map $\Sigma^{\infty}_{G}X \to A_G(X)$ whose cofiber deloops to the equivariant "stable $h$-cobordism space" for a smooth $G$-manifold $M$. Non-equivariantly, Waldhausen’s original vision for $A$-theory was an interpretation that initiated work in "brave new algebra" that happens on the level of spectra. Moreover, he gave an interpretation of $A$-theory analogous to the theory of rings where we take the $K$-theory of $(\mathbb S[ \Omega X])$, thinking of this as a "spherical group ring" in analogy with $\mathbb Z[\pi_1 X]$ . A natural question is whether or not there is a similar story for $A_G(X)$, and we propose a model that gives a positive answer to this question. As an application, we construct an equivariant trace map to a version of equivariant topological Hochschild Homology possessing the correct properties in analogy with the identification of $THH(\mathbb S[\Omega X])$ as the free loop space of $X$.

Recently, Annala, Hoyois, and Iwasa have defined and studied the $\mathbf{P}^1$-homotopy theory, a generalization of $\mathbf{A}^1$-homotopy theory that does not require $\mathbf{A}^1$ to be contractible, but only requires pointed $\mathbf{P}^1$ to be invertible. This makes it applicable to non-$\mathbf{A}^1$-invariant cohomology theories such as Hodge, de Rham, and prismatic. I will recall basic facts in their theory, and construct the $\mathbf{P}^1$-motivic Gysin map, thus giving a uniform construction for the Gysin maps of the above cohomology theories that are automatically functorial. If time permits, I will also explain its basic property, as well as its applications such as the Atiyah duality and the Steinberg relation in $\mathbf{P}^1$-homotopy theory.

Adams spectral sequence has been the most important computational tool in the stable homotopy theory whereas chromatic homotopy theory has allowed us to detect large scale patterns in the homotopy groups. It’s only natural to wish for a tool that will allow us to do both these procedures simultaneously. In this talk, I will show that by doing chromatic homotopy theory in ($H\mathbb{F}_{p}$ based) synthetic spectra, a gadget interpolating between topology and algebra which can also be thought of as a categorification of Adams spectral sequences, we can achieve that! This will naturally lead us to the study of tensor ideals of compact objects of synthetic spectra. I will end by stating and proving a version of Periodicity Theorem in this setting.

We give a gentle introduction to the theory of Du Bois complexes (which is the cdh sheafification of the Kähler differentials) and Du Bois singularities. We explain how recent development of Hodge theory leads to a better understanding of the Du Bois complexes, and to the introduction and investigation of higher Du Bois singularities. We end with some speculations of how these developments might be useful for the study of K-regularities in algebraic K-theory.

Classically, Dieudonné theory offers a linear algebraic classification of finite group schemes and p-divisible groups over a perfect field of characteristic p>0. In this talk, I will discuss generalizations of this story from the perspective of p-adic cohomology theory (such as crystalline cohomology, and the newly developed prismatic cohomology due to Bhatt--Scholze) of classifying stacks.