Tommy Lundemo: Ramified extensions of ring spectra

While several definitions attempt to capture the notions of (formally) etale and unramified morphisms of ring spectra, it has proven more difficult to provide a good distinction between tame and wild ramification in this context.
Starting from work of Rognes, I will explain how ideas from logarithmic geometry have proven useful in addressing this issue. For this, we will use that both the cotangent complex and Hochschild homology admit simultaneous logarithmic and spectral generalizations. Work by Binda, myself, Park, and Østvær relate Hochschild homology of log schemes to log differential forms. I have shown that this relationship extends to the spectral context. This reinforces long-lasting expectations regarding various extensions of ring spectra. Our motivating examples include real and complex K-theory spectra and truncated Brown–Peterson spectra. This is all closely related to work of Rognes–Sagave–Schlichtkrull and Höning–Richter, and I will make the precise relationships clear throughout the talk.

Jeremiah Heller: Power Operations and Normed Spectra

Bachmann–Hoyois introduce a theory of normed motivic spectra which play the role in motivic homotopy theory that E_oo spectra play in classical homotopy theory. In recently posted arXiv preprints, joint with Tom Bachmann and Elden Elmanto, we develop a theory of power operations on normed algebras over the mod-2 motivic cohomology spectrum and apply this theory to obtain splitting results for certain normed algebras. I'll discuss these results and some further open questions.

Denis-Charles Cisinski: The Universal coCartesian Fibration

Deep in the foundations of (higher) category theory theory is the straightening/unstraightening correspondence (the unstraightening direction being also known as the Grothendieck construction). In this talk, we will discuss a new construction of this correspondence, making it as tautological as possible (as opposed to using peculiar constructions such as homotopy coherent nerves). And we will also discuss several reformulations of it, in order to make it a robust concept that is easy to formulate in more general and more formal contexts, such as category theory interpreted in higher topoi (useful to formulate motivic/equivariant constructions) or suitable versions of dependent type theory. This is joint work with Kim Nguyen.

Rahul Gupta: Ramified class field theory and its application

The main aim of this field is to study the abelianized etale fundamental group of a quasi-projective variety in terms of a cycle-theoretic group. In this talk, we shall discuss the ramified class field theory of smooth quasi-projective varieties over finite fields. The main focus of the talk will be to use this theory to prove the failure of Nisnevich descent for the Chow groups with modulus. The talk will be based on joint works with Prof. Amalendu Krishna.

Peter Fiebig: Tilting modules and torsion phenomena

Given a root system and a prime number p we introduce a category X of “graded spaces with Lefschetz operators” over a ring A. Then we show that under a base change morphism from A to a field K this category specialises to representations of the hyperalgebra of a reductive group, if K is a field of positive characteristic, and of a quantum group at $p^l$-th root of unity, if K is the $p^l$-cyclotomic field. In this category we then study torsion phenomena (over the ring A) and construct for any highest weight a family of universal objects with certain torsion vanishing conditions. By varying these conditions we can interpolate between the Weyl modules (maximal torsion) and the tilting objects (no torsion). This construction might shed some light on the character generations philosophy of Lusztig and Lusztig–Williamson.

Ben Heuer: New approaches to p-adic non-abelian Hodge theory

I will start by giving a short introduction to p-adic non-abelian Hodge theory: In analogy to Simpson's non-abelian Hodge correspondence over the complex numbers, this aims to relate p-adic representations of the fundamental group of a smooth proper rigid space X over $\mathbb C_p$ to Higgs bundles on X. This relation can be formulated in terms of vector bundles on Scholze's pro-étale site. Based on this reformulation, I will explain how one can use analytic moduli spaces to give a new geometric formulation of the p-adic non-abelian Hodge correspondence. Surprisingly, in some aspects this seems to be better behaved than what one encounters in the complex theory.

Jay Taylor: Bounding Character Values at Regular Semisimple Elements

If $\chi : G \to \mathbb C $ is an irreducible character of a finite group $G$ then one can ask how large the absolute value $|\chi(g)| \in \mathbb R $ at a given element $g \in G$ is. In this talk we’ll try to explain the relevance of this question and discuss recent joint work with P. H. Tiep and M. Larsen obtaining new bounds in the case where $G$ is a finite reductive group and $g$ is a regular semisimple element.

Lucas Mann: A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry

We introduce a p-adic 6-functor formalism on rigid varieties and more generally Scholze's diamonds, which in particular proves Poincaré duality for étale $\mathbb F_p$-cohomology on proper smooth rigid varieties over mixed-characteristic fields. The basic idea is to employ Clausen–Scholze's condensed mathematics in order to construct a category of "quasicoherent complete topological $\mathcal O ^{+a}_{X/p}$"-sheaves on any diamond X. This category satisfies v-descent and admits the usual six functors $\otimes$, $\underline{Hom}$, $f^*$, $f_*$, $f_!$ and $f^!$ with all the expected compatibilities. One can then pass to the category of $\phi$-modules, i.e. pairs $(M, \phi_M)$ where $M$ is as before and $\phi_M : M \to M$ is a Frobenius-semilinear isomorphism. By proving a version of the $p$-torsion Riemann–Hilbert correspondence we show that classical étale $\mathbb F_p$-sheaves embed fully faithfully into the category of $\phi$-modules (identifying perfect sheaves on both sides), which finally allows us to relate the 6-functor formalism of $\phi$-modules to $\mathbb F_p$-cohomology. With this theory at hand, we also obtain a new and short proof of the primitive comparison isomorphism.

Thor Wittich: Towards all operations on Milnor–Witt K-theory

Morel gave a purely algebraic description of the graded endomorphism ring of the motivic sphere spectrum, called Milnor–Witt K-theory. We will start this talk by giving an overview over the basic theory of Milnor–Witt K-theory of fields and how it is lifted to smooth schemes. In particular we explain how this reduces the task of computing all operations $K_n^MW→K_m^MW$ of Milnor–Witt K-theory sheaves on smooth schemes to the algebraic setting we started with. Then we discuss our current progress in determining these operations.

Achim Krause: On the K-theory of Z/p^n

In recent work with Antieau and Nikolaus, we develop methods to compute algebraic K-theory of rings such as $Z/p^n$, based on trace methods and prismatic cohomology. Our methods lead to a practical algorithm, which we use to study these K-groups. The most striking pattern we discover is that these K-groups vanish in sufficiently large even degrees, which we are able to prove. In this talk, I want to explain the ingredients behind these results.

Sergei Iakovenko: Geometrizing Class Field Theory

The development of cohomological class field theory in 20th century helped to reduce many classical problems in number theory to questions about cohomology groups of certain Galois modules. I will explain that classical results such as duality theorems of Tate and explicit computations of Brauer groups of (local) number fields lead us to constructions of categories that appear naturally as values of various cohomology theories of smooth algebraic varieties over an algebraic closure of a finite field, generic fibers of Drinfeld shtukas, or in the classification of vector bundles on the Fargues–Fontaine curve.

Olivier Brunat: Basic set and unitriangularity

In this talk, we will introduce the notion of unitriangular basic sets of finite groups, and explain how we can use them to obtain useful information on the decomposition matrices of the groups. We will focus more precisely on the decomposition matrices of finite reductive groups in non-defining characteristic and of the alternating groups.