Abstracts

Tom Bachmann: Progress in unstable motivic homotopy theory

I will discuss work in various stages of progress, joint with Aravind Asok and Michael J. Hopkins. First I will explain our notion of "Gm-connectivity" (an unstable analog of effectivity) and how it can be used to formulate and prove a version of the Freudenthal suspension theorem for the motivic sphere $S^{2,1}$. I will then explore some consequences and questions naturally arising from this, including comonadicity of the suspension spectrum functor, the motivic cohomology of Eilenberg--MacLane spaces, and rational unstable motivic homotopy theory. The theory of normed spectra as formulated with Marc Hoyois, and its dual, co-normed spectra, will feature prominently in the second half.

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Federico Binda: On the p-adic weight-monodromy conjecture for complete intersections in toric varieties

We give a proof of the p-adic weight monodromy conjecture for scheme-theoretic complete intersections in proper smooth toric varieties. We rely crucially on the framework of rigid analytic and logarithmic motives, exploiting some recent results which allow us to adapt the proof of the analogous statement in the l-adic setting given by Scholze. Joint work with H. Kato and A. Vezzani.

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Jean Fasel: Formal ternary laws

In this talk, I will introduce the notion of formal ternary laws (FTL) associated to a symplectically oriented cohomology theory, which are the analogues of the formal group laws (FGL) associated to oriented cohomology theories. I will compare this notion with existing ones and provide examples of such laws. I will then explain the remaining basic problems, mainly of computational nature. Time permitting, I will discuss some applications. This is a report on joint work with D. Coulette, F. Déglise, O. Haution and J. Hornbostel.

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Eric Friedlander: Semi-topologcal theories and algebraic equivalence

This is a reflection upon work of 20 - 30 years ago related to fundamental questions concerning algebraic cycles. Semi-topological cohomology and K-theory evolved in joint work with Ofer Gabber, Christian Haesemeyer, Blaine Lawson, Barry Mazur, Andrei Suslin, Vladimir Voevodsky, and Mark Walker. We showed that these theories satisfy many good properties, but progress stalled as we were confronted by fundamental conjectures such as Grothendiek's Standard Conjecture and the Hodge Conjecture. My hope is that this semi-topological perspective can be complemented by recent homotopical and motivic techniques, leading to advances in our understanding of algebraic cycles.

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Martin Gallauer: Artin motivic tt-geometry

Reporting on joint work with Paul Balmer (UCLA), I will describe our current understanding of the tensor-triangular classification of Artin motives over arbitrary fields.

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Annette Huber: Periods and o-minimality

(joint work with Johan Commelin and Philipp Habegger) Classical periods of motives over Q can be expressed as volumes of semi-algebraic sets defined over the rationals. We provide an analogous result for exponential periods, i.e., periods of exponential motives studied by Fresan and Jossen. They are volumes of sets in a certain o-minimal structure. This hint at a possible deeper relation between the theory of motives and o-minimality. In the talk, we aim to explain these notions to a motivic audience and how the result follows.

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Nikita Karpenko: On classifying spaces of spin groups

We discuss some recent results on Chow groups of classifying spaces of spin groups.

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Håkon Kolderup: Homotopy invariance and Gersten complexes over base schemes

Voevodsky's study of cohomological properties of presheaves with transfers over a perfect field includes results on the preservation of homotopy invariance under sheafification, referred to as the strict homotopy invariance theorem. His results imply the exactness of the Gersten complex for a cohomology theory defined by a sheaf with transfers over a perfect field. In this talk we will explore similar questions over more general base schemes. A counterexample by Ayoub shows that strict homotopy invariance is no longer true over positive dimensional base schemes. In joint work with Druzhinin and Østvær we propose a modified version of strict homotopy invariance which generalizes to other bases. Finally we will have a look at how the Gersten complex behaves in this more general setting.

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Denis Nardin: The motivic spectral sequence for twisted K-theory

Twisted K-theory is a variant of algebraic K-theory built out of modules over an Azumaya algebra. It is therefore natural to ask how many properties it shares with algebraic K-theory. In this talk I will show that the situation is as good as can be expected, extending results by M. Levine and B. Kahn. In particular we will construct a spectral sequence converging to twisted K-theory whose E_2-page are certain "twisted" Chow groups. This is joint work with E. Elmanto and M. Yakerson.

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Sabrina Pauli: A quadratic enrichment of the Bernstein-Kushnirenko theorem

Let f_1,...,f_n be Laurent polynomials in n variables. The Bernstein-Kushnirenko theorem states that the number of solutions to f_1=...=f_n=0 equals the mixed volume of the Newton polytopes of the f_i. In the talk I will present a quadratic enrichment of this theorem. The proof uses methods from tropical geometry and is based on joint work with Andrés Jaramillo Puentes.

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Simon Pepin Lehalleur: Motivic mirror symmetry for Higgs bundles

The moduli spaces of semistable Higgs bundles over a smooth projective curve for two Langlands dual groups are conjecturally related by a form of mirror symmetry. In the case of SL_n and PGL_n, Hausel and Thaddeus conjectured that this should be reflected in an equality of (twisted orbifold) Hodge numbers, which was then proven by Groechenig-Wyss-Ziegler. We show that this lifts to an isomorphism of Voevodsky motives with rational coefficients. Our method is based on another approach to the Hausel-Thaddeus conjecture due to Maulik and Shen which we combine with the conservativity of realisation functors on constructible abelian motives in characteristic 0 due to Wildeshaus. I will also give an overview of some related results and conjectures on motives of moduli spaces of bundles on curves. Joint work with Victoria Hoskins (Nijmegen)

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Marco Robalo: Categorification of Donaldson-Thomas invariants

Given a (-1)-shifted derived scheme $X$ with a convenient orientation data (in the sense of Kontsevich-Soibelman), Brav-Bussi-Dupont-Joyce-Szendroi (BBDJS) constructed a perverse sheaf over $X$ and whose Euler characteristic recovers Behrend's counting of Donaldson-Thomas invariants. The BBDJS construction uses a Darboux local form for (-1)-shifted symplectic schemes: locally they are all derived critical loci of a function $f$ on a smooth scheme $U$ and the DT-invariants are obtained from the Euler characteristic of the sheaf of vanishing cycles of $f$. In this talk I will describe an ongoing joint work with B. Hennion and J. Holstein where we propose a strategy based on To\"en-Vezzosi derived foliations, to glue over $X$ a sheaf of 2-periodic dg-categories locally modeled on the categories of matrix factorisations $MF(U,f)$. In particular, our strategy allows us to recover the construction of BBDJS.

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Oliver Roendigs: Absolute purity for the motivic sphere spectrum over discrete valuation rings

This is joint work with Markus Spitzweck. Let D be a discrete valuation ring of mixed characteristic (0,p). Then the p-inverted motivic sphere spectrum satisfies absolute purity over D in the sense that the purity transformation $\mathbf{1}_k[p^{-1}] \to i^! \mathbf{1}_D[p^{-1}]\wedge T$ introduced by Deglise-Jin-Khan is an equivalence. This allows to determine its endomorphism ring as the p-inverted Grothendieck-Witt ring of D.

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Marco Schlichting: On the homology of symplectic groups over local rings

Motivated by results in A1-homotopy theory, we improve, by a factor of 2, known homology stability ranges for the integral homology of symplectic groups over commutative local rings with infinite residue field and show that the obstruction to further stability is bounded below by Milnor-Witt K-theory. Contrary to the case of special linear groups the comparison map between the obstruction groups from ordinary group homology to its A1 version is not an isomorphism, in general. Nevertheless, our results lead to a Hurewicz homomorphism from certain hermitian K-groups of local rings with infinite residue field to appropriate Milnor-Witt K-groups.

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Pavel Sechin: Morava motive of an element in Milnor K-theory modulo 2

Consider algebraic Morava K-theory K(n) with mod 2 coefficients, and the associated category of K(n)-motives of smooth projective varieties over $k$. I will explain that there are unary (i.e. invertible) K(n)-motives $L_a$ for every element a of $K^M_{n+1}(k)/2$ and an injective homomorphism of abelian groups $K^M_{n+1}(k)/2 \rightarrow \mathrm{Pic}_{K(n)}(k)$. I will explain in which varieties these unary motives are expected to appear as direct summands. Along the way I will also construct motivic decompositions of Morava motives of quadrics and study the question, for which field extensions $K/k$ the map $K(n)(X)\rightarrow K(n)(X_K)$ is surjective for all $X$. Partially joint with E. Shinder and A. Lavrenov.

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Toni Annala: Towards higher algebraic cobordism

A Levine–Morel style, geometric theory of algebraic cobordism has been constructed using derived algebraic geometry. This is a non-$\mathbb{A}^1$-invariant theory, which is expected to be the truncation of a spectrum valued invariant, so called \emph{higher algebraic cobordism}, a non-$\mathbb{A}^1$-invariant enhancement of the cohomology theory represented by $MGL$ in Morel–Voevodsky's stable $\mathbb{A}^1$-homotopy category. I will overview recent efforts by Ryomei Iwasa and I to construct such a theory. The focus will mostly be on the more recent developments, based on the pursuit of a non-$\mathbb{A}^1$-invariant motivic homotopy theory.

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Alberto Vezzani: Rigid analytic motives and applications

In this talk we will discuss some recent advances in the theory of rigid analytic motives. As an application, we show how to define and study a relative overconvergent de Rham cohomology for adic spaces in mixed characteristic, using the motivic 6-functor formalism and the language of solid quasi-coherent modules. We also extend this construction to the equi-characteristic p case, taking values on quasi-coherent sheaves over the relative Fargues-Fontaine curve, as conjectured by Fargues and Scholze. Finiteness results and the relation to classical cohomology theories are also discussed. These results are part of joint works with J. Ayoub and M. Gallauer and with A.-C. Le Bras.

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Matthias Wendt: Chow-Witt rings of flag varieties

In the talk I will explain the structure of the Chow-Witt rings of partial flag varieties in type A. The key computations concern the Witt-sheaf cohomology. To do these computations, versions of cellular complex, Künneth formula and Leray-Hirsch theorems are developed for Witt-sheaf cohomology of cellular varieties. Via the real cycle class map, the computations imply that in the singular cohomology of real flag manifolds (with integral coefficients), all torsion is 2-torsion. This is joint work with Thomas Hudson and Ákos Matszangosz.

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Kirsten Wickelgren: A quadratically enriched zeta function

The celebrated and beautiful Weil conjectures connect the Betti numbers of a complex variety whose defining equations can be reduced mod p to the number of solutions mod p using zeta functions. We define a quadratic enrichment of these zeta functions. For cellular varieties, we show a rationality result and a connection to the Betti numbers of the real points. This is joint work with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.

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