Abstracts

Tom Bachmann: Motivic spectral Mackey functors

For a suitable algebraic group G acting on a scheme S, various authors have defined the motivic stable homotopy theory of smooth S-schemes with a G-action. While this category has excellent formal properties, the construction is very involved. In this talk I will explain that if G is finite, the construction can be simplified: one can just work with the category obtained from motivic G-spaces with finite étale transfers (over S) by inverting the *trivial* motivic representation sphere. This is essentially a corollary of the motivic tom Dieck splitting theorem, established recently by David Gepner and Jeremiah Heller.

Marco D'Addezio: Torsion points of abelian varieties and F-isocrystals

I will report on a joint work with Emiliano Ambrosi. Let k be a field which is finitely generated over the algebraic closure of a finite field. As a consequence of the theorem of Lang-Néron, for every abelian variety over k which does not admit any isotrivial abelian subvariety the group of k-rational torsion points is finite. We showed that the same is true for the group of torsion points defined on a perfect closure of k. This gives a positive answer to a question posed by Hélène Esnault in 2011. To prove the theorem we translated the problem into another one on morphisms of F-isocrystals. During the talk, I will explain some ideas of our proof.

Emanuele Dotto: The Witt vectors with coefficients

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.

Bradley Drew: A universal property of Ayoub's six-functor formalism

The stable motivic homotopy category $\mathrm{SH}(S)$ over a scheme $S$ is characterized by an $\infty$-categorical universal property: each homology theory defined on the category of smooth $S$-schemes satisfying analogues of the Eilenberg-Steenrod axioms factors essentially uniquely through $\mathrm{SH}(S)$. Ayoub has equipped the categories $\mathrm{SH}(S)$ with a six-functor formalism, i.e., pullback, pushforward, and tensor operations satisfying the same formal properties as derived categories of $\ell$-adic sheaves. In this talk, we will promote the universal property of $\mathrm{SH}(S)$ to a universal property of the associated six-functor formalism: each six-functor formalism $S \mapsto \mathrm{D}(S)$ satisfying a reasonable list of axioms admits an essentially unique family of realization functors $\mathrm{SH}(S) \to \mathrm{D}(S)$ compatible with the six functors. As an application, we will discuss the construction of motivic realization functors with values in arithmetic D-modules. This is joint work with Martin Gallauer.

Fangzhou Jin: The characteristic class of a motive

We prove several Künneth formulas in motivic homotopy theory, and use them to define the characteristic class of a motive, an invariant closely related to the Euler characteristic. For cdh motives, this class lives in the Chow group of 0-cycles. We show its additivity along distinguished triangles, and deduce that the characteristic class for cdh motives is characterized by some elementary operations. If time permits, we also discuss the relative characteristic class. This is a joint work with E. Yang.

Viktoriya Ozornova: Homotopy theory for 2-categories

Grothendieck and Quillen introduced a notion of homotopy equivalences for categories using a by-now-standard tool called "nerve" of a category. This idea leads to various models of categories-up-to-homotopy. In a joint ongoing projects with Julie Bergner and Martina Rovelli, we study variants of the Roberts-Street-nerve for 2-categories and notions of homotopy equivalences arising from this nerve, with an eye towards 2-categories-up-to-homotopy.

Simon Pepin Lehalleur: On the motive of the moduli space of stable Higgs bundles on a curve

Moduli spaces of stable Higgs bundles on curves are central objects of study in non-abelian Hodge theory, geometric representation theory and mathematical physics, and their cohomology has been intensely studied. We consider the motives with rational coefficients of these moduli spaces, and prove that they are Chow motives which lie in the tensor subcategory generated by the motive of the curve, and hence in particular of abelian type. Although this is a statement about motives of smooth varieties, the proof relies crucially on the study of motives of various moduli stacks. This is joint work with Victoria Hoskins (Freie Universität Berlin).

Pavel Sechin: Algebraic Morava K-theories with applications to algebraic groups

For a fixed prime $p$ algebraic Morava K-theories $K(n)$ (where $n\ge 1$) are oriented cohomology theories which play an intermediate role between the Grothendieck K-theory and Chow groups with $p$-local coefficients. I will show that if the (pure) $K(n)$-motive of a smooth projective variety is a sum of Tate motives, then the same is true for the $K(m)$-motive, for $m < n$. As for particular examples, the obstructions for the splitting of the $K(n)$-motive of a quadric are known: they are precisely the cohomological invariants lying in degrees no greater than $n+1$. This allows to obtain bounds on the torsion of Chow groups of these quadrics in low codimensions. I will also explain how the gamma filtration on Morava K-theories and Chern classes from them can be useful for such computations.

Gouzhen Wang: Computations of prismatic cohomology and topological cyclic homology

For complete intersection rings over p-adic intergers, we use the relative THH to construct resolutions of topological cyclic homology and its periodic analogue. The resulting spectral sequences are believed to be isomorphic with the BMS spectral sequence, whose $E_2$ terms are the motivic cohomolpgy and prismatic cohomology respectively. We show that the $E_2$ terms are the Ext groups of a Hopt algebroid, with explicit formulae for the structure maps. As an example, we will give concrete descriptions of the computations in the case of local fields and compare them with known results on their TC and Galois cohomlogies.

Matthias Wendt: When does a quadratic form over a scheme split off a hyperbolic plane?

In recent work, Asok and Fasel have applied motivic homotopy methods to make significant progress on the very classical question when a vector bundle over a smooth affine scheme splits off a trivial line bundle. The goal of the talk is to show that the same methods can also be applied to similar questions for quadratic forms. On the one hand, it is actually possible to get a full isometry classification of generically split quadratic forms over smooth affine schemes of dimension at most 3. On the other hand, the computation of the first non-vanishing homotopy sheaf of the orthogonal Stiefel varieties gives rise to Euler classes controlling when generically split quadratic forms split off hyperbolic planes. For the talk, I'll formulate some of the general results that can be obtained and try to illustrate some of the phenomena with low-dimensional examples.

Zhouli Xu: Motivic $\widehat{S}/\tau$-modules and classical stable homotopy groups of spheres

I will discuss joint work with Bogdan Gheorghe and Guozhen Wang on the equivalence of stable infinity categories, between the motivic cellular $\widehat{S}/\tau$-modules over $\mathbb{C}$ and the derived category of $MU_*MU$-comodules. As a consequence, the motivic Adams spectral sequence for $\widehat{S}/\tau$ is isomorphic to the algebraic Novikov spectral sequence for the sphere. In joint work with Dan Isaksen and Guozhen Wang, we use this isomorphism of spectral sequences to compute classical stable stems at least to the 90-stem, with ongoing computations into even higher dimensions. I will also discuss ongoing work with Dan Isaksen, Hana Jia Kong and Guozhen Wang, that there exists a t-structure on the category of motivic cellular $\widehat{S}$-modules over fields of characteristic 0, with the heart equivalent to $MU_*MU$-comodules. This gives us computational tools for $\mathbb{R}$-motivic and $C_2$-equivariant stable stems.

Maria Yakerson: Modules over algebraic cobordism

When k is a field with resolution of singularities, it is known that Voevodsky’s category of motives DM(k) is equivalent to the category of modules over the motivic cohomology spectrum HZ. This means that a structure of an HZ-module on a motivic spectrum is equivalent to a structure of transfers in the sense of Voevodsky. In this talk, we will discuss an analogous result for modules over the algebraic cobordism spectrum MGL. Concretely, a structure of an MGL-module is equivalent to a structure of coherent transfers along finite syntomic maps, over arbitrary base scheme. Time permitting, we will see a generalization of this result to modules over other motivic Thom spectra, such as the algebraic special linear cobordism spectrum MSL. This is joint work with Elden Elmanto, Marc Hoyois, Adeel Khan and Vladimir Sosnilo.