Talks (titles and abstracts)

Tom Bachmann: Bousfield-Kan resolutions in motivic homotopy theory
Report on joint work with Anton Engelmann, Mike Hopkins and Klaus Mattis.
Given an adjunction F: C <-> D: G, we can approximate D "from the left" by passing to the localization L_F of C determined by the F-equivalences, and we can approximate C "from the right" by passing to comodules in D for the comonad FG. A natural question is when these two processes yield the same answer. More concretely, the adjunction determines a monad GF on C, and using it for every object X in C one obtains a cosimplicial "resolution" of X by objects in GD. When is L_F X equivalent to the totalization of this resolution?
It is well-known that this does not work in general. Nonetheless, this works for many objects and adjunctions arising in practice in classical homotopy theory, and then provides a powerful computational tool.
By revisiting and adapting the methods of Bousfield-Kan, we prove results of this kind for C the category of motivic spaces.

Hélène Esnault: tba

Jean Fasel: Classification of vector bundles on real affine algebraic varieties: does topology help?
In this talk, I will report on recent work on the classification of vector bundles on smooth affine real varieties. I will mainly discuss the problem of splitting a free factor of rank one, for vector bundles of corank 0 and 1. Along the way, I will also explain some comparison results between Chow-Witt groups and singular cohomology of the real points.

Thomas Geisser: tba

Lars Hesselholt: Homotopy invariance in topology
This is a report on joint work with Peter Haine. Let \(X\) be a topological manifold with underlying anima \(|X|\), and let \(G\) be a group in topological manifolds with underlying group in anima \(|G|\). A classical result in algebraic topology gives a canonical identification \(H^1(X,G) \simeq H^1(|X|,|G|)\) of the pointed set of isomorphism classes of \(G\)-torsors on \(X\) and the pointed set of \(|G|\)-torsors on \(|X|\). This may be viewed as a non-abelian version of the identification of sheaf cohomology and singular cohomology. In this talk, I will explain a proof, using homotopy invariance, which avoids the traditional passage through the non-abelian representable cohomology given by the pointed set \([X,BG]\) of homotopy classes of continuous maps from \(X\) to Milnor's classifying (topological) space \(BG\).

Michael Hopkins: tba

Victoria Hoskins: Motives of moduli spaces of bundles on curves
Enumerative geometry often exploits the fact that certain moduli spaces of bundles (and sheaves) have tautologically generated cohomology. In this talk I will explain a motivic incarnation of this tautological generation: that the motives of moduli spaces of (semi stable) Higgs and vector bundles on a curve with coprime rank and degree is generated by the motive of the curve. For SL-Higgs moduli spaces, which are non-tautologically generated, we additionally need motives of certain étale covers of the curve. I will explain how the fact that these moduli spaces have abelian motives can be exploited to produce motivic formulas in low rank and lift cohomological mirror symmetry to a motivic statement. This is joint work with Simon Pepin Lehalleur and partially also with Lie Fu.

Marc Hoyois: Grothendieck-Witt theory and the motivic sphere
A celebrated theorem of F. Morel identifies the zeroth homotopy group of the motivic sphere spectrum over a field with the Grothendieck-Witt group of symmetric bilinear forms. I will give some evidence for the conjecture that this already holds in non-A^1-invariant motivic spectra. In that setting, Grothendieck-Witt theory is represented by a motivic ring spectrum KO over any scheme (which is only A^1-invariant on regular Z[1/2]-schemes), whose unit map has a section on zeroth homotopy sheaves.

Adeel Khan: Motivic sheaves on algebraic stacks
I will survey recent developments on extensions of triangulated categories of motivic sheaves from schemes to algebraic stacks. In one direction, this has been applied to (virtual) intersection theory on stacks and equivariant intersection theory (largely joint work with Charanya Ravi). In another direction, it also facilitates the construction of a stacky derived Fourier duality on G_m-equivariant motivic sheaves, which we applied in joint work with Tony Feng to prove a modularity property of the higher arithmetic theta series of Feng-Yun-Zhang.

Josefien Kuijper: Uniqueness for six-functor formalisms
In recent years, efforts have been made to formalise, in the most efficient way, Grothendieck’s six operations of sheaves (tensor product and internal hom, inverse and direct image, and exceptional inverse and direct image) and their properties. In addition to these operations, one might want to encode natural isomorphisms between the inverse image and exceptional inverse image for a certain class of “étale” morphisms, and between the direct image and the exceptional direct image for a class of “proper morphisms”. These can be encoded as extra data, using (infinity,2)-categories, but this is not necessary. More recently, it has become clear that these natural isomorphisms canonically arise from a property of the six operations themselves.
We formulate this condition in an efficient way, leading to the notion of a “Nagata six-functor formalism”. We show that Nagata six-functor formalisms are uniquely determined by the data of the tensor product and the inverse image functor, giving a positive answer to a conjecture by Scholze. This talk is based on joint work with Adam Dauser.

Matthew Morrow: A motivic approach to cycles on singular varieties
On smooth algebraic varieties there is a good theory of Chow groups encoding algebraic cycles and their intersection. The theory is moreover closely related to algebraic K-theory, through the Riemann--Roch formula for K_0 and the Bloch--Quillen formula expressing the Chow groups in terms of cohomology with coefficients in sheaves of K-groups. In the 1980s Marc Levine was the progenitor of an extension of the theory to reduced singular varieties, and the Chow group of zero cycles which he studied with Weibel has proved to have particularly good properties: Marc himself established Bloch--Quillen and Roitman torsion theorems, which have since been extended to greater generality by Barbieri Viale, Binda, Krishna, Saito, Srinivas and others. In a different direction, Bloch and Esnault proposed a definition of "additive" Chow groups in 2001, designed to capture phenomena arising from non-reduced varieties; their definition has since burgeoned into the theory of Chow groups with modulus.

In this talk we will suggest a uniform approach to the problem of algebraic cycles on singular varieties, even on arbitrary Noetherian schemes over a field. Namely, we will define the motivic Chow groups of such a scheme using non-A^1-invariant motivic cohomology, which we built in earlier work by gluing together A^1-invariant motivic cohomology and de Rham/syntomic cohomology. These motivic Chow groups satisfy many of the properties one expects of such a theory, such as the Riemann--Roch relation to K_0, the existence of cycle classes (and so in principle an intersection theory on singular varieties), and the projective bundle formula. We will see that these motivic Chow group are simultaneously related to Levine--Weibel's group of zero cycles and, in the non-reduced case, to Bloch--Esnault's additive Chow groups. This is all joint work with Elden Elmanto.

Thomas Nikolaus: tba

Rahul Pandharipande: tba

Sabrina Pauli: Examples in Quadratically Refined Enumerative Geometry
Quadratically refined enumerative geometry, developed in large part thanks to the work of Marc Levine, gives a new algebraic way of thinking about enumerative geometry over various fields, including the classical complex and real numbers. In this talk I will focus on how classical enumerative questions can be refined in this way, and give some examples and techniques for computing these examples.

Arpon Raksit: tba

Anand Sawant: Cellular \(A^1\)-homology and cellular Suslin homology
I will introduce cellular \(A^1\)-homology and cellular Suslin homology and describe how these are motivic analogues for singular homology with integer and integers modulo 2 coefficients, respectively. These cellular versions are often entirely computable, in contrast with \(A^1\)-homology and Suslin homology of smooth projective varieties, and give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will discuss results and examples elaborating this analogy. The talk is based on joint work with Fabien Morel.

Vasudevan Srinivas: Some finiteness results for the etale fundamental group in positive characteristics
This talk will discuss some results on etale fundamental groups of varieties over an algebraically closed field of characteristic p > 0, based on joint work with H`el´ene Esnault and other coauthors. One result, along with Mark Schusterman, is that the tame fundamental group is finitely presented for such a variety which is the complement of an SNC divisor in a smooth projective variety. A second, along with Jakob Stix, is to give an obstruction for a smooth projective variety to admit a lifting to characteristic 0, in terms of the structure of its etale fundamental group as a profinite group. We will finally touch on some open questions.

Georg Tamme: Pro-cdh descent on derived schemes
Joint work with Shane Kelly and Shuji Saito. In general, algebraic K-theory does not send abstract blowup squares to cartesian squares. On noetherian schemes, one can fix this by taking all infinitesimal thickenings of the "center" of the abstract blowup into account. However, for general, non-noetherian schemes, this statement fails again. In the talk, I will explain a general such "pro-cdh descent" result for arbitrary qcqs derived schemes, working with derived infinitesimal thickenings of the center. As an application, we deduce a vanishing result for negative K-groups of schemes satisfying some weak noetherianity condition on the underlying topological space.

Kirsten Wickelgren: Gromov--Witten invariants in A1 homotopy theory
Gromov-Witten invariants count curves in a given homology class through an appropriate number of marked points. This talk will discuss joint work with Erwan Brugallé, Jesse Kass, Marc Levine, and Jake Solomon defining A1 Gromov--Witten invariants over non-algebraically closed fields and investigating their properties.