Schedule and Talks

Course Descriptions

Francis Brown: Topics in motives and number theory

I will give an introduction to the theory of periods from the motivic point of view and present some number-theoretic applications. Topics to be covered include: categories of realisations, Tannakian categories, motivic periods, mixed Tate motives, Beilinson's conjectures and, if time permits, a brief introduction to Mellin Motives which is an emerging topic of research.

Martin Gallauer: (Motivic) tensor-triangular geometry

This mini-course is intended as an introduction to tensor-triangular geometry. Motivation and examples will be drawn mostly from algebraic geometry, and later on specifically from the motivic theory. The only strict prerequisite is some familiarity with (very) basic category theory and algebraic geometry. Lecture notes will be made available at mrtng.gitlab.io/mttg.pdf in due course (hopefully before the workshop).

Victoria Hoskins: Motives of moduli spaces

Moduli spaces (and stacks) of vector bundles and Higgs bundles on a smooth projective curve are fundamental objects in algebraic geometry, whose rich geometry that has been intensively studied through various cohomological perspectives. I will explain some joint results with Simon Pepin Lehalleur and Lie Fu on studying the motives of these moduli spaces. Our starting point involved proving a formula expressing the motive of the stack of vector bundles on the curve with rational coefficients in terms of motives of the Jacobian and symmetric powers of the curve; the proof uses Hecke modifications and motivic descriptions of small maps. We will then see that several different moduli spaces of (Higgs) bundles on a smooth projective curve have abelian motives; for Higgs bundles, we study the fixed points for a natural scaling action and use wall-crossing arguments to prove this result. We can then harness conservatively properties for abelian motives to obtain explicit motivic formulas in low ranks and motivic lifts of known cohomological phenomena, such as chi-independence and mirror symmetry.

Alexander Vishik: Introduction to motives

Motives encode homological properties of algebraic varieties and may be considered as linearizations of the latter. I will start by briefly reminding about Chow groups and correspondences, and then will introduce the Grothendieck's category Chow(k) of Chow motives. This tensor additive category contains motives of smooth projective varieties and their direct summands and is the most "geometric" part of the motivic category DM(k) of Voevodsky which, in turn, is a tensor-triangulated category containing motives of arbitrary smooth varieties. I will introduce the latter and will discuss various tools available there, in particular, motivic homology and cohomology. There are several ways to decompose motives into simpler pieces. One such method is related to the Chow weight structure (in the sense of Bondarko). Another one is the homotopy t-structure. I will discuss both. On my way, I will recall Milnor's K-theory, Rost cycle modules and etale motives. I will then turn to the Beilinson-Lichtenbaum "conjecture" (a theorem of Voevodsky), which encodes the fundamental properties of DM(k). I will talk about "phantom" motives disappearing in the etale (=topological) realisation and will mention few open motivic conjectures. Finally, I will discuss varieties without rational points, Cech simplicial schemes and their invariants.

Contributed Talks

• Matt Broe (Boston University)

Title: The Tate conjecture for abelian fourfolds over finite fields

Abstract: We prove the Tate conjecture for abelian fourfolds over finite fields. The proof relies on techniques from Ancona’s proof of the Hodge standard conjecture for abelian fourfolds, and ultimately reduces to Markman’s results on the algebraicity of Weil classes on complex abelian varieties.

• Yorick Fuhrmann (University of Warwick)

Title: Profinite Borel completeness and smooth Artin motives

Abstract: When G is a finite group, a G-spectrum (viewed as a spectral Mackey functor) is Borel complete if it is right Kan extended from the free orbit. When G is profinite, one needs more refined notions of Borel completeness. We will explain these, express the resulting categories in terms of étale sheaves and representations, and finally relate them to certain categories of Nisnevich and étale motives controlled by the étale fundamental group of the base scheme.

• Jake Huryn (The Ohio State University)

Title: On the torsion in the Chow motive of a surface of Kodaira dimension zero

Abstract: The category of Chow motives with Z-coefficients contains nontrivial torsion objects, i.e. motives M for which (End(M),+) is a torsion group. However, they are poorly understood. For example, every known torsion motive can be built from the motives of surfaces. In this talk, I will explain some new results on torsion motives and how they relate to the integral Hodge and Tate conjectures. Notably, some of these results contradict existing statements in the literature. This is joint work with William C. Newman.

• Kenza Memlouk (University of Strasbourg)

Title: The motivic Galois group for a double zeta value

Abstract: In this talk, we consider multiple zeta values, which are periods of unramified mixed Tate motives. For a given multiple zeta value ζ, there exists a unique minimal motive so that ζ is a period of this motive. In general, this motive is very difficult to compute. In the specific case of double zeta values, we can compute such a minimal motive. We will give the Tannakian group associated to it and discuss its dimension and we will develop the technics that are involved in this computation.

• Ismael Sierra (University of Glasgow)

Title: Mixed Tate motives of number fields and Goncharov's universality conjecture.

Abstract: I will discuss ongoing work with D. Rudenko and A. Kupers on the structure of mixed Tate motives of number fields. I would explain briefly what the Goncharov conjectures predict and then state a theorem giving explicit generators and relations for the motivic Lie coalgebra of a number field. If time allows I would like to discuss some of the main ideas of the proof, involving homology of general linear groups, Steinberg modules and E_∞-algebra techniques as well as results by Borel and Yang.

• Tariq Syed (Heinrich-Heine-Universität Düsseldorf)

Title: Motivic cohomology of topologically contractible varieties

Abstract: A smooth complex variety X is called topologicaly contractible if its set of complex points, viewed as a complex manifold with analytic topology, is a contractible topological space. The computation of motivic cohomology groups of such varieties is a notoriously difficult problem and has stimulated a wealth of research over the last decades. In this short talk, we survey classical results and highlight the most recent developments on this problem. In particular, we discuss recent computations of motivic cohomology groups of cyclic coverings and explain their applications to the generalized Serre question on algebraic vector bundles (i.e., to the question whether algebraic vector bundles over topologically contractible smooth affine complex varieties are always trivial).

• Anna Ulivi (Università di Genova)

Title: Mumford-Tate conjecture for surfaces of general type with p_g=q=2.

Abstract: The Mumford–Tate conjecture admits a motivic formulation and enjoys an additive property for abelian motives: roughly speaking, if it holds for two motives, then it also holds for their direct sum. To apply this prospective to the study of algebraic surfaces we start from product-quotient surfaces, we compute a decomposition of their motives into abelian motives that satisfies MT and then use techniques from the theory of variations of Hodge structures to extend the result to broader families of surfaces. This technique was already used by Penegini-Commelin in the case of maximal albanese dimension. In this work in progress, in collaboration with Samuele Gagliardo, we aim to complete their result.