Schedule and Talks

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.