Title 

A formula for the Voevodsky motive of the moduli stack of vector bundles over a curve

Abstract

I will recall basic features of Voevodsky's category of mixed motives and explain how to define the motives of certain algebraic stacks in this context. I will then state and sketch the proof of a formula for the motive with rational coefficients of the stack of vector bundles over a smooth projective curve. This formula is compatible with classical computations of various cohomological invariants of this stack by Harder, Atiyah-Bott, Behrend-Dhillon, etc. The proof uses rigidifications of the stack by certain Quot and Flag-Quot schemes as well as a motivic version of an argument of Laumon and Heinloth on the relative cohomology of small maps. This is joint work with Victoria Hoskins (FU Berlin).

Title

Survey of the theory of multiple commutators in classical and classical-like groups

Abstract

Commutators groups play an important role in the structure of classical-like groupsandtheir algebraic K-theory. This talk will survey the theory of mixed commutator groups and multiple mixed commutator groups of congruence and relative elementary groups of classical and classical-like groups. At first only mixed commutators of 2 groups will be considered and it is assumed that at least one of the groups is absolute. We begin the survey with results of C. Jordan on the general linear group over prime fields around 1870, then results for all classical groups over all fields by E. Dickson around 1900, and then results of J. Dieudonne over division rings in the 1940's, which extend the previous results. Moving into the modern age, we begin with results of W. Klingenberg around 1960 for the general linear, symplectic and even dimensional orthogonal groups over local and semi-local rings and then their generalizations in the 1960's by H. Bass to the general linear group over rings satisfying a dimension condition (stable rank) and by A. Bak to even dimensional unitary groups over form rings satisfying a dimension condition. These results were then extended by a bunch of people to groups defined over module finite and quasi-finite rings. In 2008, A. Stepanov and N. Vavilov dropped the condition that one of the groups must be absolute and obtained partial results for the general linear group over quasi-finite rings. In 2013, complete results were obtained for arbitrary multiple mixed commutators in the same group over the same rings by H. Hazrat and the speaker. In 2017 these were extended by H. Hazrat, N. Vavilov and the speaker to even dimensional unitary groups over quasi-finite form rings.

Title 

Abstract

Title

Some implications between Grothendieck's anabelian conjectures

Abstract

In his famous letter to Faltings, Grothendieck explained a series of ideas and conjectures collected under the name of anabelian geometry. Grothendieck's picture describes how for certain varieties, called anabelian, defined over fields finitely generated over Q the étale fundamental group recovers all the geometric information about the variety. In dimension one anabelian varieties are curves with negative Euler characteristic, but in higher dimension the picture is not so clear. Some of the anabelian conjectures have been proved, most notably by Mochizuki. Among the anabelian conjectures, the so called section conjecture remains largely open. We show how the section conjecture implies a much stronger anabelian statement, from which every other anabelian statement follows immediately. We do this in the generality of Deligne-Mumford stacks, rather than varieties, and show that if the section conjecture holds for curves then it holds for some other classes of DM stacks.

Title

A Simpson correspondence for abelian varieities in postive characteristic

Abstract

Recall that the Simpson correspondence provides an equivalence between the category of local systems and that of the semi-stable Higgs bundles whose Chern class is zero, for a smooth projective variety over the complex numbers. M. Groechenig, T.-H. Chen and X. Zhu  established such a correspondence for curvers in positive characteristic. In this talk, I will present a further generalization to abelian varieties in positive characteristic. More precisely, the result states that etale locally over the Hitchin base, the moduli stack of Higgs bundles on the Frobenius twist and the moduli stack of vector bundles with flat connections on the abelian itself are equivalent.

Title

Finiteness of perfect torsion points of an abelian variety and monodromy of F-isocrystals

Abstract

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 show 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 translate the problem into a certain question on morphisms of F-isocrystals. Then we handle it by studying the monodromy groups of the F-isocrystals involved.

Title

Abstract

TBA 

Title

The (non-uniform) Hrushovski-Lang-Weil estimates.

Abstract

In 1994 using techniques from model theory and intersection theory, Hrushovski obtained a generalisation of the Lang-Weil estimates. Subsequently the estimate has found applications in group theory, algebraic dynamics and algebraic geometry. We shall discuss a geometric proof of the non-uniform version of this estimate.