Guest seminar summer 2018

Guest seminar "Arithmetic Geometry"

Time: Thursdays, 2pm-4pm

Place: SR 007/008 A6



17. May (Double Session) 


Ben Davison (Edinburgh)

Title: Integrality of BPS invariants

Abstract: BPS numbers are certain invariants that "count" coherent sheaves on Calabi-Yau 3-folds.  Because of subtleties in the definition,  especially in the presence of strictly semistable sheaves, it is not a priori clear that the numbers are in fact integers. I will present a recent proof with Sven Meinhardt of this integrality conjecture.  The conjecture follows from a stronger conjecture, namely that a certain constructible function on the coarse moduli space of semistable sheaves defined by Joyce and Song is integer valued.  This conjecture in turn is implied by the stronger conjecture that this function is in fact the pointwise Euler characteristic of a perverse sheaf.  We prove all of  these conjectures by defining this perverse sheaf, and furthermore find that the hypercohomology of this sheaf, which categorifies the theory of BPS invariants, carries a natural Lie algebra structure, generalizing the theory of symmetrizable Kac-Moody algebras.


 Dennis Gaitsgory (Harvard)

Title:  Categorical trace of Frobenius: how to rediscover Deligne-Lusztig representationsand Shtukas

Abstract: A famous construction by Grothendieck attaches to an l-adic sheaf on a scheme X over F_q a function on X(F_q), by taking traces of the Frobenius. However, we can ask: is there a formal relationship between the "category" of all l-adic sheaves on X and the vector space of all Q_l-valued functions on X(F_q). This question is important e.g. if we want to relate the geometric Langlands program and the classical one. The answer is positive, and is given by categorical trace. However, in this talk we will go yet one step further and take the 2-categorical trace. This procedure will provide a unified mechanism for recovering several constructions that were found by Drinfled in the 1970's: shtukas and Deligne-Lusztig representations.

24. May


Takehiko Yasuda (Tohoku University)

Title: The wild McKay correspondence and moduli of formal torsors.

Abstract: I will talk about a conjectural generalization of the motivic McKay correspondence to arbitrary characteristic. In this generalization, the moduli spaces of G-torsors over Spec k((t)) with G a finite group plays an important role. I will explain my recent joint works with Fabio Tonini concerning construction of this moduli space.

31. May


Shuddhodan K V (Freie Universität Berlin)

Title: Self maps of varieties over finite fields.

Abstract: Esnault and Srinivas proved that as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of a smooth proper surface over a finite field F_q is taken in the span of the Neron-Severi group inside of l-adic cohomology. In this talk we will discuss some analogous questions in higher dimensions motivated by their results and techniques.


Kivanç Ersoy (Freie Universität Berlin)

Title: Centralizers in locally finite simple groups

Abstract: Since the study of centralizers of involutions played a crucial role in the study of the classification of finite simple groups, many questions related to centralizers were naturally arised about the study of infinite simple locally finite groups. In this talk first we will start with basic constructions and examples of simple locally finite group, and describe their structural properties. Then we will discuss some problems about the centralizers of finite subgroups and fixed points of automorphisms in simple locally finite groups.

14. Juni


Shusuke Otabe (Tohoku University/Freie Universität)

Title: On a purely inseparable analogue of the Abhyankar conjecture for affine curves.

Abstract:  Let U be an affine smooth curve defined over an algebraically closed field k of positive characteristic p > 0. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck's etale fundamental group of U. In this talk, I will consider a purely inseparable analogue of this problem, formulated in terms of Nori's profinite fundamental group scheme. I will explain a partial answer to it.

18. June 





 Yohan Brunebarbe (Freiburg/Zürich)

Title: Hyperbolicity of moduli spaces of Calabi-Yau varieties with level structure.

Abstract: Moduli stacks of polarized complex varieties whose canonical bundle is trivial are known to enjoy many hyperbolicity properties. In this talk, I will explain how adding level structure to the picture yields much stronger results.

28. June 


 Emiliano Ambrosi (École Polytechnique Paris)

Title: Specialization of Néron-Severi groups in positive characteristic.

Abstract:  Let k be an infinite finitely generated field of characteristic p>0. We fix a smooth separated geometrically connected scheme X of finite type over k and a smooth proper morphism f: Y to X. In this talk we prove that there are "lots" of closed points x of X such that the fibre of f at x has the same geometric Picard rank as the generic fibre. In characteristic zero, this has been proved using Hodge theoretic methods. To extend the argument in positive characteristic we use the comparison between different p-adic cohomology theories and independence techniques. We explain some applications of the result to uniform boundedness of Brauer groups, the Tate conjecture, abelian varieties, and to proper families of projective varieties.


Sinan Unver (Berlin)

Title: Infinitesimal Chow Dilogarithm

Abstract: Let C_2 be a smooth and projective curve over the ring of dual numbers of a field k. Given non-zero rational functions f, g, and h on C_2, I will define an invariant ρ(f ∧g∧h) ∈ k. This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction I will state and prove an infinitesimal version of the strong reciprocity conjecture of Goncharov and give an extension of Park's regulator for cycles with modulus. If time permits, I will also explain what happens in the higher modulus case. 

5. July


Damian Rossler (Oxford)

Title: TBA


Tohru Kohrita (Freie Universität Berlin)

Title: Regular homomorphisms for motivic cohomology

Abstract: Let X be a smooth proper algebraic variety over an algebraically closed field. The algebraically trivial part of the divisor class group of X is canonically isomorphic to the group of rational points of the Picard variety of X. There is also a canonical homomorphism from the degree zero part of the Chow group of 0-cycles of X to the group of rational points of the Albanese variety of X. The theorem of Rojtman says that it is an isomorphism on torsion. With Samuel's notion of regular homomorphisms, one may consider similar phenomena for cycles of intermediate dimensions. In fact in the 80's, Murre proved the existence of the "universal" regular homomorphisms for cycles of codimension 2, and further proved, over the field of complex numbers, an analogue of Rojtman's theorem for codimension 2 cycles. In this talk, we explain how the notion of regular homomorphisms can be extended to the context of motivic and etale motivic cohomology. If time permits, we discuss the relation with a motivic Albanese functor and Griffiths's intermediate Jacobians.

19. July


Elden Elmanto (University of Copenhagen)

Title:  On Some Étale Descent Theorems in Motivic Homotopy Theory

Abstract: Morel-Voevodsky's motivic homotopy theory is based on the Nisnevich topology, which is coarser than the étale topology. Indeed, this choice is motivated by the fact that many motivic invariants of algebraic varieties satisfies descent for the former topology but not the latter (such as higher Chow groups and algebraic K-theory). However, invariants which do satisfy étale descent are arguably more computable and possess a tighter relationship to algebraic topology. From this point of view, one might try to compute motivic invariants by "breaking it down" in terms of a part that satisfy étale descent, and its complementary portion.

In joint work with Levine, Spitzweck and Østvær we describe the part of (many) oriented invariants which do satisfy étale descent --- they are obtained by inverting a "Bott element". Our work generalizes (in the case of regular Noetherian schemes over a DVR) earlier work of Thomason for K-theory and uses Voevodsky's resolution of the Bloch-Kato conjectures. We will also discuss integral version of these étale descent results, based on Bousfield localization at étale motivic cohomology. Time permitting, I will also discuss work in progress with Jay Shah that provides another perspective on étale descent based on equivariant homotopy theory of the group of order 2.