Algebraic Geometry and Number Theory Seminar

In this talk we will give a general setting to classical results of Magnus, Witt, Koch and Lazard and prove new ones. The talk will be devoted to the connection between integral filtrations of the free group, ideals of the integral group ring, ideals of the complete free associative algebra and kernels of homomorphisms from the free group to the ring of upper triangular unipotent matrices. Furthermore, we characterize integral filtrations as powers of the lower central series

Stringy Chern classes of singular projective algebraic varieties can be defined by some explicit formulas using a resolution of singularities. It is important that the output of these formulas does not depent on the choice of a resolution. The proof of this independence is based on the nonarchimedean motivic integration. The purpose of the talk is to explain a combinatorial computation of stringy Chern classes for singular toric varieties. As an application one obtains combinatorial formulas for the intersection numbers of stringy Chern classes with toric Cartier divisors and some interesting combinatorial identities for convex lattice polytopes.

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes, so-called tropical varieties. The fact that a tropical variety still reflects many properties of the degenerated variety allows an infusion of polyhedral and combinatorial methods into algebraic geometry. Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. In this talk, we present tropical modi cations as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input curve. Joint work with Angelica Cueto.

In a joint work with N. Gurevich we have constructed a family of Rankin-Selberg integrals representing the standard twisted L-function of a cuspidal representation of the exceptional group of type G₂. This integral representations use a degenerate Eisenstein series on the family of quasi-split forms of Spin₈ associated to an induction from a character on the Heisenberg parabolic subgroup. This integral representations are unusual in the sense that they unfold with a non-unique model. A priori this integral is not factorizable but using remarkable machinery proposed by I. Piatetski-Shapiro and S. Rallis we prove that in fact the integral does factor. As the local generating function of the local L-factor was unknown to us, we used the theory of C*-algebras in order to approximate it and perform the unramified computation. If time permits, I will discuss the poles of the relevant Eisenstein series and some applications to the theory of CAP representations of G₂.

The talk is going to be about the work carried out as part of my MSc thesis. Motivated by recent arithmetic results, we will consider new and improved results on the freeness of subgroups of free profinite groups: 1.The Intermediate Subgroup Theorem - A subgroup (of infinite index) in a nonabelian finitely generated free profinite group, is contained in a free profinite group of infinite rank. 2. The Verbal Subgroup Theorem - A subgroup containing the normal closure of a (finite) word in the elements of a basis for a free profinite group, is free profinite. These results shed light on several theorems in Field Arithmetic and may be combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a generalization of a result of Bary-Soroker, Fehm, and Wiese on the profinite freeness of subgroups arising from Galois representations. If time permits, we discuss applications of the tools developed to abstract/geometric group theory, and to torsion points on abelian varieties.

In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field. A key role is played by the computation of some Galois groups.

We consider the following problem: given a set of algebraic conditions on an n-tuple of functions and their first l derivatives, admitting finitely many solutions (in a differentiably closed field), can one give an upper bound for the number of solutions? I will present estimates in terms of the degrees of the algebraic conditions, or more generally the volumes of their Newton polytopes (analogous to the Bezout and BKK theorems). The estimates are singly-exponential with respect to n,l and have the natural asymptotic with respect to the degrees or Newton polytopes, sharpening previous doubly-exponential estimates due to Hrushovski and Pillay. No familiarity with differential algebra will be assumed. As an application, I will sketch how this result can be applied to deduce similar estimates for the number of transcendental lattice points on algebraic subvarieties of complex tori and abelian varieties, following Hrushovski and Pillay. If time permits I will also mention an application to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon.

Hochschild cohomology is the prominent cohomology theory for associative algebras. In this talk we study relations between the Hochschild cohomology modules of a commutative algebra A, and the a-adic completion operation, for an ideal a in A. We will first recall what is Hochschild (co)-homology and explain its importance, then discuss some basic results about the derived completion and derived torsion functors, and finally apply these results to the noetherian case, and deduce that Hochschild cohomology commutes with adic completion.

Homological algebra plays a major role in noncommutative ring theory. This interaction is often called "noncommutative algebraic geometry", because these homological methods allow us to treat, in an effective way, a noncommutative ring A as "the ring of functions on a noncommutative affine algebraic variety".

Some of the most important homological constructs related to a noncommutative ring A are dualizing complexes and tilting complexes over A. These are special kinds of complexes of A-bimodules. When A is a ring containing a central field K, these concepts are well-understood now. However, little is known about dualizing complexes and tilting complexes when the ring A does not contain a central field (I shall refer to this as the noncommutative arithmetic setting). The main technical issue is finding the correct derived category of A-bimodules.

In this talk I will propose a promising definition of the derived category of A-bimodules in the noncommutative arithmetic setting. Here A is a (possibly) noncommutative ring, central over a commutative base ring K (e.g. K = Z). The definition is based on resolutions of A by differential graded rings (better known as DG algebras). We choose a DG ring A', central and flat over K, with a DG ring quasi-isomorphism A' -> A. Such resolutions exist. Our candidate for the "derived category of A-bimodules" is the derived category of A'-bimodules. A recent theorem shows that this category is independent of the resolution A', up to a canonical equivalence. This justifies our definition.

Now we can define what are tilting complexes and dualizing complexes over A, in the noncommutative arithmetic setting. It seems that most of the standard properties of dualizing complexes (proved by Grothendieck for commutative rings in the 1960's, and by myself for noncommutative rings over a field in the 1990's), hold also in this more complicated setting. We can also talk about rigid dualizing complexes in the noncommutative arithmetic setting.

A key problem facing us is that of existence of dualizing complexes. When the base ring K is a field, Van den Bergh (1997) discovered a powerful existence result for dualizing complexes. We are now trying to extend Ven den Bergh's method to the noncommutative arithmetic setting. This is work in progress, joint with Rishi Vyas.

In this talk I will explain, in broad strokes, what are DG rings, DG modules, and the associated derived categories and derived functors. Also, I will try to go into the details of one or two examples, to give the flavor of this material.

For those who want to follow the talk smoothly, I recommend reading, in advance, these notes:

Introduction to Derived Categories
http://arxiv.org/abs/1501.06731 .

An object G of a triangulated category is a strong generator if there is an integer N, so that every object is obtainable from direct sums of shifts of G using no more than N triangles (and possibly taking direct summands). The smallest N which works is called the dimension of the triangulated category, and there has been much literature on this in recent years. We will review the results.

The new theorem we wish to report is that the bounded derived category of coherent sheaves on a scheme X has a strong generator, provided X is essentially of finite type over an excellent scheme of dimension no more than two.

A valued field (k,|.|) is said to be stable (this terminology has no link with model-theoretic stability theory) if every finite extension L of k is defectless, /i.e. /satisfies the equality ∑ e_vf_v= [L:k] where v goes through the set of extensions of |.| to L, and where e_v and f_v are the ramification and inertia indexes of v.

The purpose of my talk is to present a new proof (which is part of current joint reflexions with E. Hrushovski and F. Loeser) of the following classical fact (Grauert, Kuhlmann, Temkin...) : let (k,|.|) be a stable valued field, and let (r_1,...,r_n) be elements of an ordered abelian group G containing |k^*|. Let |.|' be the G-valued valuation on k(T_1,...,T_n) that sends ∑ a_I T^I to max |a_I |.r^I. Then (k(T_1,...,T_n),|.|') is stable too.

Our general strategy is purely geometric, but the proof is based upon model-theoretic tools coming from model theory (which I will first present; no knowledge of model theory will be assumed). In particular, it uses in a crucial way a geometric object defined in model-theoretic terms that Hrushovski and Loeser attach to a given k-variety X, which is called its /stable completion/; the only case we will have to consider is that of a curve, in which the stable completion has a very nice model-theoretic property, namely the definability, which makes it very easy to work with.

The Lang-Bombieri-Noguchi conjecture states that varieties of general type do not have dense sets of rational points. In the case of curves this was proved by Faltings. Also, one can prove this for subvarieties of abelian varieties. We will provide examples where one is able to prove finiteness of rational points (due to work of Martin-Deschamps and Noguchi) and where the Albanese is trivial (building on work of Bogomolov). This is joint work with Daniel Litt.

We will discuss an uniform approach to compute the equivariant K-groups of a smooth and proper spherical variety G/H. This approach is based on combining a recent result of Vezzosi and Vistoli with some old ideas of Brion.

This is a joint work with Mahir Can.