Pierre Dèbes: The inverse Galois problem and the Tchebotarev density theorem: Abstract: We will go back to the roots of the inverse Galois problem and revisit some important milestones. The goal is to explain, based on some recent work, some analytic constraints related to some number-theoretical questions around the Tchebotarev density theorem.

Shlomo Gelaki: On two properties of symmetric tensor categories: Abstract: Symmetric rigid tensor categories include the representation categories of groups. In my talk I will focus on two properties of representation categories of groups, which were proved by J-P. Serre. The first one concerns the semisimplicity of representations, and the second one concerns the spectrum of the Grothendieck ring of the category. Serre conjectured that any symmetric rigid tensor category has the first property, and asked whether any Tannakian category (a special type of a symmetric rigid tensor category) has the second property (in particular, the representation category of a modular Lie algebra). In the talk I will discuss some results, which include a proof of Serre's conjecture and a positive answer to his question, as special cases.

Robert Guralnick: Characterizations of Finite Solvable Groups and Lifting in Frattini Covers: Abstract: As a consequence of Thompson's classification of the minimal nonsolvable finite groups, one had various characterizations of finite solvable groups (e.g., every 2-generated subgroup is solvable). We will discuss other conditions which characterize finite solvable and p-solvable groups. Some of this is related to problems of lifting elements in Frattini covers with product 1 and has consequences for towers of covers of curves.

Moshe Jarden: Diamonds in the torsion of abelian varieties: Abstract: Theorem: Let A be an abelian variety over a Hilbertian field K. Then every field extension M of K contained in K(A_tor) is Hilbertian.
We explain the theorem and its proof.

Michael Larsen: Elliptic curves and finitely generated Galois groups: Abstract: Let K be a field, not locally finite, whose absolute Galois group is finitely generated. Some time ago, I conjectured that every elliptic curve over K has infinite rank. I will discuss progress on this conjecture, especially recent joint work with Bo-Hae Im, using Ramsey-theoretic methods to solve the problem when E has rational 2-torsion.

Sebastian Petersen: Independence of l-adic Galois representations over function fields

(Joint work with Gebhard Böckle and Wojciech Gajda)

Abstract: (pdf version) Let K be a field. For every rational prime l let ρ_l: Gal_K → Γ_l be a homomorphism into a group Γ_l and consider the induced homomorphism

ρ: Gal_K → ∏_l Γ_l , σ → (ρ_l(σ))_l.

Following Serre we call the family (ρ_l)_l of homomorphisms almost independent if there exists a finite separable extension E/K such that ρ(Gal_E)=∏_l ρ_l(Gal_E).
Important examples of such families are given by the representations ρ_{A, l}: Gal_K → Aut(T_l A) of Gal_K on the l-adic Tate modules of an abelian variety A over K. By a classical theorem of Serre the family (ρ_{A, l})_l is almost independent for every abelian variety A over a number field K. A recent preprint of Serre proves an analogous result for families of representations of Gal_K afforded by the l-adic ètale cohomology groups of an arbitrary separated algebraic scheme X over a number field K. Serre, Illusie and Jarden asked whether these results can be extended to the case of a ground field K which is a finitely generated extension of Q of transcendence degree > 0. In brief: We answer this question affirmatively. Furthermore we have results in the case of a finitely generated ground field K of positive characteristic. (In positive characteristic certain adaptions are necessary already in the statements.)

Lior Rosenzweig: Galois groups of random elements of linear groups: (Joint work with Alex Lubotzky); Abstract: Let A be a finitely generated subgroup of GL_n(k), where k is a finitely generated field of characteristic zero. In the talk we will discuss what type of groups can occur as Gal(k(g)/k), where g is an element of A, and k(g) is the splitting field of the characteristic polynomial of g. In particular, we will show that if the Zariski closure of A does not contain a central torus (e.g semisimple), then given a random walk on A, the behaviour of Gal(k(g)/k) is generic with respect to connected components of the Zariski closure. The proof uses the recently developed "sieve theory for groups" via expenders and property 'tau' of linear groups.

Louis Rowen: The images of non-commutative polynomials evaluated on matrices: (Joint work with Alexey Kanel-Belov and Sergey Malev); Abstract: (pdf version) We survey what is known about evaluations of a polynomial p in several non-commuting variables taken in a matrix algebra M_n(K) over a field. It has been conjectured that for any n, when p is multilinear, the image of p is either zero, or the set of scalar matrices, or the set sl_n(K) of matrices of trace 0, or all of M_n(K). The conjecture is true for n=2, and although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of M_n(K). We also give reasonable partial results for n=3.

Shmuel Weinberger: Quadratic forms over group rings and large networks: Abstract: I will try to explain an algebraic/categorical approach to thinking about how information propagates through networks (use of buzzwords intentional) and give some applications of this to quadratic forms over group rings of discrete groups.

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The 19th Amitsur Memorial Symposium הסימפוזיון התשעה עשר באלגברה לזכר שמשון אברהם עמיצור ז"ל

215 Sackler Medicine Building

Program

Talks and abstracts:

The 19th Amitsur Memorial Symposium
הסימפוזיון התשעה עשר באלגברה לזכר שמשון אברהם עמיצור ז"ל