Algebra and Number Theory Day
Algebra and Number Theory Day
Tel Aviv University
March 3, 2016
Organizers: Prof. Lior Bary-Soroker and Prof. Dan Haran
Program:
14:00-14:45 Jakob Stix (Goethe University Frankfurt): Gorenstein orders and abelian varieties over finite fields
15:00-16:00 Coffee and Informal Discussions
16:00-16:45 Mark Shusterman (Tel Aviv University): Ranks of subgroups in boundedly generated groups
17:15-18:00 Nir Avni (Northwestern Univeristy): Push forwards of smooth measures by polynomial.
Abstracts:
Jakob Stix:
Deligne proved an equivalence of categories between ordinary abelian varieties over finite fields and certain linear algebra data by means of canonically deforming to charactersitc 0. We (partly) generalize Deligne's result to arbitrary abelian varieties over finite fields replacing canonical lifts by the commutative algebra of Gorenstein orders. This is joint work with Tommaso Centeleghe.
Mark Shusterman:
A boundedly generated group, such as SL_3(Z), is a group which is a product (as a set) of cyclic subgroups. We exhibit very simple and surprisingbounds on the number of generators of any finite index subgroup of a boundedly generated group. This provides a very elementary proof of the fact that a free group is not boundedly generated, establishes a variant of a conjecture of Lubotzky (proved recently by Meiri) on the ranks of finite index subgroups of special linear groups over the integers, and proves a strong form of a conjecture of Abert, Jaikin-Zapirain, and Nikolov which asserts that the rank gradient of infinite boundedly generated residually finite groups is zero.
Nir Avni:
Title: Push forwards of smooth measures by polynomials.
Abstract: let p:A^n-->A^m be a polynomial map. In the first part of the lecture, I will explain how the singularities of the fibers of p are related to the analytic properties of the push-forwards of smooth measures by p. In the second part, I will talk about applications to counting points of varieties over finite rings, to character sums, and to random matrices.
This event is supported by Tel Aviv University, Minkowski center of Minerva, and by the Mathematical Institute at Tel Aviv (MINT).
