University of California, San Diego. Academic year: 2021-2022. Monday 2:00-3:00. Meeting ID: 939 5383 2894, Password: structures
 Date Speaker Topic November 1 Nir Avni Northwestern University Model theory of higher rank arithmetic groups   Abstract: I'll describe a new rigidity phenomenon of lattices in higher rank semisimple groups. Specifically, I'll explain why the theories of such groups can't have (finitely generated) deformations, why these groups have a very rich collection of definable subgroups, and finish by discussing a conjecture saying that being a higher rank arithmetic lattice is a first-order property. Based on joint works with Alex Lubotzky and Chen Meiri. November 8 Francois Thilmany University of Louvain on the connections between discreteness of arithmetic groups and the Lehmer conjecture.   Abstract: The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the "uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). (Joint work with Lam Pham.) November 15 Junho Peter Whang Seoul National University Diophantine study of Stokes matrices   Abstract: Stokes matrices (i.e. unipotent upper triangular matrices) and their nonlinear braid group actions arise naturally in a number of geometric and algebraic contexts. Integral Stokes matrices are often of particular interest, motivating their reduction theory. After reviewing classical work of Markoff treating the case of 3-by-3 matrices, we describe joint work with Yu-Wei Fan for the 4-by-4 case by establishing an exceptional connection to $${\rm SL}_2$$-character varieties of surfaces. This will also serve as an opportunity to present our recent work on effective finite generation of integral points on the latter moduli spaces. Time permitting, we finish by presenting new results (and problems) for Stokes matrices of larger dimension. November 22 Keivan Mallahi-Karai Jacobs University Optimal linear sofic approximation of countable groups   Abstract: Various notions of metric approximation for countable groups have been introduced and studied in the last decade, with sofic and hyperlinear approximations being two notable examples among them. The class of linear sofic groups was introduced by Glebsky and Rivera and was subsequently studied by Arzhantseva and Paunescu. This mode of approximation uses the general linear group over, say, the field of complex numbers as model groups, equipped with the distance defined using the normalized rank. Among their other interesting results, Arzhantseva and Paunescu prove that every linear sofic group is 1/4-linear sofic, where the constant 1/4 quantifies how well non-identity elements can be separated from the identity matrix. In this talk, which is based on joint work with Maryam Mohammadi Yekta, we will address the question of optimality of the constant 1/4 and report on some progress in this direction. December 6 Alex Kontorovich Rutgers University Asymptotic Length Saturation for Zariski Dense Surfaces   Abstract: The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set "saturates" (resp. "asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible). January 10 Matt Litman UC Davis Markoff-type K3 Surfaces: Local and Global Finite Orbits   Abstract: Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in $$\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$$ cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily $$W_k$$ of such surfaces, we construct finite orbits in $$W_k(\mathbb{C})$$ by studying small orbits that appear in $$W_k(\mathbb{F}_p)$$ for many values of $$p$$ and $$k$$. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran. TBA TBA UCSD TBA   Abstract: TBA.