The family of high rank arithmetic groups is a class of groups playing an important role in various areas of mathematics.
It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more.
A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity.
Model theorists define, in structures whose first-order theory is "stable" (i.e. suitably nice), a notion of independence between elements. This notion coincides for example with linear independence when the structure considered is a vector space, and with algebraic independence when it is an algebraically closed field. Sela showed that the theory of the free group is stable. In a joint work with Rizos Sklinos, we give an interpretation of this model theoretic notion of independence in the free group using Grushko and JSJ decompositions.
A free n-Engel group is the relatively free group of the variety of groups with the identical relation [x, y, y,...,y (n times)]=1. Let n>=20. We show that the free Engel group on at least two generators is not locally nilpotent. Our approach to Engel groups combines
The length of a finite group G is defined to be the maximal length of an unrefinable chain of subgroups going from G to 1. This notion was studied by many authors since the 1940s.
Recently there is growing interest also in the depth of G, which is the minimal length of such a chain. Moreover, similar notions were defined and studied for important families of infinite groups, such as connected algebraic groups and connected Lie groups.
We develop a correspondence between the study of Borel equivalence relations induced by closed subgroups of $S_\infty$, and the study of symmetric models of set theory without choice, and apply it to prove a conjecture of Hjorth-Kechris-Louveau (1998).
Abstract. Let L be a non-archimedean local field of characteristic 0. In this talk we will present a variant of the theory of (\varphi,\Gamma)-modules associated with Lubin-Tate groups, developed by Kisin and Ren, in which we replace the Lubin-Tate tower by the maximal abelian extension \Gamma=Gal (L^ab/L). This variation allows us to compute the functors of induction and restriction for (\varphi,\Gamma)-modules, when the ground field L changes. If time permits, we will also discuss the Cherbonnier-Colmez theorem on overconvergence in our setting.
Joint work with Ehud de Shalit.
We shall discuss several topics regarding symplectic measurements in the classical phase space. In particular: Viterbo's volume-capacity conjecture and its relation with Mahler conjecture, the symplectic size of random convex bodies, the EHZ capacity of convex polytopes (following the work of Pazit Haim-Kislev), and (if time permits) also computational complexity aspects of estimating symplectic capacities.
Abstract: Let V be an irreducible algebraic subvariety of C^n X C^n of
If Schanuel Conjecture holds, under some natural conditions on V, we
show that, if V is defined over the rationals, there exists a in C^n
such that (a, exp(a)) is a generic point of V.
Ergodic theoretic methods in the context of homogeneous dynamics have been highly successful in number theoretic and other applications. A lacuna of these methods is that usually they do not give rates or effective estimates. Einseidler, Venkatesh and Margulis proved a rather remarkable quantitative equidistribution result for periodic orbits of semisimple groups in homogenous spaces that can be viewed as an effective version of a result of Mozes and Shah based on Ratner's measure classification theorem.
This is the second of two lectures on the paper Einseidler,, Margulis, Mohammadi and Venkatesh https://arxiv.org/abs/1503.05884. In this second lecture I will explain how the authors obtain using property tau (uniform spectral gap for arithmetic quotient) quantitaive equidistribution results for periodic orbits of maximal semisimple groups. Surprisingly, one can then use this theorem to establish property tau...
Contingency tables are matrices with fixed row and column sums. They are in natural correspondence with bipartite multi-graphs with fixed degrees and can also be viewed as integer points in transportation polytopes. Counting and random sampling of contingency tables is a fundamental problem in statistics which remains unresolved in full generality.