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.

The basis of elements of the highest weight representations of affine Lie algebra of type A can be labeled in three different ways, my multipartitions, by piecewise linear paths in the weight space, and by canonical basis elements. The entire infinite basis is recursively generated from the highest weight vector of operators f_i from the Chevalley basis of the affine Lie algebra, and organized into a crystal called a Kashiwara crystal. We describe cases where one can move between the different labelings in a non-recursive fashion, particularly when the crystal has some symmetry.

We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in Isom(X) where X is a proper CAT(0) space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang’s finiteness theorem for certain non-positively curved metric spaces. This is a joint work with Arie Levit.

Let G be a finite group, V a finite dimensional G- module over a field F, and S(V) the symmetric algebra of V. The above problem seeks to determine when is the ring of invariants S(V)^G , a polynomial ring. In the non-modular case (i.e. char(F) being prime to order(G)), this was settled in the Shephard-Todd-Chevalley theorem. The modular case (i.e. char(F) divides order (G) ), is still wide open. I shall discuss some older results due to Serre, Nakajima , Kemper-Malle and explain some new results, mostly in dimension 3.

The sofic groups and hyperlinear groups are groups approximable by finite symmetric and by unitary groups, respectively. I recall their definitions and discuss why those classes of groups are interesting. Then I consider approximations by other classes of groups and review some results, including rather recent ones by N. Nikolov, J. Schneider, A.Thom, https://arxiv.org/abs/1703.06092 . If time permits I'll speak about stability and its relations with approximability.

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).

Title for the day: "Quantum ergodicity and spectral theory with a discrete flavour"

9:00-10:50: Shimon Brooks (Bar Ilan), "Delocalization of Graph Eigenfunctions"
14:00-15:50: Elon Lindenstrauss (HUJI), "Quantum ergodicity on graphs and beyond"

See also the Basic Notions by Elon Lindenstrauss @ Ross 70 (16:30).

Abstract for morning session:

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 dimension n. 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.

Special day on sparsification
Speakers: Ilan Newman and Yuri Rabinovich.

Part I:   10:30 - 12:30
Part II:  14:00 - 15:50

Abstract for the day:
Time permitting, we plan to discuss the following topics (in this order):

1.
* Additive Sparsification and VC dimension
* Multiplicative Sparsification
* Examples: cut weights, cut-dimension of L_1 metrics, general metrics,
                    and their high-dimensional analogues

2.

We will explain what high dimensions are, and describe some questions and answers in geometry and combinatorics of the high dimensional world.

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...

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.