The goal of the program is to survey the progress in the theory of arithmetic and Zariski-dense subgroups achieved in the last 10-15 years, including a variety of applications to algebraic and differential geometry, combinatorics and other areas. Special attention will be given to open problems and directions of further research. The program will showcase an array of techniques employed to investigate Zariski-dense subgroups but the focus will be on the use of methods from algebraic and analytic number theory and arithmetic theory of algebraic groups. These have been successfully used to tackle long-standing problems such as fake projective planes, isospectral and length-commensurable locally symmetric spaces, expanding graphs and multi-dimensional expanders, and many others, with new applications to geometry, topology and mathematical physics likely to emerge in the near future.
Also, recent results on bounded/non-bounded generation (``diophantine techniques in linear groups''), bounded generation by conjugacy classes, results of Lubotzky etal. on stability and asymptotic cohomology will be discussed. Special units of the program will be devoted to a new approach to the Bruhat-Tits theory developed by Gopal Prasad, and to groups with good reduction.
In a more geometric vein some problems for isospectral locally symmetric spaces will be discussed. These problems lead to a new form of rigidity, called eigenvalue rigidity. One of the scientific goals of the program is to understand to what extent these ideas can be applied in pseudo-Riemannian geometry in general and to the problem of classification of compact space forms in particular. A number of experts in algebraic and Lie groups, differential and algebraic geometry and adjacent areas will deliver 2- and 4-hour long lectures on various aspects of the subject, including classical rigidity, pseudo-Riemannian geometry, property (T) and its applications, construction of expander graphs, and others
Accommodation will be provided for outstation participants at our on campus guest house.
ICTS is committed to building an environment that is inclusive, non discriminatory and welcoming of diverse individuals. We especially encourage the participation of women and other under-represented groups.
Eligibility criteria: This program is addressed to active researchers (at any stage of their academic career) in the area of Lie groups and their discrete subgroups, algebraic groups and related areas of algebraic and differential geometry. Applicants should upload a CV and provide a brief description of research interests in the application. Researchers in the early stages of their careers are particularly encouraged to apply.