The goal of this program is to survey the progress made in the theory of arithmetic and Zariski-dense subgroups, including a variety of applications to algebraic and differential geometry and combinatorics, in the last 10-15 years. Special focus will be given to open problems and directions of future research in these fields.
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 methods 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 have potential applications in the fields of geometry, topology and mathematical physics.
The program will include two mini-courses. The first one led by Dr. G. Prasad, will describe the complete solution for the problem of determination of all fake projective planes posed by D. Mumford, and the arithmetic fake forms of other compact Riemannian symmetric spaces. One of the key ingredients in this work is Prasad's volume formula, an indispensable tool in the analysis of arithmetic lattices. The second course will be led by Dr. Rapinchuk and will be dedicated to the notion of weak commensurability for Zariski-dense subgroups of reductive algebraic groups and their analysis. This approach has been used to resolve problems related to iso-spectral locally symmetric spaces and has led to a new form of rigidity, called eigenvalue rigidity.
A number of experts in algebraic and Lie groups, differential and algebraic geometry and adjacent areas will deliver lectures on various aspects of the subject.
Elgibility 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 send a CV and a brief description of research interests. Researchers in the early stages of their careers are particularly encouraged to apply. Applicants should indicate if they need travel support. ICTS will cover local expenses (i.e., room and board)).