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''), bounde...
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