**Abstract. **This talk is a progress report on our worked focused on a new form of rigidity that we call eigenvalue rigidity. The latter is based on the notion of weak commensurability of Zariski-dense subgroups of semi-simple algebraic groups introduced in our joint work with G. Prasad (Publ. math. IHES 109(2009)), which provides a convenient way of matching the eigenvalues of semi-simple elements of these subgroups. A detailed analysis of this notion for arithmetic groups enabled us to resolve somelong-standing problems about isospectral compact locally symmetric spaces. Currently, there is growing evidence that some key results can be extended from arithmetic groups to arbitrary finitely generated Zariski-dense subgroups, yielding thereby certain rigidity statements, based on the eigenvalue information, in this generality (including the situations where the subgroups at hand are free groups).This work has led to new directions of research in the theory of algebraic groups, one of which is the analysis of forms of a given absolutely almost simple algebraic group that have good reduction at a given set of discrete valuations of the base field.

Video: http://www.birs.ca/events/2019/5-day-workshops/19w5040/videos/watch/201912100900-Rapinchuk.html