The most well-known mathematician with left-wing political views was [https://en.wikipedia.org/wiki/Alexander_Grothendieck#Political_activism](Alexander Grothendieck), who was an anarchist, so I guess not based by our standards.
On the maximally reactionary end, [https://en.wikipedia.org/wiki/Erich_K%C3%A4hler](Erich Kähler) volunteered for the Wehrmacht and stayed loyal to the Third Reich until at least the 1980s (I suppose no different from the government of west Germany).
Perelman himself seems to have disappeared completely from public view. Apparently, there is no even somewhat recent news about him, even in Russian.