On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture.
In August 2006, Perelman was offered the Fields Medal (“Nobel in math”) for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow”, but he declined the award, stating: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.”
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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_Kähler](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.
I’d have to check but I seem to remember that Perelman is mostly a misanthropist so calling him based is a bit meh. He’s definitely a very interesting mathematician and deserves great respect for his work though.
And as you mention Grothendieck (and since I’m studying in the university he did most of his research in lol) he’s indeed an anarchist but still, way more of a misanthropist than anything (seems to be a common factor about maths geniuses). Though compared to Perelman he had way more of what we could call a political expression.
Stanisław Lem, an extremely intelligent sci-fi author and technological philosopher, was another example of misanthropist. I think intelligent people are frequently misanthropists, and I can understand why - just look how the world looks like.
Grothendieck – besides being a math titan – was an extremely intriguing person. Actually, it is hard to be more “deviated” from regular and especially mediocre human being than he was. While in some sense anarchists, or more generally leftists, I think he was a multi-dimensional, hard to classify personality. For example, after Wikipedia: “devoted himself to political and religious pursuits (first Buddhism and later, a more Catholic Christian vision” or “Influenced by the Catholic mystic Marthe Robin who was claimed to have survived on the Holy Eucharist alone, Grothendieck almost starved himself to death in 1988.” Bruh, WTF 😅 Mathematicians are so interesting people.