On the Hodge conjecture

[This is a guest post by Burt Totaro. This blog post was initially written in a different file format and converted using AI. — T.] These are strange times for mathematicians. It now seems possible that AI companies will burn through vast resources in order to prove some new fact about the Hodge conjecture. I’d like to discuss the current status of the Hodge conjecture, in order to think about the value to the mathematical community of exploring such hard problems. Thanks to Terry Tao for suggesting this guest post. The Hodge conjecture crystallizes a big mystery: the relation between topology and algebraic geometry, or (what amounts to the same) between real and complex geometry. In short, real submanifolds are flexible, whereas complex submanifolds are quite rigid, and it is a challenge to relate the two. The setting is a “smooth complex projective variety” , a complex manifold defined by algebraic equations. We want to understand the possible shapes of complex algebraic subspaces…