The strange duality conjecture.

For C a compact Riemann surface of positive genus, the strange
duality conjecture asserts that the space of sections of certain theta bundle on moduli of vector bundles of rank r and level k on C is naturally dual to a similar space of sections for rank k and level r. This is equivalent to a geometric description of the space of non abelian theta functions as the linear span of some geometrically defined sections: the generalised theta divisors (as
emphasised in the papers of Arnaud Beauville and Mihnea Popa).

Recently, I proved this conjecture for generic C, subsequently it was proved for all C by Alina Marian and Dragos Oprea. I will talk about the (pre)history of this problem (classical relation between intersection theory of Grassmann varieties and invariant theory of the special linear groups, and its geometrization), some techniques in the proofs, and relations with Hitchin's projective connection (which need to be explored systematically).