For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical difficulties. It was not until the late 1990s that new ideas from symplectic geometry and string theory allowed rigorous definitions to be made of these “curve counting problems”.
I will try to explain a little bit about this story, focussing on the “MNOP conjecture” of Maulik-Nekrasov-Okounkov-Pandharipande. This states that 2 of these different definitions (“Gromov-Witten theory” and “stable pairs” theory) give equivalent information. I will explain how its recent proof in many cases by Pandharipande and Pixton allows computations of previously untouchable Gromov-Witten invariants, and proofs of magic predictions from string theory like the “KKV conjecture”.
The speaker is a Professor of Pure Mathematics at Imperial College, London. You can read more about him here.