I’m at a conference this week, Elliptic Integrals in Mathematics and Physics, in Ascona, Switzerland.
Elliptic integrals are the next frontier after polylogarithms, more complicated functions that can come out of Feynman diagrams starting at two loops. The community of physicists studying them is still quite small, and a large fraction of them are here at this conference. We’re at the historic Monte Verità conference center, and we’re not even a big enough group to use their full auditorium.
There has been an impressive amount of progress in understanding these integrals, even just in the last year. Watching the talks, it’s undeniable that our current understanding is powerful, broad…and incomplete. In many ways the mysteries of the field are clearing up beautifully, with many once confusingly disparate perspectives linked together. On the other hand, it feels like we’re still working with the wrong picture, and I suspect there’s still a major paradigm shift in the future. All in all, the perfect time to be working on elliptics!
Does this have any connection to the elliptic curves used by Wiles to prove Fermat’s Last Theorem? https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem
“Elliptic curve” refers to the same type of mathematical object in both contexts, yes. I think the modularity theorem he proved/others extended is somewhat relevant here, and part of why it’s possible to talk about these things in one unified formalism, but not knowing a lot about the math behind this I’m not sure whether a weaker result suffices.