Counting curves in 3-folds and 5-folds

Andrei Okounkov, December 3rd, 2013

There are many ways to count curves of given degree and genus in threefolds which,
conjecturally, are all related by simple functional relations for the corresponding
all-genera generating functions. In this talk, I will discuss a rather different conjecture
about these functions that relates them to counting curves in certain Calabi-Yau 5-folds.
In this way, the genus-counting variable q of the generating function becomes an
equivariant parameter in extra dimensions. Among other things, this gives a very
geometric way to sum up these series into a rational function of q. Joint work with
Nikita Nekrasov.