A13: Generating series for curve counting

Counts of curves can often be organized in a generating series. Important properties of the numbers such as recursions can be expressed in terms of generating series. In connection with a duality relation of elliptic curves motivated by physics, the so-called mirror symmetry, generating series of counts of curves in a surface which is a Cartesian product of an elliptic curve with a line can be expressed and studied via Feynman integrals. In this project, we will study further such generating series, using the tropical method. That is, counts of curves will first be expressed via counts of tropical curves by means of suitable correspondence theorems. The counts of tropical curves can then be studied using combinatorics. The methods that we plan to develop will therefore also be of interest to tropical geometry.