TGiF-Seminar: Tropical geometry in Frankfurt (First meeting Winter Semester 2023/24)

Adam Afandi (Universität Münster)

Abstract: By using the Gromov-Witten/Hurwitz correspondence, Okounkov and Pandharipande showed that certain generating functions of stationary descendent Gromov-Witten invariants of a smooth elliptic curve are quasimodular forms. In this talk, I will discuss the various ways one can express the discriminant modular form in terms of these generating functions. The motivation behind this calculation is to provide a new perspective on tackling a longstanding conjecture of Lehmer from the middle of the 20th century; Lehmer posited that the Ramanujan tau function (i.e. the Fourier coefficients of the discriminant modular form) never vanishes. The connection with Gromov-Witten invariants allows one to translate Lehmer’s conjecture into a combinatorial problem involving characters of the symmetric group and shifted symmetric functions.