Nate Gillman (Wesleyan University): Exact formulas for invariants of Hilbert schemes of complex surfaces

A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface $S$ and corresponding invariants of the Hilbert scheme of $n$ points on $S$. This relationship is encoded in certain infinite product $q$-series which are essentially modular forms. Here we make use of the circle method of Hardy and Ramanujan, as improved by Rademacher, to arrive at exact formulas for certain specializations of these $q$-series. This yields convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the $q$-series’ coefficients. This is joint work with Xavier Gonzalez and Matthew Schoenbauer.