Correspondences in enumerative geometry: Hilbert schemes, K3 surfaces and modular forms

Enumerative geometry is concerned with counting geometric objects on spaces defined by polynomial equations. The subject, which has roots going back to the ancient Greeks, was revolutionized by string theory in the 90s and has since become a fundamental link between algebraic geometry, representation theory, number theory and physics. With K3Mod I propose to establish a wide range of new correspondences in enumerative geometry. These link together different enumerative theories and open new perspectives to attack long-standing problems concerning the quantum cohomology of the Hilbert scheme of points on surfaces, modular properties of invariants of K3 surfaces, string partition functions of Calabi-Yau threefolds with links to Conway Moonshine, and a major case of the Crepant Resolution Conjecture.The geometry of the Hilbert scheme of points on a surface will play a central role. I aim to prove a correspondence between its Gromov-Witten theory, and the Donaldson-Thomas theory of certain threefold families. Correspondences for moduli spaces of Higgs bundles and the orbifold theory of the symmetric product of surfaces will be considered as well. This provides methods to prove that Gromov-Witten invariants of Hilbert schemes of points on K3 surfaces are Fourier coefficients of quasi-Jacobi forms, possibly leading to a complete solution of their enumerative geometry. After elliptic curves, K3 surfaces form the simplest Calabi-Yau geometry for which a complete understanding of the Gromov-Witten theory is in reach. For elliptic threefolds, I will study the relationship of their Donaldson-Thomas invariants with quasi-Jacobi forms, using both degeneration techniques and wallcrossing formulae. The research goals of this proposal will lead to exciting new connections between geometry, modular forms, and representation theory. The results will provide a clear understanding of the interplay between Hilbert schemes, K3 surfaces, and modularity in enumerative geometry.