Enumerative geometry of Hilbert schemes

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The main objective of this project is to study the geometry of moduli spaces of sheaves on low dimensional varieties and reveal possible connections of the subject with other fields of mathematics. The celebrated Gromov-Witten/Donaldson-Thomas correspondence is a conjectural correspondence between the integrals of characteristic classes of the universal ideal-sheaf over the Hilbert scheme of curves in the threefold and integrals over the moduli space of maps into the threefold. The conjecture is proven for toric threefolds and the lowest degree nontrivial characteristic classes. The project proposed is intended to extend the known results beyond the toric case and study the case of higher degree characteristic classes. Another part of the project is devoted to studying the topology of the Hilbert scheme of points on singular planar curves. A conjectural formula was proposed by the PI and V. Shende for the Poincare polynomials of the Hilbert scheme in terms of link invariants of the links of the singularities of the curve. To prove the formula is one of the goals of the project.Moduli spaces of sheaves are mathematical models for the field theories.Mathematical physics predictions, such as duality between the field theories and string theory or path integral formulas for knot invariants, can be translated into fascinating mathematical conjectures. The goal of this project is to find a proof of some of these conjectures.