Topics in Algebraic Geometry: Gromov-Witten Theory and Donaldson-Thomas Theory

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This project focuses on two related topics in algebraic geometry, the study of solutions of systems of polynomial equations, that are motivated by their connections with physical theories (string theory and gauge theory). The first topic is Gromov-Witten (GW) theory, which roughly speaking is about a systematic way of counting numbers of curves with particular constraints in a space defined by a set of polynomial equations. For example on a plane there are four conics passing three given points and two given lines in general positions. The second topic is Donaldson-Thomas (DT) theory, which studies properties of the space parametrizing sets of polynomials defining one dimensional objects with certain topological constraints in a space. In a so-called Calabi-Yau space, a series of remarkable conjectures say that counting using GW theory is equivalent to that using DT theory. Different branches of mathematics are linked together by these two theories and deep properties of geometric objects have been uncovered by calculating invariants they give rise to. In this project the PI will investigate a conjectured duality between these two theories and relate them to other branches of mathematics and physics. In more detail, the projects in this proposal are designed to study several aspects of enumerative properties of the moduli spaces of geometric objects in algebraic geometry. The first topic is Gromov-Witten theory. The PI will study birational transformations between GW invariants, calculate the quantum cohomology of symplectic Deligne-Mumford stacks, and relate the monodromy of the quantum connections to the monodromy coming from the equivalence between the derived categories. The second topic is Donaldson-Thomas theory. The PI will study local DT invariants via Berkovich analytic spaces, apply the cotangent invariants of the PI and R. Thomas to find more dualities for geometric spaces, and investigate the motivic Donaldson-Thomas invariants by the method of motivic integration.