Dualities in Enumerative Algebraic Geometry

In this project, the PI will work broadly on interlinked areas centered around modern enumerative algebraic geometry, motivated by conjectural dualities in physics. Enumerative algebraic geometry concerns counting geometric objects satisfying a given list of constraints in a space defined by polynomial equations. In good situations, these counts do not change under generically varying the set of constraints and therefore define enumerative invariants. Among the most extensively studied enumerative invariants are Gromov-Witten invariants and Donaldson-Thomas invariants, which have important applications in mathematics and physics. While Gromov-Witten invariants correspond to counts of curves in a space, Donaldson-Thomas invariants are integers counting geometric objects called vector bundles or more generally sheaves. In mathematics, both the count of curves and sheaves in a space carry crucial information about geometric properties of the space, as such properties are used to characterize and classify different spaces. On the other hand in physics, Gromov-Witten and Donaldson-Thomas invariants appear in the context of string theory, where they correspond to counts of particles in gauge theory as well as black hole microstates in quantum gravity. This award will also support graduate students working with the PI. There are four main strands of research directions the project is branched into. In the first part, the project intends to investigate conjectural dualities, establishing new bridges between enumerative algebraic geometry and quantum physics. These dualities are expected to relate Gromov-Witten and Donaldson-Thomas invariants of toric Calabi-Yau threefolds to the quantization of the mirror curves. A second subject of study is related to canonical bases in representation theory. In particular, one of the goals of the project is to use enumerative algebraic geometry of surfaces to construct canonical bases for double affine Hecke algebras and their generalizations. In a third direction, the project intends to establish connections between refined Donaldson-Thomas invariants and invariants arising in real algebraic geometry. The focus of the final part of the project will be on holomorphic symplectic geometry. In this context, the goal is to provide a categorification of Floer theory for holomorphic symplectic manifold, motivated by structures in Donaldson-Thomas theory. The unifying ingredient here is the recent advance in the understanding of wall-crossing, mirror symmetry, logarithmic geometry, and tropical geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.