Moduli Spaces and Applications of Constructible Sheaves

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This work lies at the interface of mathematics and theoretical physics, and the findings will be relevant to both communities. More specifically, the project will establish and elucidate connections between the mathematical subjects of sheaves, cluster varieties, and combinatorics, and advance physical mathematics by finding wave functions for a class of objects in string theory. The broader impacts of this project lie in the professional development of the PI's graduate students and postdocs, the dissemination of research through talks and the Northwestern Geometry/Physics seminar, and through the PI's many outreach activities.A common thread within the project is an understanding of a moduli space of objects in a category defined by a Legendrian subspace. In collaborative work with Linhui Shen, the PI will use cluster theory to determine generating functions for all-genus open Gromov-Witten invariants of certain Lagrangian three-folds filling Legendrian surfaces, given the data of a framing. These generating functions are interpreted as wave functions for branes wrapping the Lagrangians. They can be reinterpreted as Cohomological Hall invariants for a symmetric quiver determined by the framing, a phenomenon called "framing duality." A second research direction is to count the number of nodal curves of genus-g in a g-dimensional family inside a toric surface. The idea is to create a Beauville-type integrable system (curves and their Jacobians) and to use, in this novel setting, the reasoning employed by the PI and Yau to count curves on K3 surfaces, namely: find the Euler characteristic of the total space. Another viewpoint is to compute with tropical geometry. This project is joint with Helge Ruddat. Finally, using the graphical methods developed by the PI with Casals, in joint work with Ian Le, the PI plans to reframe the Deodhar decomposition diagrammatically. The techniques apply to other kinds of decompositions as well, beyond double Bruhat cells. The PI will also study the closely related topic of the skeleta of Richardson varieties.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.