Moduli Problems and Applications of Constructible Sheaves

Project Overview The project will advance mathematics by addressing problems in algebraic geometry using new methods which establish and elucidate connections between sheaves, cluster varieties, and combinatorics. The project will advance physical mathematics as well, by finding wavefunctions for a class of branes in string and M-theory. Intellectual Merit Principal Investigator Zaslow, in a number of works, spearheaded the use of constructible sheaves in symplectic geometry, mirror symmetry and cluster theory. In the present work, Zaslow will exploit these linkages to address several questions in algebraic geometry and string theory. A common thread is an understanding of a moduli space of objects in a category defined by a Legendrian subspace. In particular, Zaslow 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 (joint with Linhui Shen). These generating functions are interpreted as wavefunctions for branes wrapping the Lagrangians. They can be reinterpreted as Cohomological Hall invariants for a symmetric quiver determined by the framing, a phenomenon we call ``framing duality." * 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, reasoning employed by Zaslow with Yau to count curves on K3 surfaces: find the Euler characteristic of the total space. Another viewpoint is to compute with tropical geometry. This project is joint with Helge Ruddat. * Reframe the Deodhar decomposition diagrammatically, using the graphical methods developed by Zaslow with Casals (joint with Ian Le). The techniques apply to other kinds of decompositions as well, beyond double Bruhat cells. Also: study the skeleta of Richardson varieties, which are closely related. Broader Impacts The broader impacts of this proposal lie in the professional development of young mathematicians and in a variety of outreach efforts undertaken by PI Zaslow. Professional Development * Zaslow will continue to disseminate his research widely to young scholars, both in written form and at national and international conferences * Zaslow will continue to support and supervise the Geometry/Physics seminar at Northwestern. Outreach * Zaslow co-created the Causeway Postbaccalaureate Program at Northwestern (supported by NSF-DMS-1916410) with the goal of increasing the number of students in doctoral programs in the United States from groups historically under-represented in the mathematical sciences. Causeway will begin in 2020 and Zaslow will be its inaugural director. * Zaslow has a long history of outreach, and will continue with efforts after serving as chair. Zaslow worked in the Northwestern Bridge program for first-generation and low-income students (2012--2019); Zaslow created and co-ran the Evanston Math Circle, 2012--2018, and still contributes; Zaslow has been involved in and supported various projects toward inclusion as chair. * Zaslow is on the Board of Directors of the Institute for Mathematical and Statistical Innovation (IMSI), and was senior personnel on the establishing grant NSF-DMS-1929348.