Moduli Spaces and Applications of Constructible Sheaves

这项工作位于数学和理论物理的界面，其发现将与两个社区相关。更具体地说，该项目将建立和阐明束、簇变化和组合等数学学科之间的联系，并通过寻找弦理论中一类物体的波函数来推进物理数学。该项目的更广泛影响在于PI的研究生和博士后的专业发展，通过讲座和西北几何/物理研讨会以及PI的许多外展活动传播研究成果。项目中的一个共同主线是理解由Legendrian子空间定义的类别中的对象的模空间。在与沈林辉的合作中，PI将使用聚类理论来确定给定框架数据的某些拉格朗日三折叠填充Legendrian曲面的全属开放Gromov-Witten不变量的生成函数。这些生成函数被解释为包住拉格朗日量的膜的波函数。它们可以被重新解释为由分幅决定的对称颤振的上同调霍尔不变量，这种现象称为“分幅对偶”。第二个研究方向是计算环面内g维族中g属节点曲线的个数。我们的想法是创建一个波维尔型可积系统（曲线及其雅可比矩阵），并在这种新颖的设置中使用PI和Yau用来计算K3曲面上曲线的推理，即：找到总空间的欧拉特征。另一种观点是用热带几何来计算。这个项目是与Helge Ruddat合作的。最后，使用PI与Casals共同开发的图形方法，PI与Ian Le共同工作，计划以图表形式重新构建Deodhar分解。除了双布鲁哈特细胞，这项技术也适用于其他种类的分解。PI还将研究与之密切相关的理查森品种的骨架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.