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Moduli Spaces and Applications of Constructible Sheaves

可构造滑轮的模空间和应用

基本信息

批准号：
2104087
负责人：
Eric Zaslow
金额：
$ 41.39万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104087&HistoricalAwards=false
关键词：
Moduli Spaces Applications Constructible Sheaves

项目摘要

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.

这项工作位于数学和理论物理的界面，研究结果将与两个社区有关。更具体地说，该项目将建立和阐明层，簇品种和组合学的数学科目之间的联系，并通过寻找弦理论中一类对象的波函数来推进物理数学。该项目的更广泛的影响在于PI的研究生和博士后的专业发展，通过讲座和西北几何/物理研讨会传播研究成果，并通过PI的许多外展活动。该项目中的一个共同点是理解由Legendrian子空间定义的一类对象的模空间。在与Linhui Shen的合作中，PI将使用集群理论来确定某些拉格朗日三重填充Legendrian曲面的所有亏格开放Gromov-Witten不变量的生成函数，给定框架的数据。这些生成函数被解释为包裹拉格朗日量的膜的波函数。它们可以被重新解释为由框架决定的对称矩阵的上同调Hall不变量，这种现象称为“框架对偶性”。“第二个研究方向是计算复曲面内g维族中亏格g的节点曲线的数量。我们的想法是创建一个Beauville型可积系统（曲线和它们的雅可比行列式），并在这个新的设置中使用PI和Yau所采用的推理来计算K3曲面上的曲线，即：找到总空间的欧拉特征。另一种观点是用热带几何进行计算。该项目是与Helge Ruddat联合进行的。最后，使用PI与Casals开发的图形方法，在与Ian Le的联合工作中，PI计划以图解方式重新构建Deodhar分解。这些技术也适用于其他类型的分解，超越了双布鲁哈特细胞。 PI还将研究与Richardson品种相关的Aprieta主题。该奖项反映了NSF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。