Dualities in Enumerative Algebraic Geometry

在这个项目中，PI将广泛研究以现代枚举代数几何为中心的相互关联的领域，其动机是物理学中的猜想对偶性。枚举代数几何涉及在多项式方程定义的空间中对满足给定约束列表的几何对象进行计数。在良好的情况下，这些计数在约束集合的一般变化下不会改变，因此定义了枚举不变量。列举不变量中研究最广泛的是Gromov-Witten不变量和Donaldson-Thomas不变量，它们在数学和物理中有着重要的应用。Gromov-Witten不变量对应于空间中曲线的计数，而Donaldson-Thomas不变量是对称为矢量束或更一般的束的几何对象进行计数的整数。在数学中，一个空间中的曲线数和束数都携带着关于空间几何性质的重要信息，因为这些性质被用来描述和分类不同的空间。另一方面，在物理学中，Gromov-Witten和Donaldson-Thomas不变量出现在弦理论的背景下，它们对应于规范理论中的粒子计数以及量子引力中的黑洞微态。该奖项还将支持与PI一起工作的研究生。该项目主要分为四个研究方向。在第一部分中，该项目打算研究推测对偶性，在枚举代数几何和量子物理之间建立新的桥梁。这些对偶性有望将环面Calabi-Yau三倍的Gromov-Witten和Donaldson-Thomas不变量与镜像曲线的量化联系起来。第二个研究主题与表征理论中的规范基础有关。特别地，该项目的目标之一是使用曲面的枚举代数几何来构建双仿射Hecke代数及其推广的规范基。在第三个方向上，该项目打算建立改进的Donaldson-Thomas不变量与实际代数几何中出现的不变量之间的联系。项目最后一部分的重点将放在全纯辛几何上。在这种情况下，目标是提供全纯辛流形的花理论的分类，由Donaldson-Thomas理论中的结构驱动。这里的统一因素是最近对墙壁穿越、镜像对称、对数几何和热带几何的理解取得了进展。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。

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.