Categorical Invariants in Non-commutative Geometry

代数几何，即多项式方程解的几何研究，在过去几年中取得了重大发展。其中，最引人注目的是现代曲线计数不变量的发明，以及对它们的计算可以用与所谓镜像空间几何有关的某些微分方程的解来理解的理解。这样做的想法起源于物理学，通过弦理论和镜像对称领域，但现在是现代代数几何的主要组成部分。Costello在2005年引入了对曲线计数不变量的分类推广，定义为所有属，称为分类枚举不变量（CEIs）。尽管对它们有相当大的兴趣，但人们对它们知之甚少，主要是因为在试图计算它们时出现了一系列困难。2017年，PI与屠俊武（Junwu Tu）共同完成了通用椭圆曲线族cei的首次计算，这一计算导致了Costello、Tu和PI的工作，为cei的定义奠定了新的基础，以一种可显式计算的方式。此外，PI还将指导本科生的研究项目和培养研究生。上述计算中的一个关键因素是选择非交换霍奇滤波的分裂。cei的一般理论没有指定使用哪一种特殊的分裂，但为了获得几何上有意义的不变量，必须做出特殊的选择。在以前的工作中，选择是通过镜像对称，由a模型的辛几何来指导的。然而，许多选择似乎是临时的，而且非交换霍奇过滤的分裂的一般理论在很大程度上是缺失的。在这个项目中，PI将扩展我们对Hodge过滤的分裂在分类不变量定义中所起作用的理解。主要的应用将是计算几何空间的cei，而且用称为矩阵分解范畴的代数结构代替实际的空间。这本身是一个独立的兴趣：这些类别的不变量应该与范-贾维斯-阮-威滕不变量密切相关，但这种关系尚不明确。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。

Algebraic geometry, which is the geometric study of solutions of polynomial equations, has seen in the last few years major developments. Of these, one of the most striking is the invention of modern curve-counting invariants and the understanding that their computation can be understood in terms of solutions of certain differential equations related to the geometry of so-called mirror spaces. The ideas for doing this originated in physics, through the fields of string theory and mirror symmetry, but are now a major part of modern algebraic geometry. Costello introduced in 2005 a categorical generalization of curve-counting invariants, defined for all genera, called categorical enumerative invariants (CEIs). Despite considerable interest, little is known about them, primarily due to a range of difficulties that arise when trying to compute them. The first calculation of CEIs, for the universal family of elliptic curves, was achieved by the PI in 2017, in joint work with Junwu Tu. This computation led to work by Costello, Tu, and the PI which laid out a new foundation for the definition of CEIs, in a way that is explicitly computable. In addition, the PI will direct undergraduate research projects and train graduate students. A crucial ingredient in the above computations is a choice of a splitting of the non-commutative Hodge filtration. The general theory of CEIs does not specify which particular splitting to use, but in order to obtain geometrically meaningful invariants special choices must be made. In previous work the choices were guided, via mirror symmetry, by the symplectic geometry of the A-model. However, many of the choices appear ad hoc, and a general theory of splittings of the non-commutative Hodge filtration is largely missing. In this project the PI will expand our understanding of the role played by the spltting of the Hodge filtration in the definition of the categorical invariants. The main application will be to compute CEIs not only for geometric spaces, but also replace the actual spaces with algebraic structures called categories of matrix factorizations. This is of independent interest in itself: the invariants of these categories should be closely related to the Fan-Jarvis-Ruan-Witten invariants, but such a relationship is not explicitly known.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.