Categorical Invariants in Non-commutative Geometry

代数几何是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，最引人注目的之一是现代曲线计数不变量的发明，以及它们的计算可以根据与所谓镜像空间的几何形状相关的某些微分方程的解来理解。这样做的想法起源于物理学，通过弦理论和镜像对称领域，但现在已成为现代代数几何的主要部分。 Costello 在 2005 年引入了为所有属定义的曲线计数不变量的分类概括，称为分类枚举不变量 (CEIs)。尽管人们对它们很感兴趣，但人们对它们知之甚少，这主要是由于在尝试计算它们时出现了一系列困难。 2017 年，PI 与 Junwu Tu 合作，首次计算了通用椭圆曲线族的 CEI。这种计算促成了 Costello、Tu 和 PI 的工作，以显式可计算的方式为 CEI 的定义奠定了新的基础。此外，PI还将指导本科生研究项目并培养研究生。上述计算中的一个关键因素是非交换霍奇过滤分裂的选择。 CEI 的一般理论没有指定使用哪种特定的分割，但为了获得几何上有意义的不变量，必须做出特殊的选择。在之前的工作中，选择是通过镜像对称、A 模型的辛几何来引导的。然而，许多选择似乎都是临时的，并且很大程度上缺少非交换霍奇过滤分裂的一般理论。在这个项目中，PI 将扩展我们对霍奇过滤在分类不变量的定义中所扮演的角色的理解。主要应用不仅是计算几何空间的 CEI，而且还用称为矩阵分解类别的代数结构替换实际空间。这本身具有独立的利益：这些类别的不变量应该与 Fan-Jarvis-Ruan-Witten 不变量密切相关，但这种关系尚不清楚。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力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.