Categorical Invariants in Non-commutative Geometry

代数几何是对多项式方程的解的几何研究，在过去的几年里有了重大的发展。其中最引人注目的是现代曲线计数不变量的发明，并认识到它们的计算可以通过与所谓镜像空间的几何有关的某些微分方程解来理解。这样做的想法起源于物理学，通过弦理论和镜像对称领域，但现在是现代代数几何的主要部分。Costello在2005年提出了一种曲线计数不变量的范畴泛化，定义为所有的类，称为范畴枚举不变量(CEI)。尽管有相当大的兴趣，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与军武图合作，实现了普适族椭圆曲线的第一次CEIS计算。这一计算导致了Costello、Tu和PI的工作，他们以一种明确可计算的方式为Ceis的定义奠定了新的基础。此外，PI还将指导本科生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对易Hodge滤波的分裂。CEIS的一般理论没有指定使用哪种特殊分裂，但为了获得具有几何意义的不变量，必须做出特殊选择。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，而且关于非对易霍奇过滤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过滤的分裂在定义范畴不变量中所起的作用的理解。它的主要应用将不仅是计算几何空间的CEI，而且还将用称为矩阵因式分解范畴的代数结构来代替实际空间。这本身就具有独立的意义：这些类别的不变量应该与Fan-Jarvis-Ruan-Witten不变量密切相关，但这种关系并不明确。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.