Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry

在现代几何中，用非局部术语来考虑空间，而不是依赖于位置等经典概念，往往是有用的。辛几何——经典力学的相空间几何，其中的点“太小”而不相关，考虑称为拉格朗日子流形的对象更自然。这些对象及其相互作用由称为深谷范畴的代数结构（或“非交换空间”）编码。一个引人注目的数学猜想受到理论物理思想的启发，“同调镜像对称”，断言深谷范畴实际上通常等同于代数几何中研究的那种诚实（交换）空间。本研究项目研究具有一个或多个（交换）函数和/或在无穷远处相对于某些方向的辛流形的Fukaya范畴的版本。这些类别的丰富结构来自于附加数据，提供了理解各种几何结构对辛流形的深谷范畴的影响的新方法；这反过来应该大大扩展设置的范围，在其中，同调镜像对称可以被验证。该项目还将为一些研究生提供研究机会，总体目标是使这一研究领域更容易被更广泛的数学界所接受。从技术角度来看，这个项目的第一个目标是为部分包裹的深谷类别开发一个更好的几何设置，因为它们在镜像对称中出现，达到一个公式，其中计算是可能的，预期的结构特征是明显的。另一个主要目标是利用这些基础给出环变中（不一定是Calabi-Yau）完全交的同调镜像对称的一般证明，并研究其坐标环的正则基。最后，该项目还将通过统一部分包裹的Fukaya类别的不同提议结构，寻找镜像对称的新功能实例，以及研究几何结构和分类结构之间的相互作用，为该领域带来概念清晰度。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 modern geometry it is often useful to think of spaces in non-local terms rather than relying on classical concepts such as position. This is especially true of symplectic geometry -- the geometry of phase spaces of classical mechanics, where points are "too small" to be relevant, and it is more natural to consider objects called Lagrangian submanifolds. These objects and their interactions are encoded by an algebraic structure (or "non-commutative space") called the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry", asserts that Fukaya categories are in fact often equivalent to honest (commutative) spaces of the sort studied in algebraic geometry. This research project studies versions of Fukaya categories for symplectic manifolds equipped with one or more (commuting) functions and/or relative to certain directions at infinity. The rich structure of these categories, coming from the additional data, yields new ways of understanding the effect of various geometric constructions on the Fukaya category of a symplectic manifold; this in turn should greatly extend the range of settings in which homological mirror symmetry can be verified. The project will also provide research opportunities for several graduate students and generally aim to make this research area more accessible to the broader mathematical community. From a technical standpoint, the first goal of this project is to develop a better geometric setup for partially wrapped Fukaya categories as they arise in mirror symmetry, to arrive at a formulation where computations are possible and the expected structural features are manifest. The other main goal is to use these foundations to give a general proof of homological mirror symmetry for (not necessarily Calabi-Yau) complete intersections in toric varieties, and to study canonical bases of their coordinate rings. Finally, this project will also bring conceptual clarity to the field by unifying different proposed constructions of partially wrapped Fukaya categories, finding new instances of functoriality in mirror symmetry, and studying the interplay between geometric constructions and categorical ones.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.