Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry

在现代几何中，用非局部术语来思考空间，而不是依赖于位置等经典概念，往往是有用的。辛几何尤其如此--经典力学相空间的几何，其中的点“太小”而不相关，并且考虑称为拉格朗日子流形的对象更自然。这些对象和它们的相互作用被称为福谷范畴的代数结构（或“非交换空间”）编码。一个受理论物理思想启发的著名数学猜想“同调镜像对称”断言，福谷范畴实际上常常等价于代数几何中研究的诚实（交换）空间。本研究计画研究具有一个或多个（交换）函数和/或相对于无穷远处的某些方向的辛流形的福谷范畴。这些范畴的丰富结构，来自于额外的数据，产生了新的方法来理解各种几何构造对辛流形的福谷范畴的影响;这反过来又会大大扩展同调镜像对称性可以被验证的设置范围。该项目还将为几个研究生提供研究机会，一般旨在使这一研究领域更广泛的数学界。从技术的角度来看，这个项目的第一个目标是开发一个更好的几何设置部分包裹福谷类别，因为它们出现在镜像对称，在公式计算是可能的，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一个一般性的证明同调镜像对称（不一定是卡-丘）完全相交的环面品种，并研究规范基地的坐标环。最后，该项目还将通过统一不同的部分包裹福谷类别的建议结构，寻找镜像对称中功能性的新实例，以及研究几何结构和分类结构之间的相互作用，为该领域带来概念上的清晰性。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。

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.