Partially Wrapped Fukaya Categories and Functoriality in Mirror Symmetry

在现代几何学中，用非局部的术语来思考空间通常是有用的，而不是依赖于位置等经典概念。辛几何尤其如此--经典力学中相空间的几何，其中的点“太小”而不相关，考虑被称为拉格朗日子流形的物体更为自然。这些对象及其相互作用由一个称为Fukaya范畴的代数结构(或“非对易空间”)编码。受理论物理思想启发的一个引人注目的数学猜想--“同调镜像对称”--断言Fukaya范畴实际上常常等价于代数几何中所研究的那种诚实(交换)空间。这项研究项目研究了配备了一个或多个(交换)函数和/或相对于无穷远处的某些方向的辛流形的Fukaya范畴的版本。这些范畴的丰富结构，来自额外的数据，产生了新的方法来理解各种几何结构对辛流形的Fukaya范畴的影响；这反过来应该大大扩展了可以验证同调镜像对称性的设置范围。该项目还将为几名研究生提供研究机会，并总体上旨在使更广泛的数学界更容易接触到这一研究领域。从技术角度来看，这个项目的第一个目标是为部分包裹的Fukaya类别开发一个更好的几何设置，因为它们出现在镜像对称中，以达到一种计算可能和预期结构特征明显的公式。另一个主要目的是利用这些基础给出环丛中完全交(不一定是Calabi-Yau)的同调镜像对称性的一般证明，并研究它们的坐标环的典范基。最后，这个项目还将通过统一部分包裹的Fukaya类别的不同建议结构，找到镜像对称中的功能性的新实例，并研究几何结构和范畴结构之间的相互作用，为该领域带来概念上的清晰度。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.