Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.

非交换几何（non - commutative geometry）的概念是，人们试图用非局部的术语来思考空间，而位置等概念不再有意义，它已经成为一种很有前途的语言，可以统一不同的数学领域。在辛几何（经典力学相空间的几何）中，点“太小”而不相关，更自然的是考虑一类称为拉格朗日子流形的半维子空间。考虑这些子流形的几何，而不是点，自然会产生一个非交换空间：Fukaya范畴。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”，断言许多深谷范畴实际上等同于传统的交换空间，例如代数几何中研究的那些。本研究项目的主要目标是扩大镜像对称适用的设置范围。具体来说，目标是在足够广泛的环境中建立同调镜像对称，以展示由多项式方程系统定义的所有（交换）代数空间作为Fukaya范畴的实例。本课程的关键步骤是研究由具有一个或多个交换函数的辛流形产生的非交换几何。更具体地说，这个项目的主要目标是证明Kontsevich的同调镜像对称猜想在（可能非紧的）环变中所有完全交。这种设置中的镜像空间是所谓的环面朗多-金兹堡模型（即，配备正则函数的非紧凑环面卡拉比-丘变种）。在他们的深谷范畴的研究的一个关键因素是关于一个集合的同时可容许性的概念环单项。这为证明同调镜像对称性所需的花理论计算提供了一种新的方法。预期的结果将是一个理解镜像对称的框架，它将各种一般类型（包括非紧化类型）置于与更经典的Calabi-Yau和Fano案例相同的基础上。本项目还将研究Landau-Ginzburg模型和仿射品种的Fukaya类别的一些明显的新结构特征，这些特征表明了以前未被注意到的同调镜像对称的功能性质，以及仿射品种的计算的新方法。

The idea of noncommutative geometry, where one tries to think of spaces in nonlocal terms and concepts such as position no longer make sense, has emerged as a promising language to unify diverse areas of mathematics. In symplectic geometry -- the geometry of phase spaces of classical mechanics -- points are "too small" to be relevant, and it is more natural to consider a class of half-dimensional subspaces called Lagrangian submanifolds. Considering the geometry of these submanifolds, rather than points, naturally gives rise to a noncommutative space: the Fukaya category. A remarkable mathematical conjecture inspired by ideas from theoretical physics, "homological mirror symmetry," asserts that many Fukaya categories are in fact equivalent to conventional commutative spaces such as those studied in algebraic geometry. The main goal of this research project is to expand the range of settings to which mirror symmetry is applicable. Specifically, the goal is to establish homological mirror symmetry in a broad enough setting to exhibit all (commutative) algebraic spaces defined by systems of polynomial equations as instances of Fukaya categories. A key step in this program is to study the non-commutative geometry that arises from a symplectic manifold equipped with one or more (commuting) functions. More specifically, the main goal of this project is to prove Kontsevich's homological mirror symmetry conjecture for all complete intersections in (possibly noncompact) toric varieties. The mirror spaces in this setting are so-called toric Landau-Ginzburg models (i.e., noncompact toric Calabi-Yau varieties equipped with regular functions). A key ingredient in the study of their Fukaya categories is the concept of simultaneous admissibility with respect to a collection of toric monomials. This gives a new approach to the Floer-theoretic calculations needed to prove homological mirror symmetry. The expected outcome will be a framework for understanding mirror symmetry that places varieties of general type (including noncompact ones) on the same footing as the more classical Calabi-Yau and Fano cases. This project will also investigate some apparently new structural features of Fukaya categories for Landau-Ginzburg models and for affine varieties, which suggest previously unnoticed functoriality properties of homological mirror symmetry, as well as a new approach to computations for affine varieties.