Admissible Lagrangians, Fukaya categories, and homological mirror symmetry.

非对易几何的思想，即人们试图用非局部术语来思考空间，而位置等概念不再有意义，已经成为一种有前途的语言，可以统一数学的各个领域。在辛几何中--经典力学相空间的几何--点“太小”而不相关，考虑一类称为拉格朗日子流形的半维子空间更自然。考虑这些子流形的几何，而不是点，自然会产生一个非对易空间：福谷范畴。一个受理论物理学思想启发的著名数学猜想“同调镜像对称”断言，许多福谷范畴实际上等价于传统的交换空间，如代数几何中研究的那些空间。本研究项目的主要目标是扩大镜像对称适用的设置范围。具体地说，目标是在一个足够广泛的环境中建立同调镜像对称，以展示所有由多项式方程组定义的（交换）代数空间作为福谷范畴的实例。在这个程序中的一个关键步骤是研究从配备一个或多个（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的所有完整的交叉（可能是非紧的）环面品种。这种设置中的镜像空间是所谓的复曲面朗道-金兹伯格模型（即，非紧复曲面Calabi-Yau变种配备正规函数）。一个关键的成分，在研究他们的福谷类别是同时容许的概念，关于一个集合的环面单项式。这为证明同调镜像对称性所需的Floer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，它将一般类型（包括非紧类型）的变体置于与更经典的卡-丘和法诺情况相同的基础上。本项目还将研究Landau-Ginzburg模型和仿射簇的福谷范畴的一些明显新的结构特征，这些特征表明了以前未被注意到的同调镜像对称的函性性质，以及仿射簇计算的新方法。

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.