Relating Fukaya Categories Using Combinatorics, Operads, and Nonlinear Elliptic Partial Differential Equations

使用组合学、运算和非线性椭圆偏微分方程关联 Fukaya 类别

• 批准号: 1906220
• 负责人: Nathaniel Bottman
• 金额: $ 14.38万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2019-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906220&HistoricalAwards=false
• 关键词: Relating; Fukaya; Categories; Using; Combinatorics

Symplectic manifolds are the modern mathematical setting for classical dynamical systems. For instance, consider the motion of the Earth, the Moon, and a satellite, under the influence of gravity. The state of this system can be described by the positions and momenta of the three bodies, and the collection of all possible states is an example of a symplectic manifold. Physical laws, such as the conservation of energy and of momentum, restrict how this system can evolve. The goal of this project is to understand relationships between different symplectic manifolds, and specifically to understand how the restrictions on trajectories, such as those arising from conservation laws, can be translated from one symplectic manifold to another. While progress has been made toward this goal, the PI has proposed the first comprehensive approach. This project has a significant combinatorial component, which is an ideal point-of-entry for undergraduates. The PI is currently supervising an undergraduate research project, and aims to continue involving undergraduate and graduate students in his research program. Specifically, the PI aims to construct a single algebraic object, the "symplectic (A-infinity,2)-category Symp", which binds together the Fukaya categories of symplectic manifolds into a single structure. This extends earlier work of Wehrheim-Woodward, in which those authors associate functors between Fukaya categories to Lagrangian correspondences. Besides this central component, the PI's proposed project involves three other elements. First, the PI will compute portions of Symp in some concrete situations. The PI has begun to develop techniques for computing the functors associated to Lagrangian correspondences in the context of symplectic reduction, and plans to continue these explorations. In particular, he is working with Ritter to builds on earlier work by Ritter-Smith in order to suggest a strategy for understanding how the Fukaya category changes under complex blowup. Second, the PI will explore connections to other fields. Formulating the combinatorial structures necessary for Symp led the PI to construct the 2-associahedra, which are intricate abstract polytopes which fit in well with several existing combinatorial objects. In joint work with Alexei Oblomkov, the PI is constructing complexified versions of 2-associahedra, which form a rich new family of proper log-smooth complex varieties with a close relationship to M_{0,n}-bar. Another connection is to the theory of higher categories: Symp will be an (A-infinity,2)-category, a new algebraic structure which the PI intends to show is a convenient model for certain (infinity,2)-categories. Finally, the PI aims to understand the ramifications of the symplectic (A-infinity,2)-category for symplectic cohomology, an important invariant of a noncompact symplectic manifold. Indeed, understanding the role of unit 1-morphisms in Symp should enable the PI to equip symplectic cohomology with a chain-level algebraic structure, as conjectured by Abouzaid.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.

辛流形是经典动力系统的现代数学背景。例如，考虑地球、月球和卫星在重力影响下的运动。这个系统的状态可以用三个物体的位置和动量来描述，所有可能状态的集合就是辛流形的一个例子。物理定律，如能量守恒和动量守恒，限制了这个系统如何演化。该项目的目标是了解不同辛流形之间的关系，特别是了解如何限制轨迹，如守恒定律，可以从一个辛流形转换到另一个。虽然在实现这一目标方面取得了进展，但PI提出了第一个全面的方法。这个项目有一个重要的组合组件，这是一个理想的切入点为本科生。PI目前正在监督一个本科生研究项目，并旨在继续让本科生和研究生参与他的研究计划。具体来说，PI的目标是构建一个单一的代数对象，“辛（A-无穷大，2）-范畴Symp”，它将辛流形的福谷范畴结合在一起成为一个单一的结构。这扩展了Wehrheim-Woodward的早期工作，在该工作中，作者将福谷范畴之间的函子与拉格朗日对应联系起来。除了这一核心组成部分，PI的拟议项目还涉及其他三个要素。首先，PI将在某些具体情况下计算Symp的部分。PI已经开始在辛约化的背景下开发计算与拉格朗日对应相关的函子的技术，并计划继续这些探索。特别是，他正在与里特合作，以里特-史密斯早期的工作为基础，提出一种理解复杂爆破下福谷范畴如何变化的策略。其次，PI将探索与其他领域的联系。为Symp制定必要的组合结构，PI构建了2-associahedra，这是一种复杂的抽象多面体，适合与几个现有的组合对象。在与Alexei Oblomkov的合作中，PI正在构造2-伴随多面体的复杂化版本，它们形成了一个丰富的新的真对数光滑复簇族，与M_{0，n}-bar有密切的关系。另一个连接是更高的范畴理论：Symp将是一个（A-无穷大，2）范畴，一个新的代数结构，PI打算显示是一个方便的模型，某些（无穷大，2）范畴。最后，PI的目的是了解分支的辛（A-无穷大，2）-范畴的辛上同调，一个重要的不变量的非紧辛流形。事实上，理解单元1-态射在Symp中的作用应该使PI能够为辛上同调配备链级代数结构，正如Abouzaid所指出的那样。该奖项反映了NSF的法定使命，并被认为值得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来支持。