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

辛流形是经典动力系统的现代数学形式。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，比如能量守恒和动量守恒，限制了这个系统的演化方式。这个项目的目标是理解不同辛流形之间的关系，特别是理解轨迹上的限制，比如那些由守恒定律引起的限制，如何从一个辛流形转换到另一个辛流形。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成分，这是一个理想的切入点，本科生。PI目前正在指导一个本科生研究项目，并打算继续让本科生和研究生参与他的研究项目。具体来说，PI旨在构建一个单一的代数对象，即“辛(a -∞,2)-范畴Symp”，它将辛流形的深谷范畴结合在一起，形成一个单一的结构。这扩展了Wehrheim-Woodward早期的工作，其中那些作者将富卡亚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提出的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，提出一种策略，以理解在复杂爆炸下深谷类别是如何变化的。二是探索与其他领域的对接。通过表述Symp所需的组合结构，PI构造了2-相关面体，这是一种复杂的抽象多面体，可以很好地适应几种现有的组合对象。在与Alexei Oblomkov的合作中，PI构造了2-共轭面体的复化版本，这些复化版本形成了与M_{0，n}-bar密切相关的一个丰富的新正则对数光滑复变体族。另一个联系是更高范畴的理论：Symp将是一个(a -∞,2)-范畴，PI打算展示的一个新的代数结构是某些(a -∞,2)-范畴的一个方便模型。最后，PI旨在理解辛上同调的辛(a -∞,2)-范畴的分支，辛上同调是非紧辛流形的一个重要不变量。事实上，理解单位1-态射在Symp中的作用，将使PI能够像Abouzaid猜想的那样，为辛上同配备链级代数结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.