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，所有可能的状态的集合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进化。这个项目的目标是了解不同辛流形之间的关系，特别是了解如何将对轨迹的限制，如由守恒定律引起的限制，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，但国际和平研究所提出了第一个全面的方法。这个项目有一个重要的组合部分，这是一个理想的本科生入门点。PI目前正在指导一个本科生研究项目，目标是继续让本科生和研究生参与他的研究项目。具体地说，PI的目的是构造一个单一的代数对象，“辛(A-无穷，2)-范畴辛”，它将辛流形的Fukaya范畴结合到一个单一结构中。这推广了Wehheim-Woodward早期的工作，在该工作中，作者将Fukaya范畴之间的函子联系到Lagrangian对应。除了这一核心组成部分外，国际和平倡议提出的项目还包括其他三个要素。首先，PI将在一些具体情况下计算symp的部分。PI已经开始开发在辛约化的背景下计算与拉格朗日对应有关的函子的技术，并计划继续这些探索。特别是，他正在与里特合作，建立在里特-史密斯早期工作的基础上，以便提出一种策略，以了解福屋类别在复杂井喷下如何变化。其次，PI将探索与其他领域的联系。构造符号所需的组合结构使PI构造了2-结合体，它是复杂的抽象多面体，与现有的几个组合对象很好地吻合。在与Alexei Oblomkov的合作中，PI正在构造2-结合面的复化形式，它形成了一族丰富的新的适当的对数光滑复变数族，与M_{0，n}-bar关系密切。另一个联系是高级范畴理论：symp将是一个(A-无穷，2)-范畴，PI想要展示的一种新的代数结构对于某些(无穷，2)-范畴是一个方便的模型。最后，PI的目的是理解辛(A-无穷，2)-辛上同调范畴的分支，辛上同调是非紧辛流形的一个重要不变量。事实上，了解单位1-态射在符号中的作用应该使PI能够像Abouzaid猜测的那样，为辛上同调配备链级代数结构。这一奖项反映了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。

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.