Higher Algebraic Structures in Symplectic Geometry and Applications

辛几何是现代数学的一个分支，位于许多其他子领域的交叉点，包括微分拓扑、代数几何、动力系统和理论物理。它的方程描述了许多自然现象，包括经典力学的运动，但它的数学结构惊人地丰富和微妙，需要大量的新工具。该项目试图构建新的数学对象，称为“辛不变量”，并将它们应用于几个几何问题，以帮助理解辛动力学何时会将一种形状扭曲成另一种形状。该项目还将通过组织大型研讨会和讲习班，指导研究生，以及通过计算机可视化和3D打印制作抽象几何可视化的新方法，为传播和教育做出贡献。更具体地说，这个项目将侧重于开发花理论和辛场论中的高等代数结构。专家们知道这些结构已经有一段时间了，但直到最近才被证明在嵌入障碍物和其他几何问题中发挥了强大的作用。PI将继续研究“更高的辛能力”，开发新的算法来促进计算，并探索由此产生的枚举组合学。该项目还将扩展这些高等代数结构的基础和范围，包括它们在Weinstein几何中的作用以及与Fukaya类别和同调镜像对称的联系。此外，PI将继续开发研究辛几何数值模拟的框架，重点放在理论思想和实用的计算机软件上。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 geometry is a branch of modern mathematics lying at the crossroads of many other subfields including differential topology, algebraic geometry, dynamical systems, and theoretical physics. Its equations describe many natural phenomena, including the motions of classical mechanics, yet its mathematical structures are surprisingly rich and subtle, necessitating a wide array of new tools. This project seeks to construct new mathematical objects called "symplectic invariants", and apply them to several geometric problems to help understand when symplectic dynamics can distort one shape into another. This project will also contribute to dissemination and education by organizing large-scale seminars and workshops, mentoring graduate students, and crafting new ways to visualize abstract geometry via computer visualizations and 3D printing.More specifically, this project will focus on exploiting higher algebraic structures in Floer theory and symplectic field theory. These structures have been known to experts for some time, but only recently have been shown to play a powerful role in embedding obstructions and other geometric problems. The PI will continue their investigation of "higher symplectic capacities", developing new algorithms to facilitate computations and exploring the resulting enumerative combinatorics. This project will also expand the foundations and purview of these higher algebraic structures, including their role in Weinstein geometry and connections with Fukaya categories and homological mirror symmetry. Furthermore, the PI will continue to develop a framework for studying numerical simulations in symplectic geometry, with an emphasis on both theoretical ideas and practical computer software.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.