CAREER: Higher Algebra and Symplectic Geometry

辛几何是一种对运动规律进行有效编码的几何，它规定了物理学中的经典问题。在过去的几十年里，由于被称为Fukaya范畴的强大代数工具的出现，该领域出现了大量的活动。与此同时，“谱代数”--一种将传统的加法和乘法概念与更现代的研究任意高维形状的工具相结合的代数--的新语言的发展，使我们能够利用代数直觉来组织复杂的结构。这两个故事情节非常富有成效，但还没有交叉授粉。这个项目的目的是建立一座人们期待已久的桥梁，不仅产生研究辛几何的谱方法，而且建立研究谱代数的辛工具。该项目还将支持旨在丰富数学界并使其多样化的许多教育举措。这些活动包括为学生和由学生创建数学播客，为学生学习感兴趣的当代数学技巧举办研讨会，以及旨在培养新兴数学家社区的各种联合课程活动。该项目的技术和协作核心是将破碎的全纯对象的模堆叠上的可分解结构形式化。通过证明在Liouville扇区的Floer理论中出现的全纯物体的通常的模数会引起模堆叠上的可分解滑轮中编码的变形问题，该项目旨在为Liouville扇区构建频谱包裹的Fukaya类别，作为这些变形问题的解决方案。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 geometry that efficiently encodes laws of motion dictating classical problems in physics. The last few decades have seen a burst of activity in the field thanks to the emergence of powerful algebraic tools called Fukaya categories. At the same time, the development of a new language for “spectral algebra”—an algebra that mixes traditional notions of adding and multiplying with more contemporary tools for studying shapes of arbitrarily high dimensions—has allowed us to organize sophisticated structures using algebraic intuitions. These two storylines have been highly fruitful, but have yet to cross-pollinate. This project aims to construct a long-sought-after bridge, to not only produce spectral methods for studying symplectic geometry, but to establish symplectic tools for studying spectral algebra. The project will also support numerous educational initiatives aimed at enriching and diversifying the mathematics community. These include the creation of a math podcast for and by students, a workshop for students to learn contemporary mathematical techniques of interest, and various co-curricular activities aimed at fostering communities of emerging mathematicians. The technical and collaborative heart of the project is the formalization of factorizable structures on moduli stacks of broken holomorphic objects. By showing that the usual moduli of holomorphic objects appearing in the Floer theory of Liouville sectors give rise to deformation problems encoded in factorizable sheaves on the moduli stacks, the project aims to construct spectral wrapped Fukaya categories for Liouville sectors as solutions to these deformation problems.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.