CAREER: Higher Algebra and Symplectic Geometry

职业：高等代数和辛几何

• 批准号: 2044557
• 负责人: Hiro Tanaka
• 金额: $ 40万
• 依托单位: Texas State University - San Marcos
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2044557&HistoricalAwards=false
• 关键词: CAREER; Higher; Algebra; Symplectic; Geometry

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.

辛几何是一种有效地编码物理学中经典问题的运动定律的几何。由于被称为福谷范畴的强大代数工具的出现，在过去的几十年里，该领域的活动激增。与此同时，一种新的“谱代数”语言的发展--一种将传统的加法和乘法的概念与更现代的工具相结合的代数，用于研究任意高维的形状--使我们能够利用代数直觉来组织复杂的结构。这两个故事情节已经非常富有成效，但尚未交叉授粉。该项目旨在构建一个长期寻求的桥梁，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。该项目还将支持许多旨在丰富和多样化数学界的教育举措。其中包括为学生创建数学播客、为学生学习感兴趣的当代数学技术的研讨会，以及旨在培养新兴数学家社区的各种课外活动。该项目的技术和协作核心是在破碎的全纯对象的模栈上的可因子分解结构的形式化。通过证明在刘维扇区的Floer理论中出现的全纯对象的通常模引起了在模栈上的可因子化层中编码的变形问题，这个项目的目标是为刘维扇区建立光谱包裹的福谷类别，作为这些变形问题的解决方案。这个奖项反映了美国国家科学基金会的法定使命，并且通过利用基金会的智力价值进行评估，被认为是值得支持的和更广泛的影响审查标准。