New Directions in Monopole Floer Homology

单极子同源性的新方向

• 批准号: 2203498
• 负责人: Francesco Lin
• 金额: $ 28.22万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203498&HistoricalAwards=false
• 关键词: New; Directions; Monopole; Floer; Homology

The goal of topology is to identify which features of a shape do not change under a continuous deformation, with concrete applications in many areas of science such as condensed matter physics, cosmology, data analysis, and biology. As one can infer information about the shape of a drum by listening to the way it sounds, one can define topological invariants of spaces of dimension 3 and 4 by studying the solutions of certain partial differential equations naturally arising in gauge theory, the geometric language in which the fundamental laws of the Standard Model of particle physics are formulated. This project focuses on the Seiberg-Witten equations which, because of their geometric nature, provide a perfect vantage point to probe the interactions of topology with neighboring subjects such as hyperbolic geometry, spectral theory, and complex analysis. A key objective is the exploration of new avenues of investigation at the interface with these fields of mathematics. Towards this end, this project will also create many research opportunities both at the undergraduate and graduate level.The PI will study monopole Floer homology, with the following main goals: explore the interactions with hyperbolic geometry in three dimensions using tools from spectral geometry, the theory of elliptic partial differential equations, and the Selberg trace formula; develop computational tools for the maps induced by general negative definite cobordisms using index theory and functoriality properties of coupled Morse homology; investigate possible relations with classical topics in algebraic geometry such as the study of the singularities of the theta divisor of a Riemann surface; and use Pin(2)-symmetry to understand potential geometric characterizations of rational homology spheres with small Floer homology, in the spirit of the L-space conjecture.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.

拓扑学的目标是确定形状的哪些特征在连续变形下不会改变，在凝聚态物理、宇宙学、数据分析和生物学等许多科学领域都有具体应用。正如我们可以通过听鼓的声音来推断有关鼓的形状的信息一样，我们可以通过研究规范理论中自然产生的某些偏微分方程的解来定义 3 维和 4 维空间的拓扑不变量，规范理论是一种几何语言，粒子物理标准模型的基本定律是在其中表述的。该项目重点研究 Seiberg-Witten 方程，由于其几何性质，该方程为探索拓扑与双曲几何、谱理论和复分析等邻近学科的相互作用提供了完美的有利位置。一个关键目标是探索与这些数学领域相结合的新研究途径。为此，该项目还将为本科生和研究生创造许多研究机会。 PI将研究单极子Floer同调性，主要目标如下：利用谱几何、椭圆偏微分方程理论和Selberg迹公式等工具探索三维双曲几何的相互作用；使用指数理论和耦合莫尔斯同调的函子性质，开发由一般负定配边引起的映射的计算工具；研究与代数几何中经典主题的可能关系，例如黎曼曲面 theta 除数奇点的研究；并本着 L 空间猜想的精神，使用 Pin(2) 对称性来理解具有小弗洛尔同调性的有理同调球的潜在几何特征。该奖项反映了 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。