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.

拓扑的目的是确定在连续变形下形状的哪些特征不会改变，并在许多科学领域（例如冷凝物质物理学，宇宙学，数据分析和生物学）中使用具体应用。因为人们可以通过聆听听起来的方式来推断有关鼓的形状的信息，因此可以通过研究仪表理论中自然产生的某些部分偏微分方程的解决方案（在仪表上的几何语言中自然产生）来定义尺寸3和4的拓扑不变性。制定了粒子物理标准模型的基本定律。该项目的重点是塞伯格（Seiberg）的方程式，由于其几何特性，它提供了一个完美的有利位置，可以探究拓扑与邻近主题（例如双曲几何形状，光谱理论和复杂分析）的相互作用。一个关键目标是探索与这些数学领域的界面上新的调查途径。为此，该项目还将在本科和研究生层面创造许多研究机会。PI将使用以下主要目标研究Monopole Floer同源性：探索使用光谱几何学工具，探索三个维度在三个维度中的相互作用，椭圆形偏微分方程的理论和Selberg痕量公式；使用索引理论和耦合的摩尔斯同源性索引理论和功能性特性，开发由一般负面确定的共同体引起的地图的计算工具；研究与代数几何形状中的经典主题的可能关系，例如研究riemann表面的theta分裂的奇异性；并使用PIN（2） - 对称性以L-Space的精神来了解具有小型同源性的理性同源性领域的潜在几何特征。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的评估来支持的。智力优点和更广泛的影响审查标准。