RUI: Legendrian Submanifolds in Contact and Smooth Topology

RUI：接触和光滑拓扑的勒让德子流形

• 批准号: 1406093
• 负责人: Joshua Sabloff
• 金额: $ 14.32万
• 依托单位: Haverford College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406093&HistoricalAwards=false
• 关键词: RUI; Legendrian; Submanifolds; Contact; Smooth

Symplectic topology is a rich field of mathematics with roots in classical physics that has blossomed into a central mathematical field that combines features of geometry (the science of measurement) and topology (the study of the shape of space). This field has a variety of applications including fluid mechanics, differential equations, and the study of the possible shapes of the 3-dimensional space and the 4-dimensional space-time in which we live. The goal of this project is to achieve a better understanding of how symplectic and contact topology sit between geometry and topology, thereby strengthening the foundation for the aforementioned applications. The project's research activities will increase participation and mentoring of students from undergraduate institutions in the critical STEM pipeline. The project's activities will also encourage the exchange of ideas between faculty, graduate students, and undergraduates, thereby providing additional means of bringing undergraduates into the research process. Research with undergraduates will also serve as a pedagogical laboratory for integrating ideas arising in mathematical research into the PI's courses at all levels of the curriculum.Approaching symplectic topology (and its sister field contact topology) through a topological lens has given rise to a young and thriving discipline with interesting questions that explore the boundary between flexibility (when the symplectic world behaves topologically) and rigidity (when the symplectic world behaves geometrically). This project sets forth a program to answer fundamental flexibility and rigidity questions about Legendrian and Lagrangian submanifolds. A number of the projects are concrete and easy to explain, and hence appeal to the imagination of a wide mathematical audience. The proposed research is framed by three themes. The first is a focus on the global properties of the space of Legendrian submanifolds, with specific goals of introducing new quantitative techniques into the study of Lagrangian cobordisms and beginning the study of homotopy groups of spaces of higher dimensional Legendrians. The second theme seeks to link Legendrian and smooth topology, using the Lagrangian cobordism relation to give meaning to certain quantum knot invariants and the conormal construction to connect Legendrian and smooth invariants. The final theme emphasizes investigations into the scope and structure of Legendrian invariants, with one project, in particular, poised to uncover a new type of algebraic pattern for Legendrian Contact Homology.

辛拓扑是一个丰富的数学领域，其根源在于经典物理学，它已经发展成为一个中心数学领域，结合了几何学（测量科学）和拓扑学（空间形状的研究）的特征。 这个领域有各种各样的应用，包括流体力学，微分方程，以及我们生活的三维空间和四维时空的可能形状的研究。这个项目的目标是更好地理解辛拓扑和接触拓扑如何介于几何和拓扑之间，从而加强上述应用的基础。 该项目的研究活动将增加本科院校学生在关键STEM管道中的参与和指导。 该项目的活动还将鼓励教师，研究生和本科生之间的思想交流，从而提供更多的手段，使本科生进入研究过程。 与本科生的研究也将作为一个教学实验室，将数学研究中产生的想法融入PI的课程的各个层次。（和它的姐妹场接触拓扑）通过拓扑透镜已经产生了一个年轻和蓬勃发展的学科与有趣的问题，探索灵活性之间的边界（当辛世界表现为拓扑时）和刚性（当辛世界表现为几何时）。这个项目提出了一个程序来回答勒让德和拉格朗日子流形的基本灵活性和刚性问题。一些项目是具体的，易于解释，因此呼吁广大数学观众的想象力。拟议的研究由三个主题构成。 首先是一个专注于整体性质的空间的勒让德子流形，具体目标是引入新的定量技术的研究拉格朗日配边和开始研究同伦群的空间高维勒让德。 第二个主题旨在连接勒让德和光滑拓扑，使用拉格朗日配边关系来赋予某些量子结不变量的意义，并使用余正规构造来连接勒让德和光滑不变量。 最后一个主题强调调查的范围和结构的Legendrian不变量，与一个项目，特别是，准备揭示一种新的代数模式的Legendrian接触同源。