RUI: Legendrian Submanifolds in Contact and Smooth Topology

辛拓扑学是一个丰富的数学领域，起源于经典物理学，现已发展成为一个结合几何(测量科学)和拓扑学(研究空间形状)特征的中心数学领域。这个领域有各种各样的应用，包括流体力学、微分方程式，以及研究我们生活的3维空间和4维时空的可能形状。这个项目的目标是更好地理解辛拓扑和接触拓扑学是如何位于几何和拓扑学之间的，从而加强上述应用程序的基础。该项目的研究活动将增加来自本科院校的学生在关键的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，从而提供更多将本科生带入研究过程的手段。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整合到PI的所有课程中。通过拓扑透镜探索辛拓扑(及其姊妹场接触拓扑学)已经催生了一门年轻而蓬勃发展的学科，它带有有趣的问题，探索灵活性(当辛世界表现为拓扑学时)和刚性(当辛世界表现为几何行为时)之间的边界。这个项目提出了一个程序来回答有关Legendrian子流形和Lagrangian子流形的基本灵活性和刚性问题。许多项目都是具体的，很容易解释，因此吸引了广泛的数学观众的想象力。这项拟议的研究由三个主题构成。第一个是关于Legendrian子流形空间的整体性质的研究，其具体目的是将新的定量方法引入到拉格朗日余边空间的研究中，并开始研究高维Legendrian子流形空间的同伦群。第二个主题试图将Legendrian拓扑学与光滑拓扑学联系起来，利用拉格朗日协边关系赋予某些量子纽结不变量以意义，并利用正则结构将Legendrian不变量与光滑不变量联系起来。最后一个主题强调对传奇不变量的范围和结构的研究，特别是一个项目，准备揭示传奇接触同调的一种新类型的代数模式。

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.