RUI: Legendrian Submanifolds in Contact and Smooth Topology

辛拓扑是一个丰富的数学领域，根源于经典物理学，现已发展成为一个结合了几何学（测量科学）和拓扑学（空间形状的研究）特征的中心数学领域。 该领域具有多种应用，包括流体力学、微分方程以及对我们生活的 3 维空间和 4 维时空可能形状的研究。该项目的目标是更好地理解辛拓扑和接触拓扑如何位于几何和拓扑之间，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，从而为本科生参与研究过程提供额外的手段。 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的各个级别的课程中。通过拓扑透镜研究辛拓扑（及其姐妹场接触拓扑）已经产生了一个年轻而蓬勃发展的学科，其中提出了有趣的问题，探索灵活性（当辛世界表现出拓扑性时）和刚性（当辛世界表现出几何性时）之间的界限。该项目提出了一个程序来回答有关勒让德和拉格朗日子流形的基本灵活性和刚性问题。许多项目具体且易于解释，因此吸引了广大数学读者的想象力。拟议的研究由三个主题构成。 第一个重点是勒格朗德子流形空间的全局属性，具体目标是将新的定量技术引入拉格朗日配边的研究，并开始研究高维勒格伦空间的同伦群。 第二个主题试图将勒让德和光滑拓扑联系起来，使用拉格朗日配边关系赋予某些量子结不变量以意义，并使用共正规结构来连接勒让德和光滑不变量。 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，准备揭示传奇联系同调的一种新型代数模式。

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.