RUI: Legendrian Submanifolds in Contact and Smooth Topology

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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.