Lagrangian fillings of Legendrian submanifolds

This project studies contact and symplectic geometry, an area of mathematics studying key physical phenomena. In particular, the project develops new results and techniques to mathematically model, manipulate and classify the behavior of rays of light, in the form of geometric optics, and the evolution of temperature and equilibrium states, in the form of thermodynamics. The novel mathematical methods developed in this project rely on a geometric approach, based on the manipulation of a new type of diagrams introduced by the Principal Investigator, that bypasses brute force computations in favor of quicker and clearer qualitative understanding. The results of this project lead to more efficient predictions of the behavior of rays of light and the evolution of heat, thus allowing for a wider range of scientific applications. The project also contains a number of such applications, including results in algebra and combinatorics, as well as mathematical physics, via the quantization of classical systems. The broader impacts of this project include a series of intensive workshops and seminars aimed at training the current generation of early career mathematicians, and two new research monographs streamlining some of the most useful and recent developments in the area, making them accessible and usable to the broader scientific community. In a nutshell, the two major goals of the project are the study of Legendrian submanifolds and their Lagrangian fillings, and the development of applications of Lagrangian submanifolds. The project addresses geometric and algebraic properties of the moduli space of Lagrangian fillings of Legendrian submanifolds, and the construction and manipulation of combinatorial structures from Lagrangian submanifolds. The project includes new applications of these goals to cluster algebras, mathematical physics and algebraic combinatorics. The advent of sheaf-theoretic techniques has now opened a wide variety of questions on Legendrian submanifolds, specifically via the study of their moduli spaces of Lagrangian fillings. In this project, we study the classification of Lagrangian fillings of a given Legendrian submanifold, including the derived geometry of their moduli, incarnated by the space of sheaves singularly supported on that Legendrian. In addition, we develop of a series of constructions connecting the symplectic geometry of Lagrangian submanifolds to other areas of mathematics, including algebraic combinatorics and string theory. Specifically, the bridge between 3D plabic graphs and Lagrangian fillings, the combinatorics of higher-dimensional rulings for Legendrian fronts, and the construction of spectral networks from Betti Lagrangians. 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.