Relating Fukaya Categories Using Combinatorics, Operads, and Nonlinear Elliptic Partial Differential Equations

Princeton University, Princeton NJ

Investigators

Abstract

Symplectic manifolds are the modern mathematical setting for classical dynamical systems. For instance, consider the motion of the Earth, the Moon, and a satellite, under the influence of gravity. The state of this system can be described by the positions and momenta of the three bodies, and the collection of all possible states is an example of a symplectic manifold. Physical laws, such as the conservation of energy and of momentum, restrict how this system can evolve. The goal of this project is to understand relationships between different symplectic manifolds, and specifically to understand how the restrictions on trajectories, such as those arising from conservation laws, can be translated from one symplectic manifold to another. While progress has been made toward this goal, the PI has proposed the first comprehensive approach. This project has a significant combinatorial component, which is an ideal point-of-entry for undergraduates. The PI is currently supervising an undergraduate research project, and aims to continue involving undergraduate and graduate students in his research program. Specifically, the PI aims to construct a single algebraic object, the "symplectic (A-infinity,2)-category Symp", which binds together the Fukaya categories of symplectic manifolds into a single structure. This extends earlier work of Wehrheim-Woodward, in which those authors associate functors between Fukaya categories to Lagrangian correspondences. Besides this central component, the PI's proposed project involves three other elements. First, the PI will compute portions of Symp in some concrete situations. The PI has begun to develop techniques for computing the functors associated to Lagrangian correspondences in the context of symplectic reduction, and plans to continue these explorations. In particular, he is working with Ritter to builds on earlier work by Ritter-Smith in order to suggest a strategy for understanding how the Fukaya category changes under complex blowup. Second, the PI will explore connections to other fields. Formulating the combinatorial structures necessary for Symp led the PI to construct the 2-associahedra, which are intricate abstract polytopes which fit in well with several existing combinatorial objects. In joint work with Alexei Oblomkov, the PI is constructing complexified versions of 2-associahedra, which form a rich new family of proper log-smooth complex varieties with a close relationship to M_{0,n}-bar. Another connection is to the theory of higher categories: Symp will be an (A-infinity,2)-category, a new algebraic structure which the PI intends to show is a convenient model for certain (infinity,2)-categories. Finally, the PI aims to understand the ramifications of the symplectic (A-infinity,2)-category for symplectic cohomology, an important invariant of a noncompact symplectic manifold. Indeed, understanding the role of unit 1-morphisms in Symp should enable the PI to equip symplectic cohomology with a chain-level algebraic structure, as conjectured by Abouzaid. 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.