LEAPS-MPS: Integrable systems and Hamiltonian torus actions in symplectic geometry

Amherst College, Amherst MA

Investigators

Abstract

Symmetries play an important role in understanding any complex system, and this project will further the understanding of symmetries in a field of mathematics called symplectic geometry. Tie a ball to a string, hold the string up, and give the ball a push. Can you predict how the ball will swing? Though it has now extended far beyond its roots, symplectic geometry was originally developed to study classical dynamical systems, such as this pendulum. To predict its motion, it is useful to notice that while the position and velocity of the pendulum change over time, the values of both the total energy and the angular momentum stay constant. This concept can be extended beyond this specific example to a broad class of systems called integrable, which are the focus of this project. Roughly speaking, a system is integrable if it possesses a large number of quantities which do not change as the system evolves. These conserved quantities reflect symmetries of the system and make the dynamics easier to understand. The PI will investigate the interactions between symplectic geometry, symmetries, and integrable systems. This project naturally interacts with other disciplines, such as physics, control theory, and computer modeling. Surprisingly, sufficiently symmetric integrable systems, though complex, can often be described by relatively simple objects, such as polygons. Studying these objects opens up many opportunities for undergraduate research projects, which the PI will support and mentor, providing hands-on experience in mathematical research to the next generation of scientists and engineers. Furthermore, understanding conserved quantities in dynamical systems has a variety of applications, especially related to creating accurate and robust computer models, which is a useful application in its own right, and in turn opens up even more excellent opportunities for undergraduate research. The symmetries discussed above are modeled by Hamiltonian group actions, and this project investigates the interactions between Hamiltonian torus actions and integrable systems. On a symplectic manifold of dimension 2n, an integrable system is the data of an independent smooth functions which are pairwise Poisson-commuting. On the other hand, a Hamiltonian action of a k-dimensional torus is generated by k such functions, and the action is said to be complexity-one if k is one less than n. Thus, a complexity-one torus action is only one function short of being an integrable system. This research is centered around integrable systems where the first n-1 functions generate a complexity-one torus action, which strike a delicate balance: they are rigid enough to admit powerful invariants, yet flexible enough to appear in many examples. The project has four main goals: (1) Determine conditions on when a complexity-one torus action can be extended to an integrable system of a certain class by constructing one additional function; (2) Develop new invariants of integrable systems and use these invariants to obtain classifications; (3) Construct new examples of integrable systems with desired properties by deforming well-understood systems, typically while preserving an underlying complexity-one action; (4) Apply results to questions related to Hamiltonian displaceability, symplectic capacities, and Floer theory. 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.