Moduli Problems in Contact Geometry

Columbia University, New York NY

Investigators

Abstract

Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as springs, planetary motion, and wave propagation. The equations of motion in classical mechanics are determined by the notion of a conserved quantity, energy. A related quantity is action, which is minimized by solutions to the equations of motion. For a closed system, such as the Kepler problem whose solutions describe paths of planets orbiting the sun, the energy is the sum of the kinetic and potential energy in the system, and the action is given by the (minimized) mean value of kinetic minus potential energy. Symplectic and contact structures emerge as we investigate these systems by unpacking the information hidden in the notions of energy and action. Understanding the evolution and distinguishing transformations of these systems led to the development of global invariants of symplectic and contact manifolds. The PI plans to continue her work in providing foundations and applications for contact invariants stemming from nonequivariant and (circle) equivariant constructions of contact homology. Contact homology is built out of closed orbits of Reeb vector fields and counts of solutions to a nonlinear Cauchy-Riemann equation which interpolates between closed Reeb orbits. Reeb vector fields are Hamiltonian-like vector fields, whose flow lines are solutions to Hamilton's equations of motion, as they conserve energy. Closed Reeb orbits are of particular interest because they can be used to describe the motion of a planet orbiting a star, the closed trajectories of a satellite, and other local distance minimizing "loops." Contact homology can be used to better distinguish contact structures on a given smooth manifold and to extract dynamical information pertaining to the possible Reeb vector fields which can be associated to a fixed contact structure. The PI's work has shown that these invariants capture more phenomena than previously expected. She also plans to study additional structural properties of these contact and related symplectic invariants, Legendrian knots in closed contact manifolds, dynamics of Seifert fiber spaces, and symplectic embeddings. The PI plans to continue her outreach programs which have increased the access and success of underrepresented students and faculty in mathematics as well as foster greater interest in geometry and topology amongst local high school students and undergraduates. 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.