Holomorphic Curves in Embeddings and Dynamics

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Systems that evolve over time are known as "dynamical systems" and appear in a wide range of fields including biology, economics, physics, and engineering.  The equations that describe these systems are often difficult to solve directly, so new tools and techniques are needed to study them.  This project is centered around a tool called "contact homology" that has produced powerful insights into many kinds of dynamical systems in recent years.  Contact homology provides a framework for finding features of a dynamical system that are conserved even as the parameters of the dynamical system are allowed to vary.   In this way, complicated systems can be studied by continuously changing the system into a more basic form, and then using knowledge of the dynamics in this simpler state to learn about the structure of the original system.  This project aims to further develop the foundations of contact homology and to use it to produce new insights about dynamics.  An example of an expected application is the discovery of new periodic trajectories for many dynamical systems, which are configurations of the system that re-occur infinitely often in time. An expected educational impact is the fostering of new opportunities for research training for undergraduates. In this area the PI plans to continue his efforts, which have led to two publications by undergraduates in peer-reviewed journals in the last several years.  The PI also plans to continue his commitment to reach broader segments of the public, for example through participation in prison education initiatives and through outreach to local high schools. In a different direction, the PI plans to continue conversations with researchers in other fields of science to find new collaborative opportunities. The PI will disseminate the results of the project as broadly as possible, for example through organizing conferences and through effective use of online tools.  To elaborate on the scientific merit of the project, previous work of the PI used a special kind of contact homology, called embedded contact homology (ECH), to show that any vector field of Reeb type on a closed three-manifold has at least two distinct periodic orbits.  In fact, evidence suggests that stronger results hold, and in the current project the PI and collaborators plan to show in many cases that a Reeb vector field with more than two closed orbits has infinitely many.  Insights into Reeb dynamics also come from the closely related field of symplectic embedding problems, and this project will pursue several lines of inquiry in this direction, for example one thread involves better understanding the number theoretic aspects of four-dimensional embedding problems while another involves finding new obstructions to symplectic embeddings in higher dimensions.  New combinatorial tools are expected to be useful for these investigations, so the PI plans to continue developing an irrational version of the classical Ehrhart theory familiar from the theory of lattice point enumeration.  More speculative directions of the project involve establishing new formulas for the asymptotics of the ECH spectrum and further developing the foundations of embedded contact homology by applying new tools for regularizing moduli spaces of pseudoholomorphic curves.