Dynamics, Embeddings, and Continuous Symplectic Geometry

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

This is a project about systems that evolve in time, called dynamical systems. Dynamical systems are fundamental across the sciences, but the mathematics underlying their study is very difficult, many important questions remain open, and new ideas are needed. A main theme of this project is to use a geometric approach to bring fresh insight to this field. Recently, the PI and collaborators used techniques from symplectic geometry to settle a longstanding question about certain dynamical systems. They showed that a particular dynamical system would take an infinite amount of energy to produce and used this to establish a new dichotomy in the set of systems: the finite-energy ones, and the rest. This dichotomy will enable a deeper understanding of the classification of dynamical systems and should allow for the resolution of several questions that have attracted wide interest. This project involves further research in this direction. One goal of the project is to find a further dichotomy for the finite-energy systems. By bringing a geometric approach to bear on questions about dynamical systems that have resisted solution by other methods, the project aims to showcase the power of these kinds of techniques. At the same time, an idea at the heart of the approach comes from studying when one symplectic shape can be deformed into a subset of another, and research in these directions should deepen our understanding of symplectic geometry. The project also has a substantial education component. The PI has previously worked to broaden participation in mathematics, foster equity, and support educational efforts in the field through prison education, research training in geometry at the high school and undergraduate levels, and course design. The project supports this work and related activities. This project involves a series of research investigations touching on the related fields of symplectic dynamics, symplectic embeddings, and continuous symplectic geometry. The main emphasis is on questions in low dimensions where powerful gauge theoretic techniques are available. The PI and collaborators recently settled the longstanding "Simplicity Conjecture" partly by contributing to the theory of periodic Floer homology spectral invariants. The project aims to further develop this theory and solve several longstanding questions; for example, to prove an analogue of the Simplicity Conjecture in higher genus. The project also aims to clarify the dynamics of Reeb flows, study infinite staircases in symplectic embedding problems, investigate questions about topological symplectic manifolds, and study higher-dimensional symplectic packing problems. 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.