Dynamics Around Translation Surfaces

This award will support a project in dynamical systems. The mathematical field of dynamical systems seeks to understand how a system behaves as time evolves; it is an important subfield of mathematical analysis, which enjoys connections and applications to many other areas of the mathematical sciences. The systems at the heart of this project are connected to physics and geometry. A concrete example is that of a point mass traveling inside a polygon, which has elastic collision when it hits the sides. One focus of the project is to understand how prevalent randomness is in these systems. The PI will also investigate the structure of paths in related dynamical systems and aims to deepen our understanding of the connection between geometric and dynamical properties. This project will also stimulate the growth of the next generation of mathematicians by providing graduate student research opportunities. This project is concerned with two closely related dynamical systems: flows on translation surfaces and the SL(2,R) action on the space of translation surfaces. It seeks to better understand when the spectrum of (the one-parameter unitary group coming from) a flow on a translation surface is continuous (aside from a simple eigenvalue of 0). It seeks to understand the dynamics of the strictly upper triangular subgroup of SL(2,R) on spaces of translation surfaces. In particular, whether there are situations where the orbit closures are always constrained and how wild averages along orbits can behave. 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.