Ergodic Theory of Billiards and Quadratic Differentials

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The principal investigator will study problems in billiards which model important questions from Kolmogorov-Arnold-Moser theory, as well as quadratic differentials and Teichmuller theory. He will investigate whether convergence of ergodic averages for measured foliations can be estimated if one knows the orbit of the Teichmuller geodesic it determines. Another important question addressed in this project concerns the extremal length geometry of Teichmuller space. Specifically, are there a finite number of curves whose extremal lengths determine the Riemann surface? Invariant annular bands arise in the Kolmogorov-Arnold-Moser theory of turbulence. Iterations of functions on these bands can be modeled by angles and locations of bounces of a billiard ball on a circular table. For these reasons, the principal investigator might shed light on relevant questions of turbulence (and the shape of ships' hulls) by answering questions described by the game of billiards. For example, how many of the possible directions that one might hit a billiard ball carry the ball in a periodic trajectory, so that the ball eventually returns to the exact same spot travelling in the exact same direction? Does the location of the ball or shape of the table make a difference?