Columbia Mathematics Department Colloquium


Winning sets of Diophantine measured foliations.


Howard Masur

U. of Chicago



 In the 1960's W.Schmidt invented a game now called a Schmidt game to be played in Euclidean space with the Euclidean metric. "Winning sets" for this game have various nice properties; one of which is full Hausdorff dimension. The main motivating example of a winning set considered by  Schmidt are those reals which have  bounded coefficients in their continued fraction expansion, or equivalently reals  badly approximable by rationals. They also correspond to geodesics that stay in a compact subset of the moduli space H^2/SL(2,Z), the moduli space of genus 1 surfaces. One can formulate  a similar condition for a measured foliation on a surface of genus g>1 to be badly approximated by simple closed curves. These measured foliations correspond to Teichmuller geodesics that stay in a compact subset of the moduli space of genus g.   After giving the background on winning sets, and the classical continued fraction example, I will discuss the theorem, joint with Jon Chaika and Yitwah Cheung that the set of  foliations is Schmidt winning as a subset of PMF, Thurston's sphere of measured foliations.


Wednesday, May 2nd, 5:00 - 6:00 p.m.
Mathematics 520
Tea will be served at 4:30 p.m.