## Columbia Mathematics Department Colloquium

*Winning sets of Diophantine measured foliations.*

by

# Howard Masur

## U. of Chicago

Abstract:

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.