Dynamics on polygonal billiard tables is best understood by unfolding the table and studying the resulting flat surface. The moduli space of flat surfaces carries a natural action of SL(2,R) and all the questions about Lie group actions on homogeneous spaces reappear in thisnon-homogeneous setting in an even more interesting way.Closed SL(2,R)-orbits give rise to totally geodesicsubvarieties of the moduli space of curves, calledTeichmueller curves. Their classifcation is a major goal over the coming years. The applicant's algebraic characterization of Teichmueller curves plus the comprehension of the Deligne-Mumford compactification of Hilbert modular varitiesmake this goal feasible.on polygonal billiard tables is best understoodunfolding the table and studying the resultingsurface. The moduli space of flat surfaces carriesaction of SL(2,R) and all the questions aboutgroup actions on homogeneous spaces reappear in this homogeneous setting in an even more interesting way.SL(2,R)-orbits give rise to totally geodesicof the moduli space of curves, calledcurves. Their classifcation is a major goalthe coming years. The applicant's algebraic characterization Teichmueller curves plus the comprehension of the Mumford compactification of Hilbert modular varities this goal feasible.