FLAT SURFACES

"SL(2,R)-action on flat surfaces and geometry of extremal subvarieties of moduli spaces"

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 Obiettivo del progetto (Objective)

'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 this non-homogeneous setting in an even more interesting way. Closed SL(2,R)-orbits give rise to totally geodesic subvarieties of the moduli space of curves, called Teichmueller 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 varities make this goal feasible. on polygonal billiard tables is best understood unfolding the table and studying the resulting surface. The moduli space of flat surfaces carries action of SL(2,R) and all the questions about group actions on homogeneous spaces reappear in this homogeneous setting in an even more interesting way. SL(2,R)-orbits give rise to totally geodesic of the moduli space of curves, called curves. Their classifcation is a major goal the coming years. The applicant's algebraic characterization Teichmueller curves plus the comprehension of the Mumford compactification of Hilbert modular varities this goal feasible.'