2014/11/06 by Alex Wright, Wright, Alex · 3 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1411.1827
openalex publication_date 2014/11/06 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Translation surfaces can be defined in an elementary way via polygons, and\narise naturally in in the study of various basic dynamical systems. They can\nalso be defined as Abelian differentials on Riemann surfaces, and have moduli\nspaces called strata that are related to the moduli space of Riemann surfaces.\nThere is a GL(2,R) action on each stratum, and to solve most problems about a\ntranslation surface one must first know the closure of its orbit under this\naction. Furthermore, these orbit closures are of fundamental interest in their\nown right, and are now known to be algebraic varieties that parameterize\ntranslation surfaces with extraordinary algebro-geometric and flat properties.\nThe study of orbit closures has greatly accelerated in recent years, with an\ninflux of new tools and ideas coming diverse areas of mathematics.\n This survey is an invitation for mathematicians from different backgrounds to\nbecome familiar with the subject. Little background knowledge, beyond the\ndefinition of a Riemann surface and its cotangent bundle, is assumed, and top\npriority is given to presenting a view of the subject that is at once\naccessible and connected to many areas of mathematics.\n