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Polynomial Pick forms for affine spheres, real projective polygons, and surface group representations in PSL(3,R).

Friday 31st July 2015 - 09:00 to 10:00
INI Seminar Room 1
Abstract: (Joint work with David Dumas.) Discrete surface group representations into PSL(3, R) correspond geometrically to convex real projective structures on surfaces; in turn, these may be studied by considering the affine spheres (an interpretation of the Hitchin system of equations in this case) which project to the convex hull of their universal covers. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found, which translates to a version of Stokes data. We begin by describing the basic objects and context and conclude with a sketc h of some of the useful technique and an application.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons