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Explore advanced mathematical concepts in dynamical systems through this joint IAS/PU Groups and Dynamics Seminar lecture examining the equidistribution of expanding horospheres in the space of translation surfaces. Learn about translation surfaces, which are closed surfaces obtained by gluing edges of polygons in parallel, and discover how the group GL_2(R) acts on collections of translation surfaces of fixed genus g. Investigate the probability measures obtained by rotating a fixed translation surface S with uniform angles and multiplying by diag(e^t, e^-t), equivalent to expanding pieces of horospherical orbits. Understand the proof of equidistribution for sequences of these measures as t approaches infinity, which resolves a conjecture by Forni and extends previous results by Eskin and Mirzakhani that demonstrated similar outcomes using Cesà ro averages. Examine practical applications of these theoretical findings to billiards with rational angles, connecting abstract mathematical theory to geometric problems in dynamical systems.
