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Mathematics Colloquium


Date:  Tuesday, February 09, 2016

Title:  Billiards in polygons, translation surfaces, and moduli spaces

Abstract:  A nice example of a dynamical system is a billiard flow in a polygon. Most work has concentrated on the case that the vertex angles are multiples of pi. An unfolding process leads to considering the straight line flow on what is called translation surface. A translation surface can be also be thought of as a holomorphic 1-form on a compact Riemann surface. The classic example is billiards in a rectangle which leads to the straight line flow on a flat torus. I will talk about basic questions one asks about straight line flows. The solutions to these questions most often relies on considering a given translation sutface as a point in a moduli space of all translation surfaces and then considering the orbit of the point under the action of the group SL(2,R) on the moduli space. I will talk at the end about recent spectacular work of Eskin-Mirzakhani-Mohammadi on this subject and some of what it says about billiards. This talk is meant to be a gentle introduction to the subject, accessible to graduate students.

Speaker:  Howard Masur
Institution:  University of Chicago


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