Ergodic Theory of Rational Billiards | |
Student No.： | 50 |
Time： | Tue/Thu 10:10-12:00, 2016-06-07 ~ 2016-07-26（No classes on June 9\23\28） |
Instructor： | Yitwah Cheung [San Francisco State University] |
Place： | Conference room 3, floor 2, Jin Chun Yuan West Building |
Starting Date： | 2016-6-7 |
Ending Date： | 2016-7-26 |
Description:
I will use the study of billiards in polygons as a motivating example for this introductory course in Ergodic Theory. A remarkable observation from the 1980's is that when the angles of the polygon are rational multiples of pi, the theory of (closed) Riemann surfaces can be brought to bear. This naturally leads to the study of Teichmulller flows. These flows have many similarities with flows on homogeneous spaces, which in turn have many connections to other areas of mathematics, such as Diophantine approximation. In this course I will use concrete examples to illustrate this rich interplay between geometry, number theory and dynamics.
Prerequisite:
Familiarity with basic topology and Lebesgue measure will be assumed. Familiarity with differentiable manifolds and Riemann surfaces is helpful, but not a necessity.
Reference:
1. A. Wright, From Rational Billiards to Moduli Spaces, Bull. Amer. Math. Soc., 2016 http://web.stanford.edu/~amwright/BilliardsToModuli.pdf
2. H. Masur and S. Tabachnikov, Rational billiards and Flat surfaces, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089.
3. A. Zorich, Flat surfaces, Frontiers in number theory, physics and geometry. I, Springer, Berlin, 2006, pp.437-583.