Title: Calculus of the number of periodic billiard orbits in a polygon
Speaker: Professor Gregory Galperin
Speaker Info: Eastern Illinois University
Consider a polygonal table with pockets (small circles) at all of its vertices and the billiard trajectories (orbits) living on this table: if a biiliard particle enters a pocket it disappears. It turns out, all such orbits must be periodic. How many of them (up to parallelism) could be there on that table?Date: Tuesday, May 23, 2000
The following results will be presented at the talk: (1) the number of such orbits is finite for any table; (2) only "short" trajectories live on "rational" polygons, which gives a possibility to estimate this number from above in terms of the areas of the table and a pocket. Unfortunately, the multiplicative constant in the upper bound in the "rational" case is very sensitive to small perturbations of the table.
The very recent result gives a STABLE upper bound for the number of periodic orbits both for rational and irrational polygons: the multiplicative constant does not depend on the polygon at all and is less than 2.5!
The last result is based on the so-called "Uncertainty Principle" for polygonal billiards discovered by the speaker.
No preliminary knowledge is required. Students are encouraged to attend the talk.