**Title:** Calculus of the number of periodic billiard orbits in a polygon

**Speaker:** Professor Gregory Galperin

**Speaker Info:** Eastern Illinois University

**Brief Description:**

**Special Note**:

**Abstract:**

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?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.

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