# Schedule of the event

The 30^{th} Annual Texas A&M High School Mathematics Contest is scheduled
for Saturday, November 13, 2021.

Due to the COVID-19 Emergence all events will be online.

This year the contest is free for all participants.

Note that The POWER TEAM EXAM will be posted on Friday, November 5.

## SCHEDULE for November 13.

9:00 am - SUBMISSION OF POWER TEAM RESPONSES ARE DUE.

3:00 - 4:00 Public Lecture (via ZOOM)

### Title: Polygonal Billiards

### Speaker: Professor Boris Hanin (Princeton University)

### Abstract:

*Imagine a polygonal (e.g. triangular, rectangular, pentagonal, etc) billiard table and consider the idealized situation in which there is no friction (and no pockets). In this situation, the initial position x of a billiard ball and its initial velocity v determine a trajectory that goes on forever. Specifically, the ball moves in straight line segments across the interior of the billiard table and bounces off the sides according to the familiar Snell's Law in which the angle of incidence is equal to the angle of reflection (and the ball's speed is preserved).*

In this talk, I will carefully introduce this setup and ask two famous questions: for a given polygonal billiard table P does there exist a starting point x inside of P and an initial velocity v for a billiard ball so that after some finite number of bounces it returns precisely to the initial position x with the initial velocity v (and hence its trajectory becomes periodic)? If such a periodic trajectory does exist, then can we describe all possible (x,v) pairs that lead to a periodic trajectory?

Even when P is a triangle this problem is unsolved, despite the effforts of many mathematicians. I will describe two basic ideas, unfolding and gluing, that help give intuition for what is happening and, time permitting, will also explain how to solve several special cases.4:00 - 4:30 AWARDS CEREMONY (via Zoom)