DescriptionMathematical billiards is a beautiful subject. It is concerned with a "billiard table", think of a bounded domain (let's say, without holes) and its boundary a piecewise smooth simple closed curve, and the trajectory of a billiard ball in this domain following a straight line until it hits the boundary, where it is elastically reflected to follow a new straight line. However, in difference to billiard tables in real life, we assume that the billiard ball doesn't slow down and that it continues its path forever within the billiard table. It is not surprising that the study of this system leads to many interesting and exciting geometric problems.For example, it is still an open problem whether every triangular billiard table admits a "periodic" billiard trajectory, i.e., a billiard path which returns back into itself after finitely many reflections. Another result, closely related to billiards, is Poncelet's Theorem, which is a beautiful geometric statement about two nested ellipses. There are also relations between billiard trajectories in the square and number theory (continued fractions), between polygonal billiard tables and so-called flat surfaces of higher genus, and so on and so on... Billiards can be integrable or chaotic. One can study billiards in special types of billiard tables like convex domains or polygonal domains. One can also study billiards in more general geometries like the hyperbolic plane. Another variant are so-called outer billiards, which are played outside the billiard table. One starts with a point outside the billiard table and chooses a trajectory tangent to the billiard table and doubles the distance from the starting point to the point of tangency to get a new point, draws again a tangent from this new point to the billiard table and continues this procedure on and on... A long-standing question was whether there are outer billiard trajectories which are unbounded (which is an idealised version of the question about the stability of the solar system). This problem was finally settled by Richard Schwartz in 2007. The great thing about billiard is that the system is easy to understand and that one can obtain very beautiful and stunning results without too much theory. We will read some chapters of the books by Serge Tabachnikov and Richard Schwartz, which should be inspiring enough to branch into different project directions. Some Resources
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