Advisor: Vladimir Baranovsky

Co-Advisors: Professor Yifeng Yu, Professor Jack Xin

The billiard table with a nowhere differentiable boundary is not well defined; the law of reflection holds a no point of the boundary.  Denoting the Koch snowflake by KS, the billiard Omega(KS) is a canonical example of such a table and the focus of the talk.  We will show that KS being approximated by a sequence of rational polygons and Omega(KS) being tiled by equilateral triangles both allow us to construct what we call a sequence of compatible periodic hybrid orbits.  Under certain situations, such sequences have interesting limiting behavior indicative of the existence of a well-defined billiard orbit of Omega(KS).  In addition to this, we provide a topological dichotomy for a sequence of compatible orbits.  Other important properties and interesting results will be discussed, especially with regards to the possible presence of self-similarity in what we propose to be a well-defined periodic hybrid of the Koch snowflake fractal billiard Omega(KS).  Finally, we will briefly discuss future research problems.

I will review briefly some recent developments in financial mathematics research, put them in a historical context, and then discuss the modeling and analysis of systemic risk phenomena.