We give a review of Euclidean and pseudo-Euclidean billiards in the plane and in d-dimensional space. If the billiard table is bounded by confocal quadrics, periodic trajectories can be expressed in algebro geometric terms based on work of Poncelet, Cayley, and others. In particular, we consider a billiard problem for compact domains on a hyperboloid of one sheet bounded by confocal quadrics using the pseudo-Euclidean metric. Using a matrix factorization technique of Moser and Veselov, the billiard is shown to be integrable in the sense of Liouville. Further, we derive a Cayley condition for the billiards under consideration and explore geometric consequences. This is joint work with Milena Radnovic (USYD).
Sean Gasiorek is a Postdoctoral Research Associate at the University of Sydney. He received his Ph.D. from UC Santa Cruz in 2019 and moved to Sydney in August 2019.