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One of the simplest dynamical systems is the square billiard, which has a complete theory now. If we change the square to a (say) rhombus, we know much, much less, including even the case of the simplest 60-120 degree rhombus. Similarly, the cube surface consists of 6 squares, but again we know much, much less about the geodesic flow on the cube surface. One good reason is the appearance of singularities, or ``chaos' (=highly sensitive dependence on the intial condition), which is missing in the square billiard.

Wth my Ph.D. student Michael Donders recently we made some progress in these long-standing open problems. In my lecture I will report on these new results.