Mathematical Sciences: Ergodic Theory and Topological Dynamics

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The situation of a group acting on a space, for instance the group of real numbers acting by time evolution on the state space of a physical system, is encountered quite often in mathematics and its applications. One is interested in what happens overall, on average, in the long run, which may be quite difficult to infer solely from what happens over short intervals of time. A seemingly straightforward example figures prominently in the project of Professor Veech, namely the flow described by billiard balls careening about inside a triangular region. Trajectories will either essentially fill out the whole triangle over time or be periodic, coming back on themselves after a finite number of bounces. Classifying and describing the periodic trajectories requires some surprisingly sophisticated mathematics. In particular, it turns out to be advantageous to relate billiard flows to flows and other group actions on more exotic geometric objects. At a technical level, this has to do with the ergodic theory and dynamics of the action of SL(2,R) on Teichmueller space. Professor Veech is studying the prevalence of points in this space whose isotropy subgroups for the action are lattices. Certain such points arise from triangular billiards. The difficult problem of length spectrum for a flow, i.e. of what lengths of periodic orbits can occur, will be studied in various geometric contexts.