Mathematical Sciences: Combinatorial Topology and Surface Dynamics

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The investigator will work on three projects. The first is a continuation of work with Mladen Bestvina (U.C.L.A.) on automorphisms of the free group. They hope to prove a classification theorem for subgroups of polynomial growth automorphisms and thereby complete the proof of the Tits Alternative for Out(Fn). Secondly, the investigator hopes to prove that there are no minimal homeomorphisms of the once punctured plane, which would answer a question of Besicovitch and of Herman. Finally, the investigator, in collaboration with Bruce Kitchens (I.B.M.), will explore the relationship between two natural definitions of topological entropy for homeomorphisms of non- compact spaces. The topology of surfaces is a highly developed subject, addressing properties which remain invariant when a surface is stretched or twisted without tearing it, so-called rubber sheet geometry. The theory of dynamical systems is another highly developed subject, addressing questions about transformations of a space into itself, such simple things as rotations of spheres, but also far more complicated transformations. For transformations of surfaces, the topology is intimately involved and says a great deal about what kinds of transformations are possible. Extracting this information by combining the topology of surfaces with the theory of dynamical systems is a sophisticated endeavor which forms a large part of this project.