Mathematical Sciences: Combinatorial Topology and Surface Dynamics

调查员将从事三个项目。第一个是与Mladen Bestvina（加州大学洛杉矶分校）关于自由群的自同构的工作的延续。他们希望证明多项式生长自同构子群的一个分类定理，从而完成对Out（Fn）的Tits Alternative的证明。其次，研究者希望证明一旦被刺穿的平面不存在最小同胚，这将回答Besicovitch和Herman的一个问题。最后，研究者将与Bruce Kitchens （ibm）合作，探讨非紧空间同胚拓扑熵的两种自然定义之间的关系。表面的拓扑学是一门高度发达的学科，研究的是当表面被拉伸或扭曲而不撕裂时保持不变的特性，即所谓的橡胶片几何。动力系统理论是另一个高度发达的学科，它解决了空间转化为自身的问题，如球体旋转这样简单的事情，但也有更复杂的转化。对于曲面的变换，拓扑结构是密切相关的，并且说明了很多关于什么样的变换是可能的。通过结合曲面拓扑和动力系统理论来提取这些信息是一项复杂的努力，它构成了这个项目的很大一部分。

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.