Mathematical Sciences: Combinatorial Topology and Surface Dynamics

调查员将从事三个项目。 第一个是 继续与Mladen Bestelf（U.C.L.A.）合作对 自由群的自同构 他们希望证明一个 多项式增长子群的分类定理 自同构，从而完成证明的山雀 Out（Fn）的替代方案。 其次，调查员希望证明 有一次穿孔没有极小同胚 飞机，这将回答贝西科维奇和赫尔曼的问题。 最后，调查员与布鲁斯·基托合作， （I.B.M.），将探索两种自然的 的同胚的拓扑熵的定义 紧致空间 曲面拓扑学是一门高度发展的学科， 寻址属性保持不变，当表面 拉伸或扭曲而不撕裂它，所谓的橡胶片 几何 动力系统理论是另一个高度 发展的主题，解决有关的问题，一个 空间本身，像球体旋转这样简单的事情，但是 也有更复杂的转换。 对于转换， 表面，拓扑结构是密切相关的，并说了很多 关于什么样的转变是可能的。 提取此 通过将曲面的拓扑结构与理论相结合， 动力系统是一个复杂的奋进，形成一个 这个项目的大部分。

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.