Mathematical Sciences: Combinatorial Topology and Surface Dynamics

数学科学：组合拓扑和表面动力学

• 批准号: 9204292
• 负责人: Michael Handel
• 金额: $ 8.85万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1996-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204292&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Combinatorial; Topology; Surface

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.

调查员将致力于三个项目。 第一个是与 Mladen Bestvina（加州大学洛杉矶分校）关于自由群自同构的工作的延续。 他们希望证明多项式增长自同构子群的分类定理，从而完成 Out(Fn) 的 Tits Alternative 的证明。 其次，研究者希望证明曾经被刺穿的平面不存在最小同胚，这将回答贝西科维奇和赫尔曼的问题。 最后，研究人员将与 Bruce Kitchens (IBM) 合作，探索非紧空间同胚的拓扑熵的两个自然定义之间的关系。 表面拓扑是一门高度发达的学科，它解决的是当表面被拉伸或扭曲而不撕裂时保持不变的特性，即所谓的橡胶板几何形状。 动力系统理论是另一个高度发展的学科，它解决空间向自身变换的问题，例如球体旋转等简单的事情，但也涉及更复杂的变换。 对于曲面的变换，拓扑结构密切相关，并且充分说明了可能的变换类型。 通过将表面拓扑与动力系统理论相结合来提取这些信息是一项复杂的工作，也是该项目的重要组成部分。