Geometry, group theory, and dynamics

[摘要]获奖：DMS 1510034，首席研究员：Christopher J. Leininger为了研究一个进化的物理或几何系统的长期行为，可以观察它的“横截面”。例如，为了分析在封闭系统中流动的流体，我们可以在流体的特定位置放置示踪剂，然后检查当它返回时（如果）它是如何混合、扭曲或变化的。我们还可以尝试预测“首次返回”到起始位置（称为横截面）的行为如何影响或预测其他横截面的行为。该项目的目标之一是研究某些抽象数学系统的演化，这些系统在许多方面类似于刚刚描述的流体流动，它们源于几何和代数方面的考虑。先前与S. Dowdall和I. Kapovich的PI工作提供了结构结果，这些结果将横截面和它们的首次回归相互联系起来，并与原始系统联系起来。PI将继续与Dowdall和Kapovich进行分析，并在不同截面和环境系统之间寻求更深层次的联系。私家侦探与道尔和卡波维奇的合作，无论是过去还是未来，都受到W.瑟斯顿、D.弗里德和C.麦克马伦在更古典背景下的基础性工作的激励，但在根本和引人注目的方面存在分歧。PI还将继续与Agol和Margalit一起分析经典背景，进一步研究可能提供统一理论的联系和推广。这个项目涉及几何、拓扑学、群论和动力学的各个方面。中心对象是平面同胚和自由群自同构的列车轨道图。PI将继续与Dowdall和Kapovich合作，通过2-配合物上的特殊半流分析自由环群。除了加强不同截面之间的联系，例如证明单点的完全不可约性是所有或没有部分共有的性质，他还将所有单点与整个群在树上的作用联系在一起，并将Cannon-Thurston映射彼此连接起来。利用Margalit和Agol-Margalit， PI将继续分析伪anosov同胚的悬浮流的经典设置，试图理解模空间的所有系统。在他与Kent， Bestvina-Bromberg-Kent等人的合作中，PI将继续他对映射类群和凸紧性的子群的几何分析。

AbstractAward: DMS 1510034, Principal Investigator: Christopher J. Leininger To study the long-term behavior of an evolving physical or geometric system one can look at its "cross sections." For example, to analyze a fluid flowing throughout a closed system, we might deposit tracers into the fluid at a particular location then examine how it has been mixed, distorted, or changed when (and if) it returns. We can also try to predict how the behavior of the "first-return" to the starting location-called a cross section-might influence or predict the behavior at some other cross section. One goal of this project is to study the evolution of certain classes of abstract mathematical systems, similar in many ways to the fluid flow just described, which arise from geometric and algebraic considerations. Previous work of the PI with S. Dowdall and I. Kapovich provides structural results which relate the cross sections and their first-returns to one another and to the original system. The PI will continue this analysis with Dowdall and Kapovich and pursue deeper connections between different cross sections and with the ambient system. The PI's work with Dowdall and Kapovich, past and future, is motivated by the foundational work of W. Thurston, D. Fried, and C. McMullen in a more classical setting, but diverges in fundamental and striking ways. The PI will also continue his analysis of the classical setting with Agol and Margalit investigating further connections and generalizations that might provide a unified theory.This project involves aspects of geometry, topology, group theory, and dynamics. The central objects are surface homeomorphisms and train track maps for free groups automorphisms. The PI will continue his work with Dowdall and Kapovich, analyzing free-by-cyclic groups via special semi-flows on 2-complexes. In addition to strengthening the connection between different cross sections, for example proving that full-irreducibility for monodromies is a property shared by all or none of the sections, he will also tie together all the monodromies with an action of the entire group on a tree and connect the Cannon-Thurston maps to each other. With Margalit and Agol-Margalit, the PI will continue to analyze the classical setting of suspension flows of pseudo-Anosov homeomorphisms in an attempt to understand all systoles of moduli spaces. In his work with Kent, Bestvina-Bromberg-Kent, and others, the PI will continue his geometric analysis of subgroups of the mapping class group and convex cocompactness.