权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Geometry, group theory, and dynamics

几何、群论和动力学

基本信息

批准号：
1510034
负责人：
Christopher Leininger
金额：
$ 33.61万
依托单位：
University of Illinois at Urbana-Champaign
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-06-15 至 2019-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510034&HistoricalAwards=false
关键词：
Geometry group theory dynamics

项目摘要

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.

AbstractAward：DMS 1510034，首席研究员：Christopher J. Leininger为了研究一个不断演化的物理或几何系统的长期行为，可以观察其“横截面”。“例如，要分析流经封闭系统的流体，我们可能会在流体的特定位置放置存款示踪剂，然后检查它是如何混合的，扭曲的，或者当它返回时（如果）发生变化。我们还可以尝试预测“第一次返回”到起始位置的行为-称为横截面-可能会影响或预测其他横截面的行为。该项目的一个目标是研究某些抽象数学系统的演化，这些系统在许多方面与刚刚描述的流体流动相似，它们来自几何和代数考虑。 PI与S的先前工作。道达尔和我。Kapovich提供了结构性的结果，这些结果将横截面和它们的第一次返回相互联系起来，并与原始系统联系起来。 PI将继续与Dowdall和Kapovich进行分析，并在不同横截面之间以及与环境系统之间进行更深入的联系。PI与Dowdall和Kapovich的工作，过去和未来，是由W。Thurston，D. Fried和C.麦克马伦在一个更经典的设置，但分歧的根本和惊人的方式。 PI还将继续他的分析与Agol和Margalit研究进一步的连接和推广，可能提供一个统一的理论的经典设置。这个项目涉及几何，拓扑，群论和动力学方面。中心对象是自由群自同构的曲面同胚和列车轨道图。 PI将继续与Dowdall和Kapovich合作，通过2-复体上的特殊半流分析自由循环群。除了加强不同截面之间的联系，例如证明monodromies的完全不可约性是所有截面共享或没有共享的属性，他还将所有monodromies与整个组在树上的动作联系在一起，并将Cannon-Thurston地图相互连接。随着Margalit和Agol-Margalit，PI将继续分析伪Anosov同胚的悬流的经典设置，试图理解模空间的所有系统。在他的工作与肯特，Bestvina-Bromberg-肯特，和其他人，PI将继续他的几何分析子群的映射类组和凸余紧性。