Topology and Geometry from 3-Manifolds to Free Groups

从三流形到自由群的拓扑和几何

• 批准号: 2102018
• 负责人: Samuel Taylor
• 金额: $ 22.33万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2102018&HistoricalAwards=false
• 关键词: Topology; Geometry; Manifolds; Free; Groups

Many problems throughout mathematics (as well as the other sciences) come down to needing to understand the inner workings of complex geometric or dynamical systems. One fruitful approach is to address the question, “Given the most salient or easily measured features of an object, how can we predict its global shape or long term behavior?” The research in this proposal is directed at addressing this question in the low dimensional setting, focusing on the topology and geometry of three-dimensional spaces with connections to dynamics. At its core, the investigator will employ two basic techniques. The first is to model a complex system with its basic building blocks. For example, his work with Landry and Minsky shows how dynamical properties of important three-dimension flows are combinatorially encoded by a triangulation of the underlying space. The second is to use probabilistic techniques that aim to understand the “typical” behavior within a complicated dynamical system. This is the focus of his work with Gekhtman and Tiozzo on properties of random symmetries of negatively curved spaces. In conjunction with this research, the proposal also seeks to support student involvement in related fields within the Philadelphia area through new local seminars and the continued mentoring of graduate students and postdocs.In greater detail, the proposed research projects divide into three themes. The first is to study the topology and geometry of three-manifolds using a new polynomial invariant of veering triangulations. The polynomial extends McMullen’s Teichmuller polynomial to more general pseudo-Anosov flows, detects faces of the Thurston norm ball, and packages relative growth rates of the associated flow. The second set of projects seeks to extend these insights to study automorphisms of surface groups and free groups. This includes a project, joint with Dowdall, to build a canonical ideal simplicial complex associated to a hyperbolic free-by-cycle group that encodes the various splitting of the group. The final set of projects concerns dynamics in group theory. First, the investigator will prove general central limit theorems for the distribution of geometric quantities in finitely generated groups when sampling with respect to the word metric. Second, in collaboration with Gupta, the PI plans to solve a conjecture of Handel–Mosher by demonstrating that a typical free group automorphism and its inverse have distinct stretch factors. These projects will also shed light on dynamical properties of Teichmuller and Outer space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

整个数学（以及其他科学）中的许多问题都归结为需要了解复杂的几何或动态系统的内部工作。一种富有成果的方法是解决这个问题，“鉴于对象的最显着或易于测量的特征，我们如何预测其全球形状或长期行为？”该提案中的研究旨在在低维度的环境中解决这个问题，重点介绍了三维空间的拓扑和几何形状以及与动力学的联系。研究人员将采用两种基本技术。首先是建模具有基本构建块的复杂系统。例如，他在兰德里（Landry）和明斯基（Minsky）的工作表明，重要三维流动的动态特性如何通过对基础空间的三角剖分来组合编码。第二个是使用旨在理解复杂动态系统中“典型”行为的概率技术。这是他与Gekhtman和Tiozzo一起工作的重点，即负弯曲空间的随机对称性的属性。结合这项研究，该提案还旨在通过新的本地准则来支持学生参与费城地区相关领域的参与，并更详细地详细介绍了研究生和博士学位，拟议的研究项目将其分为三个主题。首先是使用新的多项式三角剖分研究三个序列的拓扑和几何形状。多项式将McMullen的Teichmuller多项式扩展到更通用的伪-Anosov流，检测到瑟斯顿正常球的面，并包装相关流的相对生长速率。第二组项目旨在扩展这些见解，以研究表面群体和自由群体的自动形态。这包括一个与Dowdall的项目，以建立与编码该组各种分裂的双曲线自由周期组相关的规范理想的简单复合物。最后一组项目涉及小组理论中的动态。首先，研究者将证明一般的中央限制定理，用于相对于指标一词进行采样时，最终产生的组中几何量的分布。其次，与Gupta合作，PI计划通过证明典型的自由团体自动形态及其倒数具有不同的伸展因素来解决Handel – Soher的概念。这些项目还将阐明Teichmuller和外太空的动态特性。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估，被视为珍贵的支持。

项目成果

Random trees in the boundary of outer space

外太空边界的随机树

• DOI: 10.2140/gt.2022.26.127
• 发表时间: 2022
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Kapovich, Ilya;Maher, Joseph;Pfaff, Catherine;Taylor, Samuel J
• 通讯作者: Taylor, Samuel J

Orientable maps and polynomial invariants of free-by-cyclic groups

可定向映射和自由循环群的多项式不变量

• DOI: 10.1016/j.aim.2023.108872
• 发表时间: 2023
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Dowdall, Spencer;Gupta, Radhika;Taylor, Samuel J.
• 通讯作者: Taylor, Samuel J.

Central limit theorems for counting measures in coarse negative curvature

粗负曲率计数测度的中心极限定理

• DOI: 10.1112/s0010437x22007680
• 发表时间: 2022
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Gekhtman, Ilya;Taylor, Samuel J.;Tiozzo, Giulio
• 通讯作者: Tiozzo, Giulio

Flows, growth rates, and the veering polynomial

流量、增长率和转向多项式

• DOI: 10.1017/etds.2022.63
• 发表时间: 2022
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: LANDRY, MICHAEL P.;MINSKY, YAIR N.;TAYLOR, SAMUEL J.
• 通讯作者: TAYLOR, SAMUEL J.