Topology and Geometry from 3-Manifolds to Free Groups

数学（以及其他科学）中的许多问题都归结为需要理解复杂几何或动力系统的内部工作原理。一个富有成效的方法是解决这个问题，“给定一个物体最显著或最容易测量的特征，我们如何预测它的整体形状或长期行为？” 在这个建议的研究是针对解决这个问题，在低维设置，侧重于三维空间的拓扑结构和几何与动力学的连接。 在其核心，调查员将采用两个基本技术。 第一种是用基本的构建块来建模复杂的系统。例如，他与兰德里和明斯基的工作表明，重要的三维流动的动力学特性是如何通过底层空间的三角测量组合编码的。 第二种是使用概率技术，旨在了解复杂动态系统中的“典型”行为。 这是重点，他的工作与Gekhtman和Tiozzo的财产随机对称的消极弯曲空间。 在这项研究的同时，该提案还寻求通过新的本地研讨会和继续指导研究生和博士后来支持学生参与费城地区的相关领域。更详细地说，拟议的研究项目分为三个主题。 第一个是利用转向三角剖分的一个新的多项式不变量来研究三维流形的拓扑和几何。 该多项式扩展McMullen的Teichmuller多项式更一般的伪Anosov流，检测瑟斯顿规范球的面，并包相关的流的相对增长率。 第二组项目旨在扩展这些见解，研究曲面群和自由群的自同构。 这包括一个项目，联合Dowdall，建立一个典型的理想单纯复杂的双曲自由循环组，编码组的各种分裂。 最后一组项目涉及群论中的动力学。首先，调查员将证明一般的中心极限定理的分布几何量的抽样时，在n-生成组的字度量。第二，与古普塔合作，PI计划通过证明一个典型的自由群自同构及其逆具有不同的拉伸因子来解决Handel-Mosher的猜想。 这些项目还将揭示Teichmuller和外层空间的动力学特性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.