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了解复杂几何或动力系统的内部工作原理。一种卓有成效的方法是解决这个问题，“给出一个物体最显著或最容易测量的特征，我们如何预测它的全局形状或长期行为？”这项提案中的研究旨在解决低维环境下的这个问题，重点是与动力学有关的三维空间的拓扑和几何。在其核心，调查员将使用两种基本技术。第一种方法是使用基本的构建块对复杂系统进行建模。例如，他与Landry和Minsky的合作展示了重要的三维流的动力学特性是如何通过对底层空间的三角剖分进行组合编码的。第二种是使用概率技术，旨在理解复杂动力系统中的“典型”行为。这是他与Gekhtman和Tiozzo在负曲空间的随机对称的性质方面工作的重点。与这项研究相结合，该提案还寻求通过新的本地研讨会以及继续指导研究生和博士后，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。第一种是利用一种新的转向三角剖分多项式不变量来研究三维流形的拓扑和几何。该多项式将McMullen的Teichmuller多项式推广到更一般的伪Anosov流，检测瑟斯顿范数球的面，并包装相关流的相对增长率。第二组项目试图将这些洞察力扩展到研究表面群和自由群的自同构。这包括一个项目，与Dowdall联合，建立一个典型的理想单纯复形，它与一个双曲自由循环群有关，它编码了群的各种分裂。最后一组项目涉及群论中的动力学。首先，研究者将证明有限生成群中几何量分布的一般中心极限定理。其次，PI与古普塔合作，计划通过证明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的一个猜想。这些项目还将揭示泰希穆勒和外层空间的动力学特性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.