Deformation Spaces of Geometric Structures

在这个项目中，PI将研究几何结构的变形空间。曲面是局部看起来像二维平面的空间。例如足球或甜甜圈的表面。曲面在拓扑学、复杂分析和动力学等数学领域中自然产生。经典的泰希穆勒理论是一个研究固定曲面上所有几何形状的空间的数学领域。它还与其他科学领域相互作用，例如，通过它与理论物理中的弦理论的联系。PI的研究在于对TeichMuller理论的两种推广。曲面的三维推广被称为三维流形，这些流形就像我们生活的三维空间一样。PI将继续一个长期项目，对三维流形的可能形状进行分类和理解。在高等泰希穆勒理论中，PI将研究高维空间上几何结构的变形空间，重点是理解这些空间上的度量或距离函数。PI还将通过参与密歇根大学基于探究的学习中心、本科生课程开发、担任数学期刊编辑、组织会议以及指导本科生、研究生和博士后助理教授来为数学界做出贡献。更高级的TeichMuller理论将双曲群的几何表示研究为半单李群，通常排名至少为2。它的指导灵感来自泰希穆勒理论，总体目标是创建一个具有经典泰希穆勒理论的一些美感和深度的一般理论。PI将使用动力学和几何工具来研究Hitchin组件上的压力指标。特别是，PI将研究增广的Teichmuller空间和由Loftin引入的增广的Hitchin分量之间的类比。PI还将研究有限区域射影曲面的变形空间、与Hitchin表示有关的Liouville流、允许指定类型的Anosov表示的群的类以及Anosov表示的空间的结构。在Klein群领域，PI将使用在结束分层猜想证明中开发的工具来建立具有自由不可分解基本群的双曲三维流形的组合模型流形。这个项目有望对瑟斯顿的蒙皮映射有一个更好的理解，并允许PI和他的合著者建立一个迭代的有界映象定理，该定理被瑟斯顿用于Haken 3-流形的几何化定理的原始证明中，但其证明仍然难以捉摸。PI还将使用动力学工具研究几何有限群的变形空间，目标是证明极限集的Hausdorff维度的解析性并产生压力度量。该奖项反映了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。

In this project the PI will investigate deformation spaces of geometric structures. A surface is a space which looks locally like the two-dimensional plane. Examples are the surface of a football or a donut. Surfaces arise naturally in the mathematical fields of topology, complex analysis, and dynamics. Classical Teichmuller theory is an area of mathematics that studies the space of all geometric shapes on a fixed surface. It also interacts with other scientific fields, e.g. through its connections with string theory in theoretical physics. The PI's research lies in two types of generalizations of Teichmuller theory. Three-dimensional generalizations of surfaces are called three-manifolds, and these are locally like the three-dimensional space we live in. The PI will continue a long-term project to classify and understand possible shapes of three-dimensional manifolds. In Higher Teichmuller Theory, the PI will study deformation spaces of geometric structures on higher-dimensional spaces with an emphasis on understanding the metrics, or distance functions, on these spaces. The PI will also contribute to the mathematical community through involvement in the Inquiry Based Learning Center at the University of Michigan, curriculum development for undergraduate courses, serving as editor of mathematical journals, organizing conferences, and mentoring undergraduate students, graduate students and postdoctoral assistant professors.Higher Teichmuller theory studies geometric representations of hyperbolic groups into semi-simple Lie groups, usually of rank at least two. It is guided by inspiration from Teichmuller theory and the overall goal is to create a general theory with some of the beauty and depth of classical Teichmuller theory. The PI will use dynamical and geometric tools to study pressure metrics on Hitchin components. In particular, the PI will investigate an analogy between augmented Teichmuller space and an augmented Hitchin component introduced by Loftin. The PI will also study deformation spaces of finite area projective surfaces, Liouville currents associated to Hitchin representations, the class of groups which admit Anosov representations of specified type and the structure of spaces of Anosov representations. In the field of Kleinian groups, the PI will use tools developed in the proof of the Ending Lamination Conjecture to build combinatorial model manifolds for hyperbolic three-manifolds with freely indecomposable fundamental group. This project is expected to yield a finer understanding of Thurston's skinning map and to allow the PI and his co-authors to establish an iterated bounded image theorem, which was used by Thurston in the original proof of the Geometrization Theorem for Haken 3-manifolds, but whose proof remains elusive. The PI will also use dynamical tools to study deformation spaces of geometrically finite groups, with the goal of proving analyticity of the Hausdorff dimension of the limit set and producing pressure metrics.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.