Deformation spaces of geometric structures

PI将研究n维流形上几何结构的变形空间。流形是一个数学对象，它在局部看起来像普通的欧几里得空间，但可以有更高或更低的维度。二维流形被称为曲面（例如，足球、甜甜圈或椒盐卷饼的表面），而我们的宇宙是三维流形的一个例子。流形上可能的几何结构的变形的研究自然出现在许多数学领域以及物理学中。PI还将通过参与密歇根大学的探究式学习中心为数学界做出贡献，他参与了MACSS学者计划的形成，该计划旨在支持数学，计算机科学和统计学专业的低收入学生，本科课程的课程开发，担任数学期刊的编辑，组织会议，并指导本科生，研究生和博士后助理教授。具体而言，该项目侧重于研究曲面上的高阶结构的Hitchin分量和三维流形上的双曲结构空间。 希钦分量是高等泰希穆勒空间的一个杰出的例子，它把闭曲面群表示成半单李群。与经典泰希穆勒理论惊人的相似性已经被发现，包括从经典泰希穆勒理论推广Weil-Petersson度量的压力度量。闭曲面的Teichmuller空间关于Weil-Petersson度量的度量完备化是增广的Teichmuller空间，可以看作模空间的轨道泛覆盖。PI将研究希钦组成部分的度量完成相对于压力度量，并制定一个几何理论的增强希钦组成部分平行的经典理论。Thurston终结层猜想的证明为双曲三维流形及其变形空间的研究提供了许多新的工具。在终结层猜想的原始证明中，得到了双曲3-流形同伦等价于可定向闭曲面的一致bilipschitz模型。PI将为任何双曲3流形开发统一的bilipschitz组合模型，该双曲3流形具有非线性生成的、自由不可分解的基本群。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

The PI will investigate deformation spaces of geometric structures on n-dimensional manifolds. A manifold is a mathematical object which looks locally like ordinary Euclidean space, but can have higher or lower dimension. Two-dimensional manifolds are called surfaces (e.g., the surface of a football, a donut or a pretzel), while our universe is an example of a 3-dimensional manifold. The study of deformations of possible geometric structures on manifolds arises naturally in many fields of mathematics as well as in physics. The PI will also contribute to the mathematical community through involvement in the Inquiry Based Learning Center at the University of Michigan, his involvement in the formation of the MACSS scholars program which is designed to support low-income students majoring in Mathematics, Computer Science and Statistics, curriculum development for undergraduate courses, serving as editor of mathematical journals, organizing conferences, and mentoring undergraduate students, graduate students and postdoctoral assistant professors.Specifically, the project focuses on the study of the Hitchin component of higher rank structures on surfaces and on space of hyperbolic structures on 3-manifolds. The Hitchin component is the pre-eminent example of a Higher Teichmuller space of representations of a closed surface group into a semi-simple Lie group. Striking analogies with classical Teichmuller theory have been discovered, including pressure metrics which generalize the Weil-Petersson metric from classical Teichmuller theory. The metric completion of the Teichmuller space of a closed surface, with respect to the Weil-Petersson metric, is the augmented Teichmuller space, which one may view as the orbifold universal cover of moduli space. PI will study the metric completion of the Hitchin component with respect to a pressure metric and to develop a geometric theory of the augmented Hitchin component which parallels the classical theory. The proof of Thurston’s Ending Lamination Conjecture developed many new tools for the study of hyperbolic 3-manifolds and their deformation spaces. In the original proof of the Ending Lam- ination Conjecture one obtains uniform bilipschitz models for hyperbolic 3-manifolds homotopy equivalent to an orientable closed surface. PI will develop uniformly bilipschitz combinatorial models for any hyperbolic 3-manifold with finitely generated, freely indecomposable fundamental group.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.