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，它局部看起来像普通的欧几里德空间，但可以有更高或更低的维度。二维流形被称为表面(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。研究流形上可能的几何结构的变形是在许多数学和物理领域中自然产生的。PI还将通过参与密歇根大学基于探究的学习中心，参与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，为数学界做出贡献。具体而言，该项目专注于曲面上更高级别结构的希钦分量和三维流形上的双曲线结构空间的研究。Hitchin分支是闭曲面群表示为半单李群的高阶TeichMuller空间的杰出例子。已经发现了与经典泰希穆勒理论惊人的相似之处，包括从经典泰希穆勒理论推广的Weil-Petersson度规的压力度规。关于Weil-Petersson度量，闭曲面的Teichmuller空间的度量完备性是增广的Teichmuller空间，它可以看作模空间的orforold泛覆盖。PI将研究Hitchin分量相对于压力度量的度量完备性，并发展与经典理论平行的增广Hitchin分量的几何理论。瑟斯顿结束分层猜想的证明为研究双曲三维流形及其变形空间发展了许多新的工具。在结束LAMING猜想的原始证明中，我们得到了等价于可定向闭曲面的双曲3-流形同伦的一致的bilipschitz模型。PI将为任何具有有限生成的、自由不可分解的基本群的双曲3-流形开发统一的bilipschitz组合模型。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。

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.