Deformation spaces of geometric structures

几何结构的变形空间

• 批准号: 1306992
• 负责人: Richard Canary
• 金额: $ 23.66万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1306992&HistoricalAwards=false
• 关键词: Deformation; spaces; geometric; structures

Prof. Canary proposes to study deformation spaces of geometric structures arising from representations of hyperbolic groups into semi-simple Lie groups. Many of these investigations are motivated by the classical study of the Teichmuller space of hyperbolic (or conformal) structures on a surface. For example, the Hitchin component of the space of (conjugacy classes of) representations of a closed surface group into PSL(n,R) has been shown to have many striking resemblances to Teichmuller space. Prof. Canary, in collaboration with Bridgeman, Labourie and Sambarino, has recently developed a mapping class group invariant metric on each Hitchin component which restricts to the Weil-Petersson metric on the Fuchsian locus. Prof. Canary proposes to investigate the properties of this metric and use it as a tool to understand the geometry of the Hitchin component. More generally, he proposes to study metrics on spaces of Anosov representations of word hyperbolic groups into semi-simple Lie groups. Prof. Canary also proposes to study the topology and geometry of deformation spaces of hyperbolic 3-manifolds. The Ending Lamination Theorem provides a classification of the manifolds in these deformation spaces, but as the invariants in this classification are not continuous, it does not provide a parameterization. Recent work has shown that the topology of these spaces has a rich structure and many intriguing questions remain to be studied.Prof. Canary will study deformation spaces of geometric structures on manifolds. A 2-dimensional manifold is a space which looks locally like the 2-dimensional plane, e.g. the surface of a basketball or a pretzel. A geometric structure gives a way of measuring distances and angles on the manifold. It is natural to then study spaces of geometric structures (or shapes) on a fixed manifold. The classical Teichmuller theory studies all hyperbolic geometric structures on a fixed surface. Teichmuller theory has played a central role in several mathematical fields, e.g. complex analysis and dynamics, as well as in physics, especially in string theory. Similarly, a 3-manifold is a space that looks locally like 3-dimensional Euclidean space. The universe we live in is an example of a 3-dimensional manifold. Professor Canary proposes to study deformation spaces of hyperbolic geometric structures on 3-manifolds. More generally, Prof. Canary will study both the geometry and the topology of deformation spaces of geometric structures arising from representations of groups into semi-simple Lie groups. In addition, Prof. Canary will continue his commitment to undergraduate education, by continuing his involvement with curriculum development in courses using inquiry-based learning techniques, and his work mentoring graduate students and postdoctoral assistant professors.

Canary教授建议研究从双曲群到半单李群的表示所产生的几何结构的变形空间。许多这些调查的动机是经典的研究Teichmuller空间的双曲（或共形）结构的表面上。例如，一个闭曲面群在PSL（n，R）中的表示空间（共轭类）的希钦分支已经被证明与泰希穆勒空间有许多惊人的相似性。Canary教授与Bridgeman，Labourie和Sambarino合作，最近在每个Hitchin分量上开发了一个映射类组不变度量，该度量限制于Fuchsian轨迹上的Weil-Petersson度量。Canary教授建议研究这个度量的属性，并将其用作理解Hitchin组件几何形状的工具。更一般地说，他提议研究词双曲群到半简单李群的Anosov表示空间上的度量。Canary教授还建议研究双曲三维流形的变形空间的拓扑和几何。结束层合定理提供了这些变形空间中流形的分类，但由于这种分类中的不变量是不连续的，因此它不提供参数化。最近的工作表明，这些空间的拓扑结构丰富，还有许多有趣的问题有待研究。Canary教授将研究流形上几何结构的变形空间。二维流形是一个局部看起来像二维平面的空间，例如篮球或椒盐卷饼的表面。几何结构提供了一种测量流形上的距离和角度的方法。然后研究固定流形上的几何结构（或形状）空间是很自然的。经典的Teichmuller理论研究固定曲面上的所有双曲几何结构。 泰希穆勒理论在许多数学领域，如复分析和动力学，以及物理学，特别是弦理论中发挥了核心作用。类似地，三维流形是局部看起来像三维欧几里得空间的空间。我们生活的宇宙是三维流形的一个例子。Canary教授建议研究3-流形上双曲几何结构的变形空间。更一般地说，Canary教授将研究几何结构的变形空间的几何和拓扑，这些变形空间是由半单李群的表示产生的。此外，Canary教授将继续致力于本科教育，继续参与课程开发，使用基于探究的学习技术，指导研究生和博士后助理教授。