FRG: Collaborative Research: Deformation Spaces of Geometric Structures

FRG：协作研究：几何结构的变形空间

• 批准号: 1065939
• 负责人: Daryl Cooper
• 金额: $ 11.97万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065939&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Deformation; Spaces

When can a global topology support a local structure modeled on a classical geometry? A "classical geometry" means the structure of a manifold invariant under a transitive action of a Lie group. For a fixed topology, the space of such structures is a natural object, with a rich geometry and symmetry of its own, associated with the topology and the homogeneous space. The study of deformation spaces of geometric structures was initiated by Charles Ehresmann in the 1930's. It unifies what have been disparate areas of research in the 19th century (crystallography, holomorphic differential equations and conformal mapping, development of projective geometry and non-Euclidean geometry). The subject became prominent through the influence of William Thurston in the 1970's. The prototype of this theory is the space of hyperbolic geometry structures on a closed surface, which by the classical uniformization theorem, identifies with the Teichmueller space of the surface. Our project explores three recent developments - geometric structures on 3-manifolds, higher Thurston-Teichmueller theory, and Anosov representations - where the rich structure of Teichmueller space generalizes to deformation spaces of more complicated geometries. The intimate relations of this subject with many other fields of mathematics underscores its central role in mathematics.This project synthesizes disparate mathematical subjects: the topology of manifolds, various kinds of geometry, algebra and dynamics. The Moebius band is an example of a two dimensional manifold with only one side. It describes, for example, the collection of all straight lines in the plane. The universe we live in is an example of a three dimensional manifold. The position and velocity of a satellite or missile is described by a point in a six-dimensional manifold. Different kinds of geometries distinguish special properties of manifolds. The Moebius band is naturally described using the projective geometry inspired by the work of Renaissance painters. Cartographers used conformal geometry to produce more accurate maps of the world. Differential geometry enabled Einstein to develop his theory of gravitation. Chemists use the algebra of groups to classify crystals. The periodic table of chemical elements is intimately connected to the group of rotations of space. Much of this mathematical landscape remains unexplored. Using modern computers, students can contribute to this investigation. The exploration of explicit examples and their interactions provides problems for talented students, inviting them to the excitement of mathematical research.

全球拓扑何时可以支持以经典几何形状建模的本地结构？ “经典的几何形状”是指在谎言群体的及传统作用下流形不变的结构。对于固定拓扑，这种结构的空间是一个自然物体，具有丰富的几何形状和对称性，与拓扑和均匀空间相关。 查尔斯·埃斯曼（Charles Ehresmann）在1930年代启动了几何结构变形空间的研究。它统一了19世纪的研究领域（晶体学，全态微​​分方程和保形映射，投射几何形状的发展和非欧几里得几何形状）。威廉·瑟斯顿（William Thurston）在1970年代的影响下，这个主题变得很突出。 该理论的原型是封闭表面上双曲几何结构的空间，通过经典统一定理，它可以与表面的Teichmueller空间相同。我们的项目探讨了三个最新的发展 - 3个manifolds的几何结构，较高的瑟斯顿·蒂奇默勒理论和Anosov代表 - Teichmueller空间的丰富结构概括为更复杂的几何形状的变形空间。 该主题与许多其他数学领域的亲密关系强调了其在数学中的核心作用。该项目综合了不同的数学主题：流形的拓扑结构，各种几何形状，代数和动力学。 Moebius频带是仅一侧的二维流形的一个例子。例如，它描述了平面中所有直线的集合。 我们生活的宇宙是三维流形的一个例子。 六维流形中的一个点描述了卫星或导弹的位置和速度。不同种类的几何形状区分了歧管的特殊特性。 Moebius带自然地使用了受文艺复兴时期画家作品启发的投射几何形状进行描述。制图师使用保形几何形状生成世界上更准确的地图。差异几何形状使爱因斯坦能够发展他的引力理论。化学家使用组代数来对晶体进行分类。化学元件的元素表与空间旋转群密切相关。 这种数学景观的大部分仍未开发。 使用现代计算机，学生可以为这项调查做出贡献。 探索明确的例子及其互动为才华横溢的学生提供了问题，邀请他们激发数学研究的兴奋。