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.

全局拓扑何时才能支持以经典几何体为模型的局部结构？“经典几何”是指在李群的传递作用下流形不变量的结构。对于固定的拓扑结构，这种结构的空间是一个自然对象，具有丰富的几何和对称性，与拓扑和齐次空间相联系。几何结构形变空间的研究是由查尔斯·埃雷斯曼在20世纪30年代由S开创的。它统一了19世纪不同的研究领域(结晶学、全纯微分方程和保角映射、射影几何和非欧几里德几何的发展)。这一理论的原型是闭曲面上的双曲几何结构空间，根据经典的一致化定理，它与曲面上的Teichmueller空间相同。我们的项目探索了三个最新的发展--三维流形上的几何结构，更高的瑟斯顿-泰希穆勒理论和Anosov表示--其中Teichmueller空间的丰富结构推广到了更复杂几何的变形空间。这门学科与许多其他数学领域的密切关系突出了它在数学中的核心作用。这个项目综合了不同的数学学科：流形的拓扑学、各种几何、代数和动力学。莫比乌斯带是只有一条边的二维流形的一个例子。例如，它描述了平面上所有直线的集合。我们生活的宇宙是一个三维流形的例子。卫星或导弹的位置和速度由六维流形上的一个点来描述。不同类型的几何区分了流形的特殊性质。Moebius乐队是受文艺复兴时期画家的作品启发而用射影几何自然地描述的。制图师使用保角几何绘制出更精确的世界地图。微分几何使爱因斯坦发展了他的引力理论。化学家使用群代数来对晶体进行分类。化学元素的元素周期表与空间的旋转群密切相关。这一数学图景的大部分仍未被探索。使用现代计算机，学生们可以为这项调查做出贡献。显性例子的探索及其相互作用为有才华的学生提供了问题，邀请他们参与数学研究。