FRG: Collaborative Research: Deformation Spaces of Geometric Structures

什么时候全局拓扑可以支持在经典几何上建模的局部结构？“经典几何”是指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，这类结构的空间是一个自然对象，具有自身丰富的几何和对称性，与拓扑结构和同质空间相关联。几何结构变形空间的研究是由Charles Ehresmann在20世纪30年代发起的。它统一了19世纪不同领域的研究（晶体学、全纯微分方程和共形映射、射影几何和非欧几里得几何的发展）。这个主题在20世纪70年代受到威廉·瑟斯顿的影响而变得突出。该理论的原型是封闭曲面上的双曲几何结构空间，根据经典均匀化定理，该空间与曲面的Teichmueller空间一致。我们的项目探讨了三个最近的发展- 3流形上的几何结构，更高的Thurston-Teichmueller理论和Anosov表示-其中Teichmueller空间的丰富结构推广到更复杂几何形状的变形空间。这门学科与许多其他数学领域的密切关系强调了它在数学中的核心作用。这个项目综合了不同的数学科目：流形的拓扑，各种几何，代数和动力学。莫比乌斯带是一个只有一面的二维流形的例子。例如，它描述了平面上所有直线的集合。我们生活的宇宙就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中的一个点来描述。不同的几何形状区分了流形的特殊性质。莫比斯带自然是用受文艺复兴时期画家作品启发的射影几何来描述的。制图师使用共形几何绘制出更精确的世界地图。微分几何使爱因斯坦得以发展他的万有引力理论。化学家用群的代数来对晶体进行分类。化学元素周期表与空间的旋转群密切相关。这一数学领域的许多领域仍未被探索。使用现代计算机，学生可以为这项调查做出贡献。对明确的例子及其相互作用的探索为有才华的学生提供了问题，邀请他们参与数学研究的兴奋。

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.