FRG:Collaborative Research: Deformation spaces of geometric structures

全局拓扑何时可以支持以经典几何为模型的局部结构？“经典几何”是指在李群的传递作用下流形不变的结构。对于一个固定的拓扑，这种结构的空间是一个自然的对象，具有丰富的几何和对称性，与拓扑和齐次空间相关联。 对几何结构的变形空间的研究是由Charles Ehresmann在20世纪30年代开始的。它统一了世纪不同的研究领域（晶体学、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）。这一主题在20世纪70年代受到威廉·瑟斯顿的影响而变得突出。 这个理论的原型是闭曲面上的双曲几何结构空间，通过经典的一致化定理，它与曲面的Teichmueller空间一致。我们的项目探索了三个最新发展--三维流形上的几何结构、更高的Thurston-Teichmueller理论和Anosov表示--其中Teichmueller空间的丰富结构可以推广到更复杂几何的变形空间。 这个主题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主题：流形的拓扑结构，各种几何，代数和动力学。莫比乌斯带是只有一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。 我们生活的宇宙是三维流形的一个例子。 卫星或导弹的位置和速度由六维流形中的一点来描述。不同种类的几何区分特殊性质的流形。 莫比乌斯带自然是用受文艺复兴画家作品启发的投影几何来描述的。制图师使用保形几何来制作更精确的世界地图。微分几何使爱因斯坦发展了他的引力理论。化学家用群的代数对晶体进行分类。化学元素周期表与空间旋转群密切相关。 这一数学景观的大部分尚未探索。 利用现代计算机，学生可以为这项调查作出贡献。 明确的例子和他们的相互作用的探索提供了有才华的学生的问题，邀请他们的数学研究的兴奋。

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.