GEOMETRIC STRUCTURES AND SURFACES

该项目将集中于流形几何结构的模。这一学科起源于19世纪的几条线索：(1)晶体的对称性，它发展成为晶体学群理论；(2)保角映射，以及相应的全纯微分方程和它们的单群，由此产生了经典的均匀化理论；(3)经典曲面微分几何，推广到黎曼几何，后来推广到受爱因斯坦引力理论影响的洛伦兹几何。几何本身的概念是由Felix Klein代数化的，他将几何解释为在李群的传递作用下不变的性质。本文从基本群和李群的角度探讨了在拓扑流形上放置几何结构的可能方法。目标是描述给定的拓扑（通常由组合对象描述）何时可以给定给定的几何形状。在一些情况下，存在完整的几何结构分类，模空间本身既具有丰富的几何结构，又具有自身的对称性，从而导致复杂的动力系统。该项目的具体目标和范围涉及平面投影和保形结构，特别是在三维空间。最近对完全仿射3流形的分类导致了许多关于更一般结构的问题。该提案特别询问哪些群体承认适当的仿射行动；目前已知的双曲群是自由群。方法涉及动力系统、代数群论、拓扑与几何群论和表示理论的结合。其他重要的问题还包括：寻找3流管的障碍物，使其能够容纳射影结构；目前只知道一个这样的封闭3流形。计算机实验可以在这个方向上提供帮助，实验几何实验室的项目可以提供几何结构的新例子。

The project will concentrate on the moduli of geometric structures on manifolds. This subject arose from several threads in the 19th century: (1) symmetries of crystals, which developed into the theory of crystallographic groups; (2) conformal mapping, and the corresponding holomorphic differential equations and their monodromy groups, which led to the classical theory of uniformization; (3) classical differential geometry of surfaces, generalized to Riemannian geometry, and later to Lorentzian geometry influenced by Einstein's theory of gravitation. The notion of a geometry itself was algebraicized by Felix Klein, who interpreted a geometry as the properties invariant under a transitive action of a Lie group. The present study investigates the possible ways of putting a geometric structure on a topological manifold in terms of the fundamental group and the Lie group. The goal is to describe when a given topology (often described by a combinatorial object) can be given the given geometry. In several cases, there is a complete classification of geometric structures, and the moduli space itself enjoys both a rich geometry of its own and symmetries of its own, which lead to intricate dynamical systems.The specific goals and scope of this project involve flat projective and conformal structures, especially in dimension three. The recent classification of complete affine 3-manifolds leads to many questions about more general structures. In particular the proposal asked which groups which admit proper affine actions; at present the only known hyperbolic groups are free groups. The methods involve a combination of dynamical systems, algebraic group theory, topology and geometric group theory and representation theory. Other important questions include finding obstructions for a 3-manifold to admit a projective structure; at present only one such closed 3-manifold is known. Computer experiments can help in this direction, and projects in the Experimental Geometry Lab can suggest new examples of geometric structures.