Topology and Dynamics of Moduli Spaces of Geometric Structures

小行星9803518 流形M上的局部齐次几何结构是M的开子集上的局部坐标系，其值在齐次空间G/H中。 在M的重叠面片上，坐标通过李群G中的变换相关联。 在Cartan和Ehresmann之后，坐标图全球化为M的泛覆盖空间U到G/H的“发展”，定义了M的基本群F的同态（实现为U的甲板变换群）。 一个形式的模空间Def（M，G/H）的几何结构在M上建模G/H的分类结构等价。 Def（M，G/H）自身局部同胚于同态的等价类的模空间Hom（F，G）/G. 这些模空间的原型是Teichmuller空间，分类为双曲结构（这里G/H是真实的双曲平面，G = PSL（2，R））。 而且M的映射类群既作用于Def（M，G/H）又作用于Hom（F，G）/G，并且Def（M，G/H）- Hom（F，G）/G的完整映射关于这些作用是等变的. 本文主要研究了F是紧致曲面的基本群的情形。 当M是闭曲面时，这些模空间具有不变的辛结构，当G是紧的时，研究者证明了在映射类群的作用下，辛测度是遍历的。 当G是SL（2，R）时，这个模空间与（可能是奇异的）双曲结构的Teichmuller空间有关，研究者利用几何结构研究了模空间上的动力学。 这种动力学研究涉及的离散映射类组的某些哈密顿流模空间上的行动。 当G/H是Minkowski 2+1-空间时，研究者（与Drumm合作）正在研究完全平坦流形的变形空间，使用Drumm论文中开发的“弯曲平面”和Margulis的思想。 当G/H是复双曲2-空间时，Xia对Hom（F，G）/G的连通分量进行了分类，并且研究人员（与Leeb和Kapovich合作）发现了每个分量中的离散嵌入。 当G/H是真实的射影平面时，结合Choi的工作，给出了变形空间Def（M，G/H）的完整描述. 这种几何结构代表了一种将拓扑学和几何学联系起来的方式。 根据Felix Klein的埃尔兰根计划，几何学是研究李群G的齐次空间G/H中的物体之间的关系，这些物体在G中的变换下是不变的。 它们包括“刚性”几何属性，如距离、角度、平行度和共线。 另一方面，拓扑学研究流形M上点的松散组织，其中唯一的关系来自连续映射。 M上的几何结构通过局部坐标图将其点从M中的开集更严格地组织到G/H中。 模空间参数化了这些结构的等价类，并与M到G的基本群的同态空间密切相关。 这些模空间本身具有丰富的几何结构，并显示出不寻常的对称性，这些对称性源于它们作为自然几何对象的起源。 对这些基本对象的更好理解一次又一次地通过在相关领域（通常是理论物理学）的见解来回报努力。 ***

9803518 Goldman A locally homogeneous geometric structure on a manifold M is a system of local coordinates on open subsets of M with values in a homogeneous space G/H. On overlapping patches of M, the coordinates are related by transformations in the Lie group G. Following Cartan and Ehresmann, the coordinate charts globalize to a "development" of the universal covering space U of M into G/H, defining a homomorphism of the fundamental group F of M (realized as the group of deck transformations of U). One forms a moduli space Def(M,G/H) of geometric structures on M modelled on G/H which classifies structures up to equivalence. Def(M,G/H) is itself locally homeomorphic to a moduli space of equivalence classes of homomorphisms Hom(F,G)/G. The prototype of these moduli spaces is the Teichmuller space, classifying hyperbolic structures (here G/H is the real hyperbolic plane and G = PSL(2,R)). Moreover the mapping class group of M acts on both Def(M,G/H) and Hom(F,G)/G, and the holonomy map from Def(M,G/H) - Hom(F,G)/G is equivariant with respect to these actions. This investigation mainly concentrates on the case when F is the fundamental group of a compact surface. When M is a closed surface, these moduli spaces carry invariant symplectic structures, and when G is compact the investigator conjectures that the symplectic measure is ergodic under the action of the mapping class group. When G is SL(2,R), this moduli space is related to Teichmuller space of (possibly singular) hyperbolic structures, and the investigator has used geometric structures to study the dynamics on the moduli space. This dynamical study relates the action of the discrete mapping class group to certain Hamiltonian flows on the moduli space. When G/H is Minkowski 2+1-space, the investigator (in collaboration with Drumm) is studying the deformation spaces of complete flat manifolds, using "crooked planes" developed in Drumm's thesis and ideas of Margulis. When G/H is complex hyperbo lic 2-space, Xia has classified the connected components of Hom(F,G)/G and the investigator (in collaboration with Leeb and Kapovich) has found discrete embeddings in each component. When G/H is the real projective plane, the work of the investigator combined with that of Choi gives a complete description of the deformation space Def(M,G/H). Such geometric structures represent a way of relating topology and geometry. According to Felix Klein's Erlangen program, geometry is the study of relations between objects in a homogeneous space G/H of a Lie group G that are invariant under the transformations in G. They include "rigid" geometric properties such as distance, angle, parallelism and collinearity. On the other hand, topology studies the loose organization of points on a manifold M, where the only relations derive from continuous mapping. A geometric structure on M more rigidly organizes its points through local coordinate charts from open sets in M to G/H. Moduli spaces parametrize equivalence classes of these structures and closely relate to spaces of homomorphisms of the fundamental group of M into G. These moduli spaces themselves enjoy rich geometric structures of their own, and display unusual symmetries arising from their origin as natural geometric objects. A better understanding of such basic objects has time and again repaid the effort through insights in related fields, typically in theoretical physics. ***