Mathematical Sciences: Geometric Structures and Discrete Groups

这种对地球上的几何结构的持续研究 流形和离散群将集中于复 双曲几何，球面CR几何，平坦洛伦兹 几何学、共形三维几何学和真实的投影 几何 这些几何形状是密切相关的，但每一个 展现出自己独特的个性。 直接构式 离散群的基本多面体（在精神上， Poincare）在几何中，其中全测地超曲面 要么不存在，要么没有足够的灵活性来构建 基本多面体，导致了半空间的替代概念 和超曲面 调查人员分析了这种 超曲面相交和可能的组合类型 由这些物体构成的多面体 他将这一理论应用于 上几何结构的模空间的几何和拓扑 一个流形，这些空间密切相关的更好地了解 流形的基本群的表示的模空间， 一种利用规范理论技术进行有益研究的结构。 这种模空间的原型是Teichmueller空间，其中一个 这个项目的目的是看看如何几何 Teichmueller空间扩展到这些更一般的模空间。 尽管我们生活在三维流形中， 我们的直觉很快就失去了作为真理或谬误的指导。 关于它们的一般几何和 拓扑性质 也不是很有帮助地告诉我们 去哪里寻找答案 一个复杂的理论已经发展出来 它正在突飞猛进地发展， 比原始的直觉更成功，最令人惊讶的特征之一 一个非欧几里德的 几何学，即所谓的双曲几何学，在这个理论中占有一席之地。 其他特殊的几何结构也已被利用， 类似的方式，有点像脚手架是用来建造一个 建筑：一个是关于流形的拓扑问题， 本质上独立于几何学的东西。 然后，一个强加于 几何结构，推导出似乎依赖于 这种几何结构（以及其中的一些，作为中间步骤， 实际上确实依赖于它），最后在某种意义上踢走了 脚手架来揭示原始问题的期望答案。 我的意思是，人们观察到，人们已经推导出一个结果， 没有明确提到几何结构， 然后成功地表明，任何其他选择的几何 结构会导致相同的最终结果。 这只是 这种研究的一个有趣的方面，因为这是真的， 所涉及的几何结构现在已经吸引了相当多的 自己的利益。

This continuing investigation of geometric structures on manifolds and discrete groups will concentrate on complex hyperbolic geometry, spherical CR geometry, flat Lorentzian geometry, conformal 3-dimensional geometry, and real projective geometry. These geometries are intimately interrelated, yet each displays its own characteristic personality. Direct constructions of discrete groups by fundamental polyhedra (in the spirit of Poincare) in geometries in which totally geodesic hypersurfaces either do not exist or are not sufficiently flexible to build fundamental polyhedra, have led to alternate notions of half-spaces and hypersurfaces. The investigator has analyzed how such hypersurfaces intersect and the possible combinatorial types for polyhedra built from these objects. He applies this theory to the geometry and topology of moduli spaces of geometric structures on a manifold, these spaces being closely related to the better known moduli space of representations of a manifold's fundamental group, a structure profitably studied using gauge-theoretic techniques. The prototype of such moduli spaces is Teichmueller space, and one objective of this project is to see how the geometry of Teichmueller space extends to these more general moduli spaces. Despite the fact that we live in a 3-dimensional manifold, our intuition quickly fails us as a guide to the truth or falsity of fairly simple questions about their general geometric and topological properties. Nor is it terribly helpful in telling us where to look for answers. An elaborate theory has been developed which is growing by leaps and bounds and has been far more successful than raw intuition, one of the most surprising features of the theory being the prominent role that a non-Euclidean geometry, so-called hyperbolic geometry, occupies in this theory. Other special geometric structures also have been exploited in similar ways, somewhat as a scaffolding is used in constructing a building: One asks a topological question about a manifold, something intrinsically independent of geometry. One then imposes a geometric structure, deduces properties that seem to depend upon this geometric structure (and some of which, as intermediate steps, actually do depend upon it), and finally in a sense kicks away the scaffolding to reveal the desired answer to the original question. By this I mean that one observes that one has deduced a result that does not mention the geometric structure explicitly, and that one then succeeds in showing that any other choice for the geometric structure would have led to the same final result. This is only one intriguing aspect of such research, for it is true that the geometric structures involved have now attracted a considerable amount of interest in their own right.