Mathematical Sciences: Geometric Structures and Discrete Groups

流形和离散群上几何结构的继续研究将集中在复双曲几何、球面CR几何、平坦洛伦兹几何、共形三维几何和实射影几何上。这些几何图形密切相关，但每一个都显示出自己独特的个性。在完全测地线超曲面不存在或没有足够的灵活性来建立基本多面体的几何中，由基本多面体直接构造离散群(本着庞加莱的精神)导致了半空间和超曲面的替代概念。研究人员分析了这些超曲面是如何相交的，以及从这些对象建立的多面体的可能组合类型。他将这一理论应用于流形上几何结构的模空间的几何和拓扑，这些空间与流形的基本群的表示的更著名的模空间密切相关，这种结构是用规范理论技术研究的有益的。这种模空间的原型是Teichmueller空间，这个项目的一个目标是研究Teichmueller空间的几何如何扩展到这些更一般的模空间。尽管我们生活在一个三维流形中，但我们的直觉很快就无法指导我们对相当简单的问题的正确或错误的判断，这些问题涉及到它们的一般几何和拓扑性质。在告诉我们在哪里寻找答案方面，它也没有太大的帮助。一种精致的理论已经被开发出来，它正在突飞猛进地发展，并且比原始的直觉更成功，该理论最令人惊讶的特征之一是非欧几里德几何，即所谓的双曲几何，在这一理论中占据着突出的角色。其他特殊的几何结构也被以类似的方式利用，有点像脚手架用于建造建筑：一个人问一个关于流形的拓扑问题，一些本质上独立于几何的东西。然后强加一个几何结构，推导出似乎依赖于这个几何结构的性质(其中一些作为中间步骤，实际上确实依赖于它)，最后在某种意义上踢开脚手架，揭示出原始问题的所需答案。我的意思是，一个人观察到，一个人已经推导出一个没有明确提到几何结构的结果，然后一个人成功地表明，对几何结构的任何其他选择都会导致同样的最终结果。这只是这类研究的一个有趣的方面，因为确实，所涉及的几何结构现在已经吸引了相当大的兴趣。

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.