Algebraic and Geometric Topology

摘要奖：DMS-0305712主要研究人员：R.L.科恩，S.P.Kerckhoff，R.J.Milgram这个项目包括在代数和几何拓扑学的广泛主题中的研究计划。主要研究人员R.L.科恩教授、S.P.Kerckhoff教授和R.J.Milgram教授都是资深拓扑学家，他们的研究兴趣涵盖了大量的拓扑学和相关领域。这些项目包括核心拓扑学问题的研究，如三维几何的研究，流形上复杂结构和相关结构的拓扑研究，以及有限群的拓扑不变量的研究。此外，它还包括拓扑学方法在机器人领域工程问题中的应用。三维流形的拓扑与微分几何密切相关，特别是与欧几里得几何、球面几何和双曲几何等齐次结构密切相关。其中，双曲几何是最普遍的。在闭的3-流形上，双曲结构如果存在，则根据Mostow刚性定理是唯一的。因此，任何几何不变量都是自拓不变量，可以用来区分不同的三维流形。当三维流形有边界时，它可以具有较大的双曲结构的参数空间；在这种情况下，结构的定性性质在参数空间上变化是特别有趣的。Kerckhoff研究了三维流形上的双曲结构，试图描述这些几何结构在一种称为Dehn手术的拓扑过程中的行为，以及在锥体上模拟奇点的情况。R·L·科恩正在进行几个项目，这些项目涉及研究源于几何的问题的代数拓扑方面。这些项目列在以下总标题下。1.复流形和辛流形中全纯曲线的模空间的拓扑。2.稳定映射类群的同伦型与Mumford猜想。3.全纯K-理论的性质及应用。R.J.米尔格拉姆将继续探讨涉及有限单群的上同调的问题，以及配置空间拓扑在机械臂设计和行为中的应用。

AbstractAward: DMS-0305712Principal Investigator: R.L. Cohen, S.P. Kerckhoff, R.J. MilgramThis project consists of research programs in a broad range oftopics in Algebraic and Geometric Topology. The principalinvestigators, Professors R.L. Cohen, S.P. Kerckhoff, andR.J. Milgram are senior topologists whose research interestscover a large number of areas of topology and related fields.Included in these projects are the study of core topologicalquestions such as the study of 3 - dimensional geometry, thestudy of the topology of complex and related structures onmanifolds, and topological invariants of finite groups. Inaddition it includes applications of topological methods toengineering questions in the field of Robotics.The topology of 3-dimensional manifolds is intimately connectedto differential geometry, particularly to homogeneous structureslike Euclidean, spherical, and hyperbolic geometry. Of these,hyperbolic geometry is the most prevalent. On a closed3-manifold a hyperbolic structure, if it exists, is unique by theMostow Rigidity Theorem. Thus, any geometric invariant is atopological invariant and can be used to distinguish different3-manifolds. When the 3-manifold has boundary, it can have alarge parameter space of hyperbolic structures; in this case, thequalitativeq properties of the structures as one varies over theparameter space are of particular interest. Kerckhoff studieshyperbolic structures on 3-manifolds, trying to describe thebehavior of these geometric structures under a topologicalprocedure, called Dehn surgery, and in the presence ofsingularities modeled on a cone. R.L. Cohen is pursuing severalprojects that involve studying algebraic topological aspects ofquestons stemming from geometry. These projects lie under thefollowing general headings. 1. The topology of moduli spaces ofholomorphic curves in complex and symplectic manifolds. 2. Thehomotopy type of the stable mapping class group and the MumfordConjecture. 3. Properties and applications of holomorphicK-theory. R.J. Milgram will pursue questions involving thecohomology of finite simple groups, as well as applications ofthe topology of configuration spaces to questions in the designand behavior of robotic arms.