Algebraic and Geometric Topology

摘要奖：DMS-0305712主要研究者：R.L.科恩，S.P. Kerckhoff，R.J. MilgramThis project consists of research programs in a wide range of topics in Algebras and Geometric Topology. 主要研究者R.L. Cohen，S. P. Kerckhoff和R.J. Milgram是资深拓扑学家，他们的研究兴趣涵盖了拓扑学和相关领域的许多领域，包括对核心拓扑问题的研究，如三维几何的研究，流形上复结构和相关结构的拓扑研究，以及有限群的拓扑不变量。 此外，它还包括拓扑方法在机器人领域的工程问题中的应用。三维流形的拓扑与微分几何密切相关，特别是与欧氏几何、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。 在闭三维流形上，如果双曲结构存在，则通过Mostow刚性定理证明它是唯一的。 因此，任何几何不变量都是拓扑不变量，并且可以用来区分不同的三维流形。 当三维流形有边界时，它可以有一个很大的双曲结构参数空间，在这种情况下，结构在参数空间上变化时的定性性质是特别令人感兴趣的。 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拓扑过程（称为Dehn手术）下的行为，并在存在以锥为模型的奇点的情况下。 R.L.科恩正在进行几个项目，涉及研究代数拓扑方面的问题源于几何。 这些项目分为以下几个大类。 1.复流形和辛流形中全纯曲线模空间的拓扑。 2.稳定映射类群的同伦型与Mumford猜想。 3.全纯K-理论的性质和应用。 R. J.米尔格拉姆将继续研究有限简单群的上同调问题，以及配置空间的拓扑在机器人手臂的设计和行为问题中的应用。

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.