The Classification Problem for Hyperbolic 3-Manifolds

DMS-0204454 Jeffrey F. Brock双曲三维流形的分类问题PI，Jeffrey Brock，将综合双曲三维流形变形理论中的各种技术来解决双曲三维流形的分类问题。 布洛克将与K. Bromberg利用双曲锥流形理论证明了每个驯服的双曲三维流形M都可以用几何有限三维流形来逼近。 这个猜想，被称为密度猜想，最近已经解决了Brock和Bromberg不确定的情况。 布洛克还将致力于完成与R. Canary和Y. Minsky证明了Thurston的终结分层猜想，该猜想预言一个驯服的双曲3-流形是由它的拓扑和它的终结不变量决定的：连接到双曲3-流形的“终结”的组合不变量。 与Bromberg，R. Evans和J. Souto，Brock将研究一个几何有限双曲3-流形序列的每个代数极限是否是其自身的拓扑极限的问题。 这个联合项目的影响Ahlfors的猜想，极限集的一个refined产生的Kleinian群eithermeasure零或充分措施的黎曼领域。 在这些领域的发展中，对数学对象进行分类与对生物、化学或物理现象进行分类起着同样的科学作用。 例如，随着人类基因组的“破解”，科学家们现在可以分离出特定的遗传原因或疾病的易感性，大大提高了科学解决这些问题的能力。 在这项研究中，Brock将致力于解决一个“通用”的三维空间类--“双曲三维流形”的分类问题。“非欧空间的几何形状与我们的欧空间相似，但它们的大尺度几何形状是指数扩张的：例如，从点源发出的光线（测地线的隐喻）是指数发散的，而不是线性发散的。 威廉P.瑟斯顿的革命性和开创性的工作在20世纪70年代和80年代表明，几乎所有的3-流形是双曲的，并提出了许多问题，双曲3-流形，因为它回答。 从他的贡献，一个引人注目的双曲三维流形的正确分类的理论图像已经成为几何和拓扑学领域研究人员的一个难题，PI和他的合作者最近的工作使这个问题的解决变得触手可及; PI将利用他NSF的支持来促进正在进行的合作，从而使一个“数据库”的双曲三维流形可供更广泛的使用，由其他数学家和物理学家一样。

DMS-0204454Jeffrey F. BrockTHE CLASSIFICATION PROBLEM FOR HYPERBOLIC 3-MANIFOLDSThe PI, Jeffrey Brock, will synthesize diverse techniques in thedeformation theory of hyperbolic 3-manifolds to address classificationproblem for hyperbolic 3-manifolds. Brock will undertake joint workwith K. Bromberg that employs the theory of hyperbolic cone-manifoldsto show that each tame hyperbolic 3-manifold M is approximated bygeometrically finite 3-manifolds. This conjecture, known as theDensity Conjecture has recently been solved by Brock and Bromberg incertain cases. Brock will also work toward completing joint workwith R. Canary and Y. Minsky to prove Thurston's ending laminationconjecture, which predicts that a tame hyperbolic 3-manifold isdetermined by its topology and its end invariants: combinatorialinvariants attached to the ``ends'' of a hyperbolic 3-manifold. Innew joint work with Bromberg, R. Evans, and J. Souto, Brock will studythe question of whether each algebraic limit of a sequence ofgeometrically finite hyperbolic 3-manifolds is itself topologicallytame. This joint project has implications for a conjecture of Ahlforsthat the limit set of a finitely generated Kleinian group has eithermeasure zero or full measure in the Riemann sphere. Classifying mathematical objects plays much the same scientific roleas classifying biological, chemical, or physical phenomena in thedevelopment of these fields. For example, with the human genome"cracked," scientists may now isolate specific genetic causes orpredispositions to diseases, greatly furthering the ability of scienceto address these problems. In the proposed research, Brock willendeavor to solve the classification problem for a "generic" class of3-dimensional spaces, the "hyperbolic 3-manifolds." Thesenon-Euclidean spaces have geometry locally like our own Euclideanspace, but their large scale geometry is expanding exponentially:for example, light rays (a metaphor for geodesics) emanating from apoint-source diverge exponentially rather than linearly. WilliamP. Thurston's revolutionary and pioneering work in the 1970's and1980's showed that almost all 3-manifolds are hyperbolic, and went onto raise as many questions about hyperbolic 3-manifolds as itanswered. From his contributions, a compelling conjectural picture ofthe right classification of hyperbolic 3-manifolds has emerged as alasting problem for researchers in the field of geometry and topology.Recent work of the PI and his collaborators has put the solution ofthis problem within reach; the PI will make use his NSF support tofacilitate ongoing collaborations to solve this fundamental problem,thereby making a "database" of hyperbolic 3-manifolds available forwider use by other mathematicians and physicists alike.