The Classification Problem for Hyperbolic 3-Manifolds

DMS-0204454 Jeffrey F.Brock双曲三维流形的分类问题Jeffrey BRock将综合双曲三维流形形变理论中的各种技术来解决双曲三维流形的分类问题。布罗克将与K.Bromberg合作，利用双曲锥流形理论证明每个驯服的双曲三维流形M都是由几何有限的三维流形逼近的。这个被称为密度猜想的猜想最近被布罗克和布朗伯格在某些情况下解决了。布罗克还将致力于完成与R.Canary和Y.Minsky的联合工作，以证明瑟斯顿的结束分层猜想，该猜想预测一个温和的双曲三维流形是由它的拓扑和它的末端不变量决定的：附加在双曲三维流形的‘末端’上的组合不变量。在与Bromberg、R.Evans和J.Souto的新合作中，Brock将研究几何有限的三维双曲流形序列的每个代数极限本身是否拓扑驯服的问题。这个联合计画暗示了Ahlfors的一个猜想，即有限生成的Klein群的极限集在黎曼球面上要么有测度零，要么有全测度。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具有很大的科学作用。例如，随着人类基因组的“破解”，科学家现在可以分离出疾病的特定遗传原因或易感性，极大地促进了科学解决这些问题的能力。在这项拟议的研究中，布罗克将努力解决一类“通用”的三维空间--“双曲三维流形”的分类问题。这些非欧几里德空间有像我们自己的欧几里得空间一样的局部几何，但它们的大尺度几何正在指数扩张：例如，从点源发出的光线(测地线的比喻)以指数而不是线性的方式发散。威廉姆·P。瑟斯顿在1970年的《S》和1980年的《S》中所做的革命性和开创性的工作表明，几乎所有的三维流形都是双曲的，并提出了与它回答的一样多的关于双曲三维流形的问题。通过他的贡献，双曲三维流形的正确分类的令人信服的猜想图景已经成为几何和拓扑学领域的研究人员的最新问题。PI及其合作者最近的工作使这个问题的解决变得触手可及；PI将利用他的NSF支持来促进正在进行的合作来解决这个基本问题，从而使双曲三维流形的“数据库”可供其他数学家和物理学家更广泛地使用。

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.