The Classification Problem for Hyperbolic 3-Manifolds

双曲3-流形的分类问题PI Jeffrey Brock将综合双曲3-流形变形理论中的各种技术来解决双曲3-流形的分类问题。Brock将与利用双曲锥流形理论的K. Bromberg进行联合工作，以证明每个驯服的双曲3流形M都是由几何有限的3流形近似的。这个猜想，被称为密度猜想，最近已经被Brock和Bromberg在不确定的情况下解决了。Brock还将致力于完成与R. Canary和Y. Minsky的联合工作，以证明Thurston的结束层积猜想，该猜想预测一个驯服的双曲3-流形是由它的拓扑和它的末端不变量决定的：连接到双曲3-流形“末端”的组合不变量。在与Bromberg， R. Evans和J. Souto的新合作中，Brock将研究几何有限双曲3-流形序列的每个代数极限本身是否具有拓扑性的问题。这一联合项目对ahlfors关于有限生成Kleinian群的极限集在黎曼球中要么是测度零，要么是满测度的猜想具有启示意义。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相同的科学作用。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倾向，极大地促进了科学解决这些问题的能力。在提出的研究中，Brock将努力解决“一般”三维空间的分类问题，即“双曲3-流形”。这些非欧几里得空间的局部几何形状与我们自己的欧几里得空间相似，但它们的大规模几何形状呈指数级扩展：例如，从点源发出的光线（测地线的隐喻）呈指数级发散，而不是线性发散。WilliamP。瑟斯顿在20世纪70年代和80年代的革命性和开创性工作表明，几乎所有的3-流形都是双曲的，并继续提出了许多关于双曲3-流形的问题。从他的贡献中，一个令人信服的双曲3流形正确分类的猜想图景已经成为几何和拓扑学领域研究人员的永恒问题。PI和他的合作者最近的工作已经使这个问题的解决方案触手可及；PI将利用他的国家科学基金会的支持来促进正在进行的合作，以解决这个基本问题，从而使双曲3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.