The Classification Problem for Hyperbolic 3-Manifolds

双曲 3 流形的分类问题

• 批准号: 0354288
• 负责人: Jeffrey Brock
• 金额: $ 6.82万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354288&HistoricalAwards=false
• 关键词: Classification; Problem; Hyperbolic; Manifolds

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.

DMS-0204454Jeffrey F. Brock双曲 3 流形的分类问题 PI Jeffrey Brock 将综合双曲 3 流形变形理论中的多种技术来解决双曲 3 流形的分类问题。 Brock 将与 K. Bromberg 合作，采用双曲锥流形理论来证明每个驯服的双曲 3 流形 M 都由几何有限 3 流形来近似。 这个猜想被称为密度猜想，最近已被布洛克和布罗姆伯格在某些案例中解决。 布罗克还将努力完成与 R. Canary 和 Y. Minsky 的联合工作，以证明瑟斯顿的最终叠层猜想，该猜想预测一个温和的双曲 3 流形是由它的拓扑和它的末端不变量决定的：附加到一个“末端”的组合不变量双曲3流形。 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是拓扑驯服的问题。 这个联合项目对 Ahlfors 的猜想有影响，即有限生成的克莱因群的极限集在黎曼球面中要么测度为零，要么完全测度。 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所起的科学作用非常相似。 例如，随着人类基因组的“破解”，科学家现在可以分离出特定的遗传原因或疾病倾向，从而极大地提高了科学解决这些问题的能力。 在拟议的研究中，布洛克将致力于解决“通用”类三维空间（“双曲 3 流形”）的分类问题。 这些非欧几里德空间的几何形状与我们的欧几里德空间类似，但它们的大尺度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）呈指数级而不是线性发散。 威廉·P. Thurston 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 3 流形都是双曲的，并提出了与它回答的有关双曲 3 流形的问题一样多的问题。 从他的贡献中，双曲 3 流形的正确分类的令人信服的猜想图已经成为几何和拓扑领域研究人员的持久问题。 PI 和他的合作者最近的工作使这个问题的解决成为可能； PI 将利用他对 NSF 的支持来促进持续的合作来解决这一基本问题，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。