Structure of hyperbolic 3-manifolds

在过去的三年里，双曲3-流形和Kleian群领域取得了长足的进展，解决了大多数主要的动机猜想，如驯服猜想、结束分层猜想和密度猜想。这些进展在很大程度上证实了我们对双曲三维流形及其变形空间的预期，并将该领域置于一个过渡和机遇的时刻。证明中介绍的技术具有进一步应用的潜力。明斯基将专注于加深我们对双曲三维流形结构和变形理论的理解，特别是应用他为解决结束分层猜想所做出的贡献的工具。这些工具提供的模型和估计应该提供一种方法来解决一些公开的问题，特别是极限集的局部连通性、来自Heegaard分解的闭流形的几何描述以及变形空间的一致性定理(其中一些工作将与Brock、Bromberg和Canary合作)。另一个应用领域(与Brock和Masur一起)涉及赋予其Weil-Petersson度量的TeichMuller空间中测地线的结构。到目前为止，这些都拒绝分析，但似乎与三维流形的几何密切相关。一百多年来，几何、拓扑和动力学之间的相互作用一直是数学和物理的一个美丽而强大的特征。动力学是对数学或物理系统的时间演化的研究，而几何学和拓扑学涉及诸如表面或更高维类似物之类的“静态”对象，通常是动态过程的背景。亨利·庞加莱已经知道，标准圆球是经典分析和几何的背景，也是一种奇特的非欧几里德几何的“无限视界”，我们现在称之为双曲空间。球面变换的动力学性质转化为该空间刚性运动的几何性质，并产生了一族对称瓦片，其结构可以用几何和拓扑方法来研究。这些系统的复杂性可以如此多地约束它们，以至于组合(或拓扑)描述足以唯一地确定它们，这就是我们所说的刚性。这种现象在整个几何学和动力学中以多种形式出现，并与系统分类、绘制稳定和不稳定区域、系统族的变形和分叉以及遍历性等概率性质有关，所有这些问题在理论数学和应用数学中都具有重要意义。这个项目研究的特定方面在某些方面是典型的，在另一些方面是特殊的。他们专注于二维和三维几何之间的错综复杂的关系，以及拓扑，特别是曲面内的曲线系统，决定几何的方式。人们也非常重视研究曲面和三维流形上的几何结构族，它们与其他动力系统族非常相似。

The field of hyperbolic 3-manifolds and Kleinian groups has seen considerable progress in the last three years, with the resolution of most of the main motivational conjectures, such as Tameness, the Ending Lamination Conjecture, and the Density conjecture. These advances confirm much about our expected picture of hyperbolic 3-manifolds and their deformation spaces, and place the field in a moment of transition and opportunity. the techniques introduced in the proofs have much potential for further applications. Minsky will focus on deepening our understanding of the structure and deformation theory of hyperbolic 3-manifold, applying in particular the tools that have come out of his contribution to the solution of the Ending Lamination Conjecture. The models and estimates provided by these tools should provide an approach to a number of open questions, notably that of local connectivity of limit sets, geometric description of closed manifolds from the Heegaard decompositions, and uniformity theorems for deformation spaces (some of this work will be in collaboration with Brock, Bromberg and Canary). Another area of applications (jointly with Brock and Masur) involves the structure of geodesics in the Teichmuller space endowed with its Weil-Petersson metric. These have up till now resisted analysis but appear to be quite intimately connected to the geometry of 3-manifolds.The interactions between geometry, topology and dynamics have been a beautiful and powerful feature of mathematics and physics for more than a hundred years. Dynamics is the study of time-evolution of mathematical or physical systems, whereas geometry and topology involve "static" objects such as surfaces or higher-dimensional analogues, often the background for a dynamical process. Henri Poincare already knew that the standard round sphere, the setting of classical analysis and geometry, functioned also as a "horizon at infinity" for an exotic non-Euclidean geometry that we now call Hyperbolic space. Dynamical properties of transformations of the sphere translate to geometric properties of rigid motions of this space, and give rise to families of symmetric tilings whose structure we can study by geometric and topological methods. The complexity of these systems can constrain them so much that a combinatorial (or topological) description suffices to determine them uniquely, and this is what we call rigidity. This phenomenon occurs in many guises throughout geometry and dynamics, and is relevant to issues such as classification of systems, mapping out regions of stability and instability, deformation and bifurcation of families of systems, and probabilistic properties such as ergodicity, all of which havesignificance in both pure and applied mathematics. The particular aspects studied in this project are typical in some ways and special in others. They focus on the intricate relationships between geometry in two and three dimensions, and also on the ways in which topology, particularly of systems of curves within surfaces, determines geometry. There is also a strong emphasis on studying families of geometric structures on surfaces and three-dimensional manifolds, which are closely analogous to other families of dynamical systems.