Hyperbolic geometry, topology and dynamics

项目负责人：Yair minsky：该项目旨在研究二维和三维双曲几何的许多不同方面，参数化这些几何系统的表示空间的结构，以及作用于它们的群的动力学。群F的SL(2，C)-字符变体，X(F)，参数化了双曲三维空间等距群SL（2，C）中F表示的共轭类。X(F)的离散忠实元素对应于双曲3流形，PI将更好地理解这些流形的拓扑和几何特征之间的相互作用，以及离散忠实轨迹本身的结构。此外，他将把X(F)作为一个整体进行研究，将其视为F的外部自同构群作用下的一个动力系统（特别是当F是一个自由群时）。最近发现了一种新的X(F)的动态分解，其结构似乎丰富且相对未被探索。此外，PI将研究二维双曲结构参数化的Teichmuller空间的几何方面，以及映射类群的自然自同构群。几何、拓扑学和动力学之间的相互作用在数学及其应用中起着核心作用。一个几何空间，例如我们自己的宇宙或某些系统的构形空间，可能允许动态现象，如流动、迭代或群体行为。这些现象的行为，以及空间的几何形状，都受到其拓扑结构的强烈影响，即几何形状覆盖的底层结缔组织。此外，我们经常发现几何和动力学在更高的抽象层次上持续存在：给定空间上所有几何结构的集合本身可以组织成一个新的“更高”空间，具有自己的几何形状和自身固有的对称性，从而产生动态结构。这些现象在不同层次上的相互作用可以丰富我们对原始系统的认识。PI自己的研究集中在这个通用模板的特定实例上，即2维和3维空间的几何和拓扑，以及它们的高参数空间的相应动力学。这种低维的设置特别适合我们的直觉，部分原因是与我们的物理世界进行直接的视觉类比，但由于各种原因，它也恰好是数学和物理的许多不同领域的交汇处，因此，通过类比和直接的数学联系，对这一领域的更全面的理解可以丰富我们对数学其他部分的方法。该奖项将由拓扑学、几何分析和分析项目共同资助。

AbstractAward: DMS-1005973Principal Investigator: Yair MinskyThe PI proposes to investigate a number of different aspects of hyperbolic geometry in 2 and 3 dimensions, the structure of the representation spaces that parametrize these geometric systems, and the dynamics of the groups that act on them. The SL(2,C)-character variety, X(F), of a group F parametrizes conjugacy classes of representations of F into SL(2,C), the isometry group of hyperbolic 3-space. Discrete faithful elements of X(F) correspond to hyperbolic 3-manifolds, and the PI will work to understand better the interaction between topological and geometric features of these manifolds, as well as the structure of the discrete-faithful locus itself. In addition, he will pursue a study of X(F) as a whole, considered as a dynamical system under the action of the outer automorphism group of F (particularly when F is a free group). A new dynamical decomposition of X(F) was recently discovered, whose structure seems to be rich and relatively unexplored. In addition the PI will study geometric aspects of Teichmuller spaces, which parametrize hyperbolic structures in two dimensions, and of Mapping Class Groups, their natural automorphism groups.The interactions between geometry, topology and dynamics play a entral role in mathematics as well as its applications. A geometric space, such as our own universe or the configuration space of some system, may admit dynamical phenomena such as flows, iterations or group actions. The behavior of these phenomena, as well as the geometry of the space, can be strongly influenced by its topological structure, namely the underlying connective tissue on which the geometry is overlaid. Furthermore, we often find that geometry and dynamics persist at higher levels of abstraction: the collection of all geometric structures on a given space can itself be organized into a new "higher" space, with its own geometry and its own inherent symmetries which give rise to dynamical structure. The interaction between these phenomena at different levels can enrich our insight about the original systems. The PI's own research focuses on particular instances of this general template, namely the geometry and topology of 2- and 3- dimensional spaces, and the corresponding dynamics for their higher parameter spaces. This low-dimensional setting is particularly amenable to our intuition and is partly motivated by direct visual analogies with our physical world, but it also happens, for a variety of reasons, to be a meeting place for a number of different areas of mathematics as well as physics, so that a fuller understanding in this domain can enrich, by analogy as well as direct mathematical connection, our approaches to other parts of mathematics. This award is to be funded jointly by the programs in Topology, Geometric Analysis, and Analysis.