Hyperbolic geometry, topology and dynamics

摘要奖：DMS-1005973首席研究员：Yair Minsky PI建议研究二维和三维双曲几何的许多不同方面，将这些几何系统参数化的表示空间的结构，以及作用于这些几何系统的群的动力学。群F的SL(2，C)特征标簇X(F)将F的表示的共轭类参数化为SL(2，C)，即双曲空间的等距群。X(F)的离散忠实元对应于双曲三维流形，PI将致力于更好地理解这些流形的拓扑特征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，他还将继续研究X(F)作为一个整体，它被认为是在F的外自同构群(特别是当F是自由群时)作用下的一个动力系统。最近发现了X(F)的一种新的动力学分解，其结构似乎丰富而相对未被探索。此外，PI还将研究TeichMuller空间和映射类群的几何方面，映射类群是它们的自然自同构群。几何、拓扑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。几何空间，如我们自己的宇宙或某个系统的构型空间，可能会允许流动、迭代或群作用等动力学现象。这些现象的行为，以及空间的几何形状，可以强烈地受到其拓扑结构的影响，即几何图形覆盖在其上的底层结缔组织。此外，我们经常发现几何和动力学在更高的抽象水平上持续存在：给定空间上所有几何结构的集合本身可以被组织成一个新的“更高”空间，具有它自己的几何和它自己的内在对称性，这就产生了动态结构。这些现象在不同层面上的相互作用，可以丰富我们对原始系统的认识。PI自己的研究集中在这种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。这种低维环境特别服从我们的直觉，部分原因是与我们的物理世界直接进行视觉类比，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汇合点，因此在这个领域更全面的理解可以通过类比和直接的数学联系来丰富我们对数学其他部分的方法。该奖项将由拓扑学、几何分析和分析方面的项目共同资助。

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.