Mathematical Sciences: Geometrical Aspects of Hyperbolic Systems

主要研究人员将致力于解决动力学问题， 系统和黎曼几何特别是他们将研究 负曲率空间上测地线流和测地线拉伸 在Tori。他们之前的一个结果证明了 一个特殊的情况下，霍普夫猜想，其中规定，一个黎曼 环面上没有共枕点的度量是平坦的。他们将 试图将他们的方法从动力系统扩展到 一般情况下。 黎曼几何试图将 流形或曲面，如局部拓扑结构 像曲率这样的性质。标准方法获得 这些关系一直是通过局部的全球解决方案， 微分方程主要研究人员采取了 不同的策略;他们使用动力系统理论定义在 歧管。这涉及到对 歧管，并且在它们的情况下利用以下结构： 这些映射的组。如果他们成功地开发了 方法，它应该适用于广泛的问题， 黎曼几何

The principal investigators will work on problems in dynamical systems and Riemannian geometry. In particular they will study geodesic flows and geodesic stretch on spaces of negative curvature and on tori. One of their previous results gave a proof of a special case of the Hopf conjecture which states that a Riemannian metric without conjugate points on the torus is flat. They will attempt to extend their methods from dynamical systems to the general case. Riemannian geometry attempts to relate global properties of manifolds or surfaces such as the topological structure to local properties such as curvature. The standard method for obtaining these relationships has been through the global solution of partial differential equations. The principal investigators take a different tack; they use the theory of dynamical systems defined on the manifold. This involves the study of iterations of mappings of the manifold and in their case takes advantage of the structure of groups of these mappings. If they are successful in developing the method, it should have application to a wide range of problems in Riemannian geometry.