Mathematical Sciences: Geometrical Aspects of Hyperbolic Systems

主要研究人员将研究动力系统和黎曼几何中的问题。他们将特别研究负曲率空间和环面上的测地线流动和测地线拉伸。他们之前的一个结果证明了Hopf猜想的一种特殊情况，即环面上没有共轭点的黎曼度规是平的。他们将尝试把他们的方法从动力系统扩展到一般情况。黎曼几何试图将流形或曲面的整体特性（如拓扑结构）与局部特性（如曲率）联系起来。得到这些关系的标准方法是通过偏微分方程的整体解。主要研究人员采取了不同的策略；他们使用在流形上定义的动力系统理论。这涉及到流形映射迭代的研究，在这种情况下利用了这些映射的群的结构。如果他们成功地发展了该方法，它将应用于黎曼几何的广泛问题。

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.