Negative and Nonpositive Curvature in Geometry, Topology and Dynamics

F.T.标准时间Farrell和P. Ontaneda计划继续他们在负弯曲流形的动力学、几何学和拓扑学方面的工作。特别是他们计划继续他们的拓扑空间的负弯曲度量的研究，这是一个年轻的领域研究流形的维度大于2。法雷尔和Ontaneda还计划继续他们的研究一般领域的流形拓扑结构及其应用的geometrics。一个黎曼度量的空间是一个规则，规定如何计算角度和长度的路径，和黎曼几何是研究这些对象。一组重要的度量是那些负弯曲的度量，也就是说，每个三角形的角度之和总是小于180度的度量。空间的形状（即其拓扑）限制了它可以支持的度量类型。例如，有些空间不支持负弯曲度规。这种类型的空间的例子是球体和（单孔）甜甜圈的表面。另一方面，两孔甜甜圈确实支持负弯曲度量。对于这种空间，所有可能的负弯曲度量的集合形成了一个新空间的点，该空间的形状（拓扑）是本项目的研究对象。

F.T. Farrell and P. Ontaneda plan to continue their work on the dynamics, geometry and topology of negatively curved manifolds. In particular they plan to continue their study of the topology of the space of negatively curved metrics, which is a young area of study for manifolds whose dimension is greater than two. Farrell and Ontaneda also plan to continue their research on the general area of manifold topology and its applications to geometry.A Riemannian metric on a space is a rule that stipulates how to calculate angles and lengths of paths, and Riemannian Geometry is the study of these objects. An important set of metrics are those that are negatively curved, that is, metrics for which the sum of the angles of every triangle is always less than 180 degrees. The shape of a space (i.e. its Topology) places restrictions on the type of metrics it can support. For instance there are spaces that don't support a negatively curved metric. Examples of this type of space are the sphere and the surface of a (one-holed) doughnut. On the other hand the two-holed doughnut does support a negatively curved metric. For spaces of this sort the collection of all possible negatively curved metrics on it form the points of a new space whose shape (topology) is the object of study for this project.