Mathematical Sciences: Topics in Dynamical Systems and Differential Geometry

这一奖项将支持三名研究人员在几个象征性、流畅和可测量的动力学领域的研究活动。符号动力学的主题包括移位的自同构、非负矩阵的反问题和一般的结构问题。在光滑动力学中，一些主题是部分双曲动力系统的遍历性和拓扑传递性，测地线流，以及非正曲率的奇异空间。在可测动力学中需要研究的课题包括限制轨道等价、非奇异和合理连接以及可测系统的自然光滑模型的构造。这个研究小组将研究动力系统。动力系统的概念是一个非常普遍的概念。我们的想法是，这个系统必须以某种方式及时演变。其目的是学习有关动力学的一般性知识，以期对这些系统的大家族进行分类。所考虑的动力系统经常在很长的时间段内表现出混乱的行为，即使它们是完全可预测的。然而，理解如何在这种情况下提取知识是当今科学面临的最重要的问题之一。这是动力系统研究的终极目标之一。

This award will support the research activities of three investigators in several areas of symbolic, smooth, and measurable dynamics. Among the topics in symbolic dynamics are automorphisms of the shift, inverse problems for non-negative matrices, and general structural questions. In smooth dynamics some of the topics are ergodicity and topological transitivity of partially hyperbolic dynamical systems, geodesic flows, and singular spaces of non-positive curvature. Among the topics to be studied in measurable dynamics are restricted orbit equivalence, non-singular and rational joinings, and the construction of natural smooth models of measurable systems. This research group will study dynamical systems. The notion of a dynamical system is a very general one. The idea is just that the system must evolve in time in some way. The object is to learn things of a general nature about the dynamics in the hope of classifying large families of these systems. The dynamical systems under consideration often behave chaotically over large time periods even when they are completely predictable. To understand how nevertheless to extract knowledge in this case is one of the most important problems facing science today. This is one of the ultimate goals in the study of dynamical systems.