Mathematical Sciences: Manifolds and Algebraic Topology

该项目涉及四名高级教员，一名初级教员和几名高级研究生进行各种各样的拓扑研究。William P. Thurston将继续研究低维流形、计算几何和几何群论。例如，他将自动群体定义为，在某种意义上，可以非常精确地，由有限状态自动机处理的群体。此外，他还证明了任何负弯曲流形的基本群都是自动的。其他几何上有趣的群体也符合这种观点，计算机算法可能会被编写出来利用这种情况。约翰·n·马瑟和维克多·j·唐纳以及瑟斯顿将讨论动力学和叶论中的问题。Donnay发现了第一个在二维球体上的遍历测地线流的例子，该球体配备了适当选择的度量。人们需要一段时间来理解这种情况的后果。William Browder和Wu-chung Hsiang在微分拓扑与变换群理论、同伦理论、代数变异与拓扑等方面进行了广泛的研究。xiang的工作也延伸到微分几何。

This project involves four senior faculty members, one junior faculty member, and several advanced graduate students in a wide variety of topological investigations. William P. Thurston will continue research on low-dimensional manifolds, computational geometry and geometric group theory. For example, he has defined an automatic group as one that can, in a sense that can be made quite precise, be treated by a finite state automaton. Moreover, he has shown that the fundamental group of any negatively curved manifold is automatic. Other geometrically interesting groups also fit this viewpoint, and computer algorithms may be written to exploit the situation. John N. Mather and Victor J. Donnay, as well as Thurston, will attack problems in dynamics and the theory of foliations. Donnay has found the first example of an ergodic geodesic flow on a two-dimensional sphere equipped with an appropriately chosen metric. The ramifications of this will take some time to understand. William Browder and Wu-chung Hsiang are engaged in wide- ranging research in differential topology and transformation group theory, homotopy theory, and algebraic varieties and topology. Hsiang's work also extends into differential geometry.