Mathematical Sciences: Manifolds and Algebraic Topology

这个项目涉及四名高级教员、一名初级教员和几名高级研究生，他们参与了各种不同的拓扑调查。威廉·P·瑟斯顿将继续研究低维流形、计算几何和几何群论。例如，他将自动机定义为在某种意义上可以被有限状态自动机处理的自动机。此外，他还证明了任何负曲线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一观点，可能会编写计算机算法来利用这种情况。约翰·N·马瑟和维克多·J·唐奈以及瑟斯顿将探讨动力学和叶层理论中的问题。唐奈在配备了适当选择的度规的二维球体上发现了第一个遍历测地线流的例子。这其中的后果还需要一些时间才能理解。William Browder和吴忠祥在微分拓扑和变换群理论、同伦理论、代数簇和拓扑学方面都有广泛的研究。向先生的工作也延伸到了微分几何。

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.