Mathematical Sciences: Manifolds and Algebraic Topology

这项计画有四位资深拓扑学家(William Browder、向武忠、John N.Mather和William P.瑟斯顿)、一位博士后助理(吴朗芳)和三位高级研究生参与。他们的研究主题相当多种多样。Browder将研究某些拓扑群在有限维空间上的自由行为。他还将学习极小曲面的不动点定理和正则性。向将致力于一些与割圆迹映射有关的问题，包括L理论中臭名昭著的诺维科夫猜想以及一些更容易处理的猜想。Mather的研究涉及无限圆柱的保面积微分同态。关于微分同胚下的不变曲线及其旋转数，有几个难猜想。瑟斯顿将致力于他著名的三维流形的几何化猜想。此外，他还将致力于自动分组、自相似平铺、一维真实和复杂动力系统以及其他主题。吴的研究涉及流形上的Ricci流方程和orborold上的Ricci流方程。这些都是拓扑学研究的主流。让研究生参与他们的学习是合适的。例如，瑟斯顿的几何化猜想是出了名的难解问题。它包含了可追溯到世纪之交的庞加莱猜想。然而，十多年来，它的碎片一直是瑟斯顿的学生撰写的无数博士论文的素材。

This project involves four senior research topologists (William Browder, Wu-chung Hsiang, John N. Mather, and William P. Thurston), one postdoctoral associate (Lang-Fang Wu), and three advanced graduate students. Their research topics are quite various. Browder will study free actions of certain topological groups on finite-dimensional spaces. He will also study fixed point theorems and regularity of minimal surfaces. Hsiang will work on a number of problems related to the cyclotomic trace map, including the notorious Novikov conjecture in L-theory as well as some more tractable ones. Mather's research deals with area-preserving diffeomorphisms of infinite cylinders. There are several hard conjectures about invariant curves under diffeomorphisms and their rotation numbers. Thurston will work toward his famous geometrization conjecture for 3-manifolds. In addition, he will work on automatic groups, self-similar tilings, one-dimensional real and complex dynamical systems, and other topics. Wu's research concerns the Ricci Flow equations on manifolds and orbifolds. All these topics are in the mainstream of research in topology. It is appropriate to involve graduate students in their study. For example, the geometrization conjecture of Thurston is a notoriously difficult problem. It subsumes the Poincare Conjecture, which dates to the turn of the century. Nevertheless, pieces of it have been the stuff of numerous Ph.D theses by Thurston's students for more than a decade.