Problems in Differential and Algebraic Topology

“高维”拓扑中的许多问题已经用以下一个或多个理论重新表述：同伦理论；代数k理论;手术L-theory;稳定伪同位素理论。为了更好地理解后三种理论，Jones（与F.T. Farrell合作）为这三种理论中的每一种都制定了一个“同构猜想”。例如，代数k理论的同构猜想给出了用G的所有次环子群的整数群环的代数k群来计算群G的整数群环的代数k群的简单公式（如果一个群是循环群的有限扩展，则它是“次环”）。对次环群的整群环的代数k群有了一定的认识；因此，上述猜想的成立对理解所有整群环的代数k理论有很大的帮助。作为该提案的一部分，Jones打算继续致力于验证G的整群环的代数k理论的同构猜想，条件是G群在一个完全的黎曼流形上，通过等距不连续地作用于处处具有非正截面曲率值的黎曼流形上。数学家通过试图找到每个空间的“遗传密码”来研究空间的结构。空间的“遗传密码”通常以与空间相关的代数小器具的形式出现；通常有一个从空间构造代数小器具的简单方法。希望两个具有相同或相同“遗传密码”的空间作为空间应该是相同或相同的。第一个与空间相关的代数小工具被称为空间的“基本群”。法国数学家亨利·庞加莱在大约100年前提出了这个想法。空间的基本群当然是其“遗传密码”的重要组成部分，但也有许多不同空间具有相同基本群的例子；在这些情况下，基本组并不是空间的完整“遗传密码”。大约50年前，数学家阿曼德·博雷尔（来自新泽西州普林斯顿高等研究院）推测，对于一类重要的空间（称为“非球面空间”），基本群确实给出了该空间的完整“遗传密码”；也就是说，两个具有相同基本群的非球面空间被强制为相同的空间。琼斯在证明博雷尔猜想方面取得了一些进展，并将继续朝这个方向努力。

DMS-0306616Lowell E. JonesMany problems in "higher dimensional" topology have been reformulated in terms of one or more of the following theories: homotopy theory; algebraic K-theory; surgery L-theory; stable pseudoisotopy theory.In an effort to better understand the latter three theories, Jones (in collaboration with F.T. Farrell) has formulated an "Isomorphism Conjecture" for each of these three theories. For example, the Isomorphism Conjecture for algebraic K-theory gives a simple recipe for computing the algebraic K-groups for the integral group ring of a group G in terms of the algebraic K-groups for the integral group rings of all the infracyclic subgroups of G. (A group is "infracyclic" if it is a finite extension of a cyclic group.) Quite a bit is know about the algebraic K-groups for integral group rings of infracyclic groups; so the truth of the preceeding conjecture would help considerably in understanding the algebraic K-theory of all integral group rings. As part of this proposal, Jones intends to continue working towards verifying the Isomorphism Conjecture for algebraic K-theory of the integral group ring of G, under the condition that the group G acts properly, discontinuously by isometries on a complete, Riemannian manifold which has non-positive sectional curvature values everywhere.Mathematicians study the structure of spaces by trying to find the "genetic code" for each space. The "genetic code" for a space usually comes in the form of an algebraic gadget associated to the space; typically there is a simple recipe for constructing the algebraic gadget from the space. The hope is that two spaces having equal or congruent "genetic codes" should be equal or congruent as spaces. One of the first algebraic gadgets to be associated to a space is called the "fundamental group" of the space. The French mathematician Henri Poincare introduced this idea about a 100 years ago. The fundamental group of a space is certainly an important ingredient in its "genetic code", but there are examples of many different spaces which have the same fundamental group; in these cases the fundamental group is not a complete "genetic code" for the space. About 50 years ago the mathematician Armand Borel (of the Institute for Advanced Study in Princeton, N.J.) conjectured that for an important class of spaces (called "aspherical spaces") the fundamental group does give the complete "genetic code" for the space; that is, two aspherical spaces with the same fundamental group are forced to be the same space. Jones has made some progress towards proving Borel's Conjecture, and will continue his work in that direction.