Mathematical Sciences: Topology and Manifolds

麦克达夫教授最近把注意力集中在 4-维流形，首先使用和发展技术 由Gromov介绍。 她已经证明，一个大类的 极小辛4-流形（包含“nice” 辛嵌入2-球）携带唯一辛 每个上同调类中的结构。 她打算继续 这个工作：例如，看看她是否可以扩展上述结果 到包含一个辛嵌入曲面的流形 更高的属，或者看看她是否能找到一个4流形， 支持不同的辛和Kahler结构在同一个 上同调类 辛4-流形的类 在Kahler曲面类和光滑4- 流形，如果辛4-流形 行为足够好，人们可能可以使用有关 他们的结构，以获得信息类的4-流形 本身 琼斯教授将继续与F. T. 法雷尔 他们最近的一项研究结果表明， 外科L-群和有理代数K-群 群环ZG可以由L-群以简单的方式计算 群环ZG '的有理K-群，其中G'是任意的 若G是余紧的，则G中的虚循环子群G' 虚连通李群的离散子群。 在 在接下来的三年里，他们希望更多地证明这一结果。 一般群G. 辛流形出现在力学研究中， 经典的和量子的 李群也普遍存在于 力学 似乎数学和理论 物理学在这个层次上并不是完全独立的学科。

Professor McDuff has recently concentrated her attention on 4-dimensional manifolds, using and developing techniques first introduced by Gromov. She has shown that a large class of minimal symplectic 4-manifolds (those containing "nice" symplectically embedded 2-spheres) carry a unique symplectic structure in each cohomology class. She intends to continue this work: for example, to see if she can extend the above result to manifolds which contain a symplectically embedded surface of higher genus, or to see if she can find a 4-manifold which supports different symplectic and Kahler structures in the same cohomology class. The class of symplectic 4-manifolds lies between the class of Kahler surfaces and the class of smooth 4- manifolds, and, if it turns out that symplectic 4-manifolds behave nicely enough, one might be able to use information on their structure to obtain information on the class of 4-manifolds itself. Professor Jones will continue his joint research with F. T. Farrell. One of their most recent results states that the surgery L-groups and the rational algebraic K-groups for the group ring ZG can be computed in a simple way from the L-groups and rational K-groups of group rings ZG', where G' is any virtually cyclic subgroup G' in G provided G is a co-compact discrete subgroup of a virtually connected Lie group. In the next three years they hope to prove this same result for more general groups G. Symplectic manifolds arise in the study of mechanics, both classical and quantum. Lie groups are also ubiquitous in mechanics. It would seem that mathematics and theoretical physics are not entirely distinct subjects at this level.