Mathematical Sciences: Topology and Manifolds

麦克达夫教授最近将她的注意力集中在4维流形上，使用并开发了由格罗莫夫最先引入的技术。她证明了一大类极小辛四维流形(那些包含“好的”辛嵌入的二维球的流形)在每个上同调类中都具有唯一的辛结构。她打算继续这项工作：例如，看看她是否能将上述结果推广到包含更高亏格的辛嵌入曲面的流形上，或者她是否能找到在同一上同调类中支持不同辛和Kahler结构的4-流形。辛4-流形类位于Kahler曲面类和光滑4-流形类之间，如果证明辛4-流形表现得足够好，人们可能能够利用关于其结构的信息来获得关于4-流形类本身的信息。琼斯教授将继续他与法雷尔的联合研究。他们的最新结果之一是，群环ZG‘的L-群和有理K-群可以简单地由群环ZG’的L-群和有理代数K-群计算出来，其中G‘是G中的任意虚循环子群G’，只要G是虚连通李群的上紧离散子群.在接下来的三年里，他们希望对更一般的群G证明同样的结果。辛流形出现在力学的研究中，无论是经典的还是量子的。李群在力学中也是普遍存在的。在这个层面上，数学和理论物理似乎并不是完全不同的学科。

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.