Mathematical Sciences: Topology and Manifolds

McDuff 教授最近将注意力集中在 4 维流形上，使用和开发了 Gromov 首次引入的技术。 她证明了一大类最小辛 4-流形（那些包含“良好”辛嵌入 2-球体的流形）在每个上同调类中都具有独特的辛结构。 她打算继续这项工作：例如，看看她是否可以将上述结果扩展到包含更高属的辛嵌入表面的流形，或者看看她是否可以找到一个在同一上同调类中支持不同辛和卡勒结构的4流形。 辛 4-流形类位于卡勒曲面类和光滑 4-流形类之间，并且，如果事实证明辛 4-流形表现得足够好，那么人们也许能够使用有关其结构的信息来获取有关 4-流形类本身的信息。 琼斯教授将继续与 F. T. Farrell 的联合研究。 他们的最新结果之一指出，群环 ZG 的手术 L 群和有理代数 K 群可以通过群环 ZG' 的 L 群和有理 K 群以简单的方式计算，其中 G' 是 G 中的任何虚拟循环子群 G'，前提是 G 是虚拟连通李群的协紧离散子群。 在接下来的三年里，他们希望为更一般的群 G 证明同样的结果。辛流形出现在经典力学和量子力学的研究中。 李群在力学中也普遍存在。 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.