Mathematical Sciences: Topology and Manifolds

数学科学：拓扑与流形

• 批准号: 9103033
• 负责人: Dusa McDuff
• 金额: $ 23.51万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9103033&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Manifolds

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.

McDuff教授最近将她的注意力集中在四维流形上，使用和发展了由Gromov首先介绍的技术。她证明了一大类极小辛4流形（包含“漂亮的”辛内嵌2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高属辛嵌入曲面的流形，或者看看她是否能找到一个在同一上同类中支持不同辛和Kahler结构的4流形。辛4流形类介于卡勒曲面类和光滑4流形类之间，如果辛4流形表现得足够好，我们就可以利用它们结构上的信息来获得关于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。他们最近的一个结果表明，群环ZG的手术l群和有理代数k群可以由群环ZG‘的l群和有理k群用一种简单的方法计算出来，其中G’是G中的任意虚循环子群G'，只要G是虚连通李群的协紧离散子群。在接下来的三年里，他们希望在更一般的g群中证明同样的结果。辛流形出现在力学研究中，无论是经典力学还是量子力学。李群在力学中也很普遍。在这个层次上，数学和理论物理似乎并不是完全不同的学科。