Mathematical Sciences: Smooth 4-Manifolds and Their Donaldson Series

9401032 Fintushel这个项目的主要目的是利用规范理论中得到的不变量来研究光滑的单连通4-流形。这有三个部分。首先是理解不变量本身的结构理论。对于简单类型的流形，这是由Kronheimer和Mrowka完成的，并由研究者和R.Stern使用完全不同的技术完成的。研究者和R.Stern还证明了一般4-流形的Donaldson不变量的爆破公式，人们期望这个公式与简单类型情形中使用的技巧一起，将导致一般的结构理论。本项目的第二部分是Donaldson不变量的计算。这位调查员打算使用他的“理性排污”技术来提供计算。这包括给出有理排污的Donaldson不变量的一般公式，这基本上是由调查员和R.Stern(一些细节仍有待检查)完成的，这导致了例如，所有椭圆表面的完全Donaldson不变量的计算和许多其他有趣的例子。项目的最后部分是产生新的例子，这些不变量可以应用于这些例子，以便在分类理论中找到一些顺序。四维流形是指局部具有4维欧几里得空间(例如，3维空间和1维时间维)的空间。这些流形在拓扑学中扮演着中心角色，因为在4维中，强大的高维外科手术技术和更特殊、更复杂的低维手术技术都被分解了。当然，这也是物理上最相关的情况，在理论物理中有大量关于四维拓扑的文献。多年来，人们从高维和低维两个方面对四维流形进行了研究，结果大多不令人满意。1984年，西蒙·唐纳森通过杨-米尔斯方程引入了一种强大的技术，该技术源于四维几何和理论物理之间的相互作用。唐纳森的技术已经产生了令人震惊的结果，一批拓扑学家和几何学家已经形成，希望使用这些技术来完全理解4维拓扑。(这与帮助催生该理论的物理学并没有太远的距离，这一点从威滕等物理学家的许多贡献中得到了证明。)特别是在过去的一年，进展非常迅速。研究者希望对理解Donaldson不变量和对4维流形的分类作出进一步的贡献。***

9401032 Fintushel The main concern of this project is to study smooth simply connected 4-manifolds by using invariants obtained from gauge theory. This has three parts. The first is to understand the structure theory of the invariants themselves. For manifolds of simple type, this has been accomplished by Kronheimer and Mrowka and, using completely different techniques, by the investigator and R. Stern. The investigator and R. Stern have also proved a blowup formula for the Donaldson invariant of a general 4-manifold, and one expects that this formula, together with the techniques used in the simple-type case, will lead to a general structure theory. The second part of this project is the calculation of Donaldson invariants. The investigator intends to use his technique of 'rational blowdowns' to provide calculations. This involves giving a general formula for the Donaldson invariant of a rational blowdown, which has essentially been done by the investigator and R. Stern (some details remain to be checked) and which leads, e.g., to the calculation of complete Donaldson invariants for all elliptic surfaces and many other interesting examples. The final part of the project is to produce new examples to which these invariants can be applied in order to find some order in the classification theory. Four-dimensional manifolds are spaces which locally have the structure of 4-dimensional Euclidean space (e.g. 3 'spatial' and one 'time' dimension). These manifolds play a central role in topology, for in 4 dimensions both the powerful techniques of higher dimensional surgery and the more special and intricate techniques of lower dimensions break down. This is, of course, also the most physically relevant situation, and there is a vast literature in theoretical physics related to 4-dimensional topology. For many years 4-manifolds were studied by applying techniques from both higher and lower dimensions, and the results were mostly unsatisfactory. In 1 984, Simon Donaldson introduced a powerful technique stemming from the interplay between 4-dimensional geometry and theoretical physics via the Yang-Mills equation. Donaldson's techniques have had startling consequences, and a school of topologists and geometers has formed in the hope of using these techniques to understand 4-dimensional topology completely. (That this is not too far removed from the physics which helped spawn the theory is evidenced by the many contributions by physicists such as Witten.) In the last year, especially, progress has been very rapid. The investigator hopes to make further contributions both to the understanding of the Donaldson invariant and to the classification of 4-dimensional manifolds. ***