Topology of Smooth 4-Manifolds

摘要：主要研究光滑4流形的拓扑结构。通过将光滑流形分解成复杂的碎片，他将使用复杂流形和辛流形的技术来理解这种分解对光滑4流形拓扑结构的限制。首席研究员希望将解决“沙尔曼问题”和芬图舍尔-斯特恩构造的技术应用于构造四维伪s坐标的问题，以及构造“三球与圆交叉”的伪副本的问题。他喜欢研究堀川曲面的拓扑，并希望对各种分支覆盖猜想给出肯定的答案。主要研究广义四维空间（4-流形）；他希望能找到新的。这些空间在物理学中自然出现：相对论（时空）和量子场论。最近，这些空间的结构有了一些令人兴奋的进展，这些进展来自于物理学的思想，但我们仍然不知道某些基本的奇异的4流形是否存在。如果它们存在，它们将是已知星系的基本组成部分。我们想研究可能的结构，使它们存在。

Proposal: DMS-9971440Principal Investigator: Selman AkbulutAbstract: The principal investigator will investigate the topology of smooth 4-manifolds. By decomposing smooth manifolds into complex pieces, he will use the techniques of complex and symplectic manifolds to understand the restriction this decomposition imposes on the topology of smooth 4-manifolds. The principal investigator hopes to apply the techniques from the solution of the `Scharlemann problem' and Fintushel-Stern constructions to the problem of constructing 4-dimensional fake s-cobordisms, and to the problem of constructing a fake copy of `three-sphere crossed with circle'. He would like to study topology of Horikawa surfaces and hopes to give affirmative answers to various branched covering conjectures.The principal investigator intends to study generalized 4-dimensional spaces (4-manifolds); he hopes to find new ones. These spaces arise naturally in physics: relativity (space-time) and quantum field theories. Recently there have been exciting developments about the structure of these spaces coming from the ideas of physics, but we still don't know if certain basic exotic 4-manifolds could exist. If they exist they will be basic building blocks of the known ones. We would like to investigate possible constructions which could give their existence.