Topology of Smooth 4-Manifolds

DMS-0204579 Selman AkbulutProposer计划通过将光滑4-流形分解为易于理解的Stein片段来研究光滑4-流形的拓扑结构（因为它们是2-圆盘上的莱夫谢茨纤维化）。他想使用复杂和辛流形的技术来理解这种分解对原始流形拓扑结构的限制。Proposer希望将这些技术应用于解决四维拓扑中的一些未解决的问题，例如寻找四维伪s-cobordisms，以及构建伪S^3 x S^1的问题。Proposer还计划研究由复代数几何引起的拓扑问题，例如来自复共轭的问题，分支覆盖和真实的代数几何中的曲线计数问题。Proposer计划通过将四维流形分解为拓扑上更容易理解的片段来研究四维流形的结构（4-流形是局部看起来像我们生活的四维欧几里得时空的空间）。这些较小的碎片具有某些“自然复杂结构”，分析技术适用于这些结构。主要研究者希望通过这些技术来构建某些被认为存在但尚未被发现的四维流形。四维流形感兴趣的原因之一是因为物理学家的10维宇宙模型由4维时空加上6维复杂部分组成。

DMS-0204579Selman AkbulutProposer plans to investigate the topology of smooth 4-manifolds by decomposing them into Stein pieces which are easy to understand (since they turn out to be Lefschetz fibrations over the 2-disk). He would like use the techniques of complex and symplectic manifolds to understand the restriction this decomposition imposes to the topology of the original manifold. Proposer hopes to apply these techniques to attack some unsolved problems of 4-dimensional topology, such as finding 4-dimensional fake s-cobordisms, and to the problem of constructing a fake S^3 x S^1. Proposer also plans to work on the topological problems arising from complex algebraic geometry, such as problems coming from the complex conjugation, branched covers and the curve counting problem in real algebraic geometry.Proposer plans to investigate structure of 4-dimensional manifolds by decomposing them into pieces that are topologically easier to understand (4-manifolds are spaces which locally look like 4-dimensional Euclidean space-time we live in). These smaller pieces carry certain `natural complex structures' which analytical techniques apply. By these techniques Principal investigator hopes to construct certain 4-manifolds believed to exists but not yet have been found. One of the reasons 4-manifolds are of interest is because of the physicists model of the 10-dimensional universe which consists of a 4-dimensional space-time plus 6-dimensional complex piece.