Topology of Smooth 4-Manifolds

selman AkbulutProposer计划通过将光滑4流形分解成易于理解的Stein块（因为它们是2盘上的Lefschetz纤振）来研究它们的拓扑结构。他想用复流形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。作者希望应用这些技术来解决一些未解决的四维拓扑问题，如寻找四维伪S -协边，以及构造伪S^3 x S^1的问题。拟研究复杂代数几何中的拓扑问题，如实际代数几何中的复共轭问题、分支复盖问题和曲线计数问题。提议者计划通过将四维流形分解成拓扑上更容易理解的部分来研究它们的结构（四维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）。这些较小的碎片带有某些“天然复杂结构”，可以应用于分析技术。通过这些技术，首席研究员希望构建某些相信存在但尚未被发现的4流形。4-流形引起兴趣的原因之一是因为物理学家对10维宇宙的模型，该模型由4维时空和6维复块组成。

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.