The Topology of Smooth 4-Manifolds

这个建议涉及低维拓扑和辛几何之间的相互作用。本文的第一个主题是研究小单连通四维流形上的奇异光滑结构。主要研究者及其合作者B。道格·帕克（Doug Park）证明了在二点、三点和四点处爆炸的复射影平面允许奇异的不可约光滑结构。PI将继续他的研究，目的是进一步理解小四维流形上的光滑结构，如CP^2，S^2 x S^2和CP^2#（-CP^2）。最初的工作表明，有可能在这些剩余的小四维流形上构造一个奇异的光滑结构。本论文的研究也将集中在光滑不可约单连通四维流形的地理问题和非零签名曲面上曲面丛的构造。这个项目可能会导致新的定理自旋和非自旋辛地理，并提供新的建设有关攻击辛Bogomolov-Miyaoka-Yau猜想。最后一个研究主题是构造Stein和强辛填充的切触三维流形，研究四维小流形上的“奇异”光滑结构，即时空上局部建模的几何对象。虽然我们知道许多光滑的四维流形都具有奇异的光滑结构，但是如果流形很小，这种结构就很难构造。著名的光滑四维庞加莱猜想说明了这一现象。PI最近开发了一种新的非常有效的技术，可以处理这些小的四维流形。PI的方法在理解光滑四维流形的分类方面显示出很大的希望。这个项目使用的思想和工具，从几个数学领域，如几何拓扑，辛几何，复代数几何，群论和规范理论。这个研究项目所涉及的问题在物理学上也有有趣的应用，例如镜像对称和弦理论。

This proposal concerns the interaction between low-dimensional topology and symplectic geometry. The first main theme of the proposed research is to study exotic smooth structures on small simply connected four-manifolds. The Principal Investigator and his collaborator B. Doug Park have shown that the complex projective plane blown up at two, three, and four points admit exotic irreducible smooth structures. The PI will continue his study with an aim of further understanding of smooth structures on small four-manifolds such as CP^2, S^2 x S^2, and CP^2#(-CP^2). The initial work suggests that it would be possible to construct an exotic smooth structure on some of these remaining small four-manifolds. The proposed research will also focus on the geography problem for smooth irreducible simply connected four-manifolds and the construction of surface bundles over surfaces with non-zero signature. This project may lead to new theorems on spin and non-spin symplectic geography, and provide new constructions relevant to attack the Symplectic Bogomolov-Miyaoka-Yau conjecture. A final theme in the proposed research is the construction of Stein and strong symplectic fillings of certain contact three-manifolds.This proposal studies the "exotic" smooth structures on small four-dimensional manifolds, i.e. the geometric objects which are locally modeled on space-time. Although it is known that many smooth four-dimensional manifolds admit the "exotic" smooth structures, such structures are very hard to construct if the manifold is small. The famous smooth four-dimensional Poincare conjecture illustrates this phenomenon. The PI recently developed a new and very effective technique that allows to tackle these small four-dimensional manifolds. The PI's approach shows a great promise in understanding the classification of smooth four-dimensional manifolds. This project uses the ideas and tools from several fields of mathematics, such as geometric topology, symplectic geometry, complex algebraic geometry, group theory and gauge theory. The problems involved in this research project also have interesting applications to physics, such as mirror symmetry and string theory.