The Topology of Smooth 4-Manifolds

这一建议涉及低维拓扑和辛几何之间的相互作用。本研究的第一个主要主题是研究小的单连通四维流形上的奇异光滑结构。首席研究员和他的合作者B.Doug Park已经证明，在两点、三点和四点爆炸的复杂射影平面允许奇异的不可约光滑结构。PI将继续他的研究，以进一步了解小四流形上的光滑结构，如CP^2，S^2 x S^2，以及CP^2#(-CP^2)。最初的工作表明，在剩余的一些小的四维流形上构建一种奇特的光滑结构是可能的。本文还将研究光滑不可约单连通四维流形的地理学问题，以及具有非零特征的曲面上曲面丛的构造问题。这个项目可能导致关于自旋和非自旋辛地理的新定理，并提供与攻击辛Bogomolov-Miyaoka-Yau猜想相关的新构造。这项研究的最后一个主题是构造某些接触三维流形的Stein和强辛填充。这一建议研究了小的四维流形上的“奇异”光滑结构，即局部模拟在时空上的几何对象。虽然众所周知，许多光滑的四维流形都承认这种“奇异”的光滑结构，但如果流形很小，这种结构就很难构造。著名的光滑四维庞加莱猜想就说明了这一现象。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。这个项目使用了几个数学领域的思想和工具，如几何拓扑学、辛几何、复代数几何、群论和规范理论。这项研究项目涉及的问题在物理学中也有有趣的应用，如镜像对称性和弦理论。

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.