The Structure of Smooth 4-Manifolds

尽管在定义简单连通光滑和辛四维流形的不变量方面取得了惊人的进展，并且发现了这些流形的重要定性特征，但我们似乎正在放弃对简单连通光滑或辛四维流形进行分类的任何希望。这门课有大量的例子，展示了各种各样的不同现象。然而，正是这种丰富性使我们几乎没有希望，甚至猜想一个分类方案。最成功的尝试是将每个四维流形的解空间与粒子物理学中出现的复杂方程组相关联：杨-米尔斯方程和塞伯格和维滕的牛顿方程。这些解空间在区分巧妙构造的四维流形时是有用的。该提案以提供进一步探索潜在分类方案的可接近问题的方式汇集了所有已知技术。第一个目标是理解为什么4-流形上的光滑结构对局部拓扑变化敏感。第二步是将局部拓扑组装成新的不变量，这些不变量将区分那些已经由主要研究者构建的具有相同Seiberg-Witten不变量的流形。这个项目的核心是确定一个与同胚光滑四维流形相关的操作序列。令人兴奋的是，所有力中最小的引力实际上可能与自然界的其他三种基本力一样强：将质子和中子结合在原子核中的强力;控制放射性衰变的弱力;以及控制电和磁的力。这三种力与引力之间的不匹配造成了理论上的噩梦;这就是我们尚未找到大统一理论的主要原因。然而，最近有人假设这种弱点是海市蜃楼;引力之所以看起来很弱，是因为它的力量在我们自己的宇宙中被稀释了，大部分引力辐射到额外的维度中。所有其他力仍然被困在我们的三维世界中，而重力可以自由地漫游其他维度。根据这一假设，可能存在与我们平行的其他世界;它们整齐地堆叠在一起，彼此互不相关，引力是它们之间唯一的作用力。 新的数学将在这个项目中产生，以进一步探索这些想法。许多相关的数学已经揭示了三维和四维的特殊性质。从一个宇宙到另一个宇宙的通道可以用一种局部变化来解释，这种局部变化被称为零同调环面上的对数变换。此操作捕获了虫洞的许多特征。这是这个项目的目的是探讨猜想，任何两个同胚光滑4-流形是相关的序列，这样的转换。在底部，这个项目的目标是开发更系统的光滑四维流形的结构，希望一个一般的图片开始出现，至少会建议一个分类方案。

Despite spectacular advances in defining invariants for simply-connected smooth and symplectic 4-dimensional manifolds and the discovery of important qualitative features about these manifolds, we seem to be retreating from any hope to classify simply-connectedsmooth or symplectic 4-dimensional manifolds. The subject is rich in examples that demonstrate a wide variety of disparate phenomena. Yet it is precisely this richness which gives us little hope to even conjecture a classification scheme. The most successful attempts have associated to each 4-dimensional manifold the solution space to complex systems of equations that arise in particle physics: the Yang-Mills equations and the monopole equations of Seiberg and Witten. These solution spaces are useful in distinguishing cunningly constructed 4-dimensional manifolds. This proposal assembles all known techniques in a manner that provides approachable questions that further explore a potential classification scheme. The first goal is to understand why smooth structures on 4-manifolds are sensitive to local topological change. The second step is to assemble the local topology into new invariants that will distinguish those manifolds with the same Seiberg-Witten invariants already constructed by the principal investigator. The core of this project is to determine a sequence of operations that relate homeomorphic smooth 4-manifolds.Excitement has been generated by the idea that the puniest of all forces, gravity, may in fact be a strong as nature's other three fundamental forces: the strong force which binds protons and neutrons together in atomic nuclei; the weak force which governs radioactive decay; and the forces that govern electricity and magnetism. The perceived mismatch between these three forces and gravity creates a theoretical nightmare; it's the principle reason we have yet to find a grand unified theory. However, it has recently been hypothesized that this weakness is a mirage; the force of gravity only appears weak because its force is diluted in our own universe and most of gravity's force radiates out into extra dimensions. All other forces remain trapped in our 3-dimensional world, while gravity is free to roam other dimensions. With this hypothesis, there could be other worlds that are parallel to our own; they all neatly stack up, each oblivious of the other, with gravity the only force that moves between them. New mathematics will be generated in this project to further explore these ideas. Much of the relevant mathematics has already exposed the special nature of dimensions three and four. The passage from one universe to another could be explained by a local change known as a logarithmic transform on a null-homologous torus. This operation captures many of the features of worm holes. It is the purpose of this project to explore the conjecture that any two homeomorphic smooth 4-manifolds are related by a sequence of such transformations. At bottom, the goal of this project is to develop more systematic constructions of smooth 4-dimensional manifolds with the hope that a general picture begins to emerge that will at least suggest a classification scheme.