The Structure of Smooth 4-Manifolds

尽管在定义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单连通光滑或辛四维流形进行分类的任何希望。这门学科有丰富的例子，展示了各种不同的现象。然而，正是这种丰富性使我们几乎没有希望推测出一种分类方案。最成功的尝试是将粒子物理学中出现的复杂方程组的解空间与每个四维流形联系起来：杨-米尔斯方程和塞伯格和威滕的单极子方程。这些解空间对于区分巧妙构造的四维流形很有用。该建议以一种提供可接近的问题的方式组装所有已知技术，以进一步探索潜在的分类方案。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合成新的不变量，将这些流形与首席研究员已经构造的Seiberg-Witten不变量区分开来。本课题的核心是确定与同胚光滑4流形相关的一系列运算。令人兴奋的是，所有力中最微小的引力，实际上可能与自然界其他三种基本力一样强大：将质子和中子束缚在原子核中的强作用力；控制放射性衰变的弱力；以及控制电和磁的力。这三种力量和重力之间的不匹配在理论上造成了一场噩梦；这就是我们还没有找到大统一理论的主要原因。然而，最近有一种假设认为，这种弱点是海市蜃楼；引力之所以显得微弱，只是因为它在我们自己的宇宙中被稀释了，而且大部分引力都辐射到了额外的维度。所有其他的力都被困在我们的三维世界里，而重力可以自由地在其他维度上漫游。根据这个假设，可能存在与我们的世界平行的其他世界；它们整齐地堆叠在一起，彼此都忽略了，重力是它们之间唯一的作用力。这个项目将产生新的数学来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过渡可以用称为零同源环面上的对数变换的局部变化来解释。该操作捕捉到了虫孔的许多特征。本课题的目的是探讨任意两个同胚光滑4流形由一系列这样的变换相联系的猜想。从根本上说，这个项目的目标是开发光滑四维流形的更系统的结构，希望一个总体的图景开始浮现，至少会提出一个分类方案。

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.