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，并发现了关于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，展示了各种各样不同的现象。然而，正是这种丰富性让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中出现的复杂方程组的解空间与每个4维流形相关联：杨-米尔斯方程以及塞贝格和维腾的单极方程。这些解空间对于区分巧妙构造的4维流形很有用。这一建议汇集了所有已知技术，提供了可进一步探索潜在分类方案的可接近的问题。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很敏感。第二步是将局部拓扑组装成新的不变量，以区分那些具有主要研究者已经构造的相同Seiberg-Witten不变量的流形。这个项目的核心是确定与同胚光滑4-流形相关的一系列操作。激动人心的想法是，所有力中最小的引力实际上可能与自然的其他三种基本力一样强大：在原子核中将质子和中子结合在一起的强力；支配放射性衰变的弱力；以及支配电和磁的力。这三种力量和地心引力之间的感知不匹配造成了一场理论噩梦；这是我们尚未找到一个大统一理论的主要原因。然而，最近有人假设，这种虚弱是一种海市蜃楼；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，而且大部分引力都辐射到了额外的维度。所有其他力量仍然被困在我们的三维世界中，而重力可以自由地在其他维度漫游。有了这个假设，可能会有其他世界与我们的世界平行；它们都整齐地堆积在一起，彼此都忽略了对方，引力是在它们之间移动的唯一力量。在这个项目中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程可以用零同调环面上的对数变换来解释。此操作捕获了蠕虫孔的许多特征。这个项目的目的是探索这样一个猜想：任何两个同胚光滑4-流形都由一系列这样的变换联系在一起。归根结底，这个项目的目标是开发光滑的4维流形的更系统的结构，希望开始出现一个总体情况，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.