The Structure of Smooth 4-Manifolds

在过去的25年里，对光滑的四维流形进行分类一直是令人兴奋的深奥数学的目标。为了解决这个问题，已经投入了大量的技术；这是几十个研究小组关注的焦点。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。这种攻击的结果是，四维流形比我们想象的要复杂得多。因此，预测分类方案是不可能的。本课题的目标是更系统地探讨分类方案的存在性和唯一性框架。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。该项目描述了光滑结构如何在已知结构中变化的基础。证明了零同源环面上的对数变换具有重要的作用。第一步是确定固定同胚型单连通光滑4流形上的两个光滑结构是否通过零同构环面上的一系列对数变换相关联。第二步是确定不可约光滑4流形的特征数，并确定这些拓扑不变量如何影响它们的Seiberg-Witten不变量。给出了一般型光滑4流形的概念，并给出了它们的Seiberg-Witten不变量的一个推测约束。这个项目概述了新的结构，表明有一般类型的流形填充由这些限制决定的区域。仔细研究一下这些结构，就会知道为什么它们是最好的。其他与不可约单连通光滑4流形特征类的潜在限制有关的问题将被研究。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。这也解释了宇宙中暗物质的缺失；它实际上存在于其他平行宇宙中。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。这些平行宇宙可以用（奇异）叶理理论来解释。对奇异叶的研究可能会构建我们看待我们自己的宇宙的方式——我们如何与可能的平行宇宙相叠加。这些奇异叶状也将为四维流形的分类提供新的见解。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。

DMS-0204041Ronald SternFor the last 25 years it has been the goal of exciting and deep mathematics to classify smooth 4-dimensional manifolds. An arsenal of techniques has been thrown at this problem; it is the focus of dozens of research groups. 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. The result of this assault is that 4-dimensional manifolds are more complicated than we ever expected. As a result, it is impossible to predict a classification scheme. It is the goal of this project to more systematically approach the existence and uniqueness framework for a classification scheme. The first goal is to understand why smooth structures on 4-manifolds are sensitive to local topological change. This project describes the underpinnings of how the smooth structures change in the known constructions. Log transformations on null-homologous tori are shown to play a significant role. The first step is to determine if two smooth structures on a fixed homeomorphism type of simply-connected smooth 4-manifold are related by a sequence of log transformations on null-homologous tori. The second step is to determine the characteristic numbers of irreducible smooth 4-manifolds and to determine how these topological invariants affect their Seiberg-Witten invariants. A notion of general-type smooth 4-manifolds is given and a conjectured restriction on their Seiberg-Witten invariants is proposed. This project outlines new constructions that show that there are general-type manifolds that fill out the regions determined by these restrictions. Careful investigation of these constructions should indicate why they are best possible. Other questions related to potential restrictions on the characteristic classes of irreducible simply-connected smooth 4-manifolds will be investigated.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. This would also account for the missing dark matter of our universe; it actually resides in other parallel universes. 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. These parallel universes may be explained by the theory of (singular) foliations. The proposed study of singular foliations may structure the way in which we view our own universe -how we stack up with possible parallel universes. These singular foliations will also provide new insight into the classification of 4- dimensional manifolds. 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.