The Structure of Smooth 4-Manifolds

DMS-0204041罗纳德·斯特恩在过去的25年里，对光滑的4维流形进行分类一直是令人兴奋的深奥数学的目标。在这个问题上已经投入了大量的技术；它是数十个研究小组的焦点。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。这次突袭的结果是，4维流形比我们预期的要复杂得多。因此，不可能预测分类方案。本项目的目标是更系统地探讨分类方案的存在和唯一性框架。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很敏感。这个项目描述了光滑结构在已知结构中如何变化的基础。零同调环面上的对数变换被证明起着重要的作用。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由零同调环面上的一系列对数变换相关。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。给出了一般类型光滑4-流形的概念，并对其Seiberg-Witten不变量提出了一个猜想限制。这个项目概述了新的构造，表明存在填充由这些限制确定的区域的通用型流形。对这些建筑的仔细调查应该会指出为什么它们是最可能的。关于不可约的单连通光滑4-流形的特征类的潜在限制的其他问题将被研究。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个基本力一样强：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。这也解释了我们宇宙中缺失的暗物质；它实际上存在于其他平行宇宙中。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。这些平行宇宙可以用(奇异)叶层理论来解释。对奇异叶的拟议研究可能会构建我们看待自己宇宙的方式--我们如何与可能的平行宇宙堆积在一起。这些奇异的叶状结构也将为4维流形的分类提供新的见解。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。

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.