FRG: Collaborative Research: The topology and invariants of smooth 4-manifolds

这个合作项目将结合低维拓扑中的众所周知的问题来研究光滑4维流形的拓扑。我们将集中于构造具有辛结构的新的光滑流形，包括Stein流形和某些接触3-流形的辛填充。基于纽结手术和Luttinger手术创建具有小欧拉特征的奇异流形的技术的最新进展将与规范理论和辛不变量的计算相结合。我们将在这些建筑中使用4维车把技术，组织原则是寻找“软木塞”和“塞子”作为一种改变光滑结构的技术。用规范理论和辛几何的方法来研究辛四维流形及其对称群的分类。时空的物理世界是一个四维空间，它的局部结构是众所周知的，但其大尺度(或拓扑)性质仍然是神秘的。这个重点研究小组将探索4维空间的全局拓扑，目的是了解哪些类型的空间(称为4维流形)可以作为数学对象存在，以及这种流形的性质是什么。特别感兴趣的是辛结构的存在和唯一性问题，以及确定给定流形的对称性的问题。该小组将研究如何通过将不同歧管的碎片粘合在一起来实现流形光滑结构的细微变化。这样的变化将通过结合几个学科的专业知识来检测，包括源自数学物理规范理论的强大技术。

This collaborative project will study the topology of smooth 4-dimensional manifolds, in connection with well-known problems in low-dimensional topology. We will focus on the construction of new smooth manifolds with symplectic structures, including Stein manifolds and symplectic fillings of certain contact 3-manifolds. Recent advances in techniques based on knot surgery and Luttinger surgery for creating exotic manifolds with small Euler characteristic will be coupled with computations of gauge-theoretic and symplectic invariants. We will make use of 4-dimensional handlebody techniques in these constructions, with an organizing principle being the search for 'corks' and 'plugs' as a technique for changing the smooth structure. Techniques of gauge theory and symplectic geometry will be used to investigate the classification of symplectic 4-manifolds and their symmetry groups.The physical world of space and time is a 4-dimensional space whose local structure is well understood but whose large-scale (or topological) properties remain mysterious. This Focused Research Group will explore the global topology of 4-dimensional spaces, with a goal of understanding what kinds of spaces (called 4-dimensional manifolds) can exist as mathematical objects, and what the properties of such manifolds are. Of particular interest will be the problem of existence and uniqueness of symplectic structures, as well as that of determining the symmetries of a given manifold. The group will investigate how subtle changes in the smooth structure of a manifold can be achieved by gluing together pieces of different manifolds. Such changes will be detected by combining expertise from several disciplines, including powerful techniques derived from gauge theories of mathematical physics.