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

这个合作项目将研究光滑四维流形的拓扑结构，与低维拓扑结构中的著名问题有关。 我们将着重于构造新的具有辛结构的光滑流形，包括Stein流形和某些切触3-流形的辛填充。 最近的技术进展的基础上结手术和Luttinger手术创造异国情调的流形与小欧拉特征将与规范理论和辛不变量的计算。 我们将在这些结构中使用四维的立体技术，组织原则是寻找“软木塞”和“塞子”，作为改变光滑结构的技术。规范理论和辛几何的技巧将被用来研究辛4-流形的分类和它们的对称群。空间和时间的物理世界是一个4维空间，它的局部结构是很好的理解，但它的大尺度（或拓扑）性质仍然是神秘的。这个重点研究小组将探索四维空间的全局拓扑，目的是了解什么样的空间（称为四维流形）可以作为数学对象存在，以及这些流形的属性是什么。特别感兴趣的是辛结构的存在性和唯一性问题，以及确定给定流形的对称性问题。该小组将研究如何通过将不同流形的碎片粘合在一起来实现流形光滑结构的微妙变化。这种变化将通过结合几个学科的专业知识来检测，包括来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.