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

该合作项目将研究光滑四维流形的拓扑，并结合低维拓扑中众所周知的问题。我们将重点讨论具有辛结构的光滑流形的构造，包括Stein流形和某些接触3流形的辛填充。基于结手术和Luttinger手术的创建具有小欧拉特征的奇异流形技术的最新进展将与规范理论和辛不变量的计算相结合。我们将在这些结构中使用四维把手体技术，其组织原则是寻找“软木塞”和“塞子”作为改变光滑结构的技术。本文将运用规范论和辛几何的技术来研究辛4流形及其对称群的分类。空间和时间的物理世界是一个四维空间，其局部结构被很好地理解，但其大规模（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结构，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，以及这些流形的性质是什么。特别感兴趣的将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的流形粘合在一起来实现流形光滑结构的细微变化。这些变化将通过结合几个学科的专业知识来发现，包括从数学物理的规范理论中衍生出来的强大技术。

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.