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

FRG：协作研究：光滑4流形的拓扑和不变量

• 批准号: 1065718
• 负责人: Cagri Karakurt
• 金额: $ 2.3万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065718&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; topology; invariants

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.

这个协作项目将研究与低维拓扑中众所周知的问题有关的平滑四维流形的拓扑。 我们将重点关注具有符号结构的新的平滑歧管的构建，包括某些接触3型Manifolds的Stein歧管和符号填充物。 基于打结手术和Luttinger手术的技术的最新进展，以创建具有较小Euler特征的外来歧管，将与仪表理论和符号不变性的计算相结合。 在这些结构中，我们将利用4维手柄技术，组织原理是寻找“软木塞”和“插头”作为改变平滑结构的技术。轨距理论和符号几何形状的技术将用于研究符号4个manifolds及其对称群的分类。时空的物理世界是一个4维空间，其局部结构被充分理解，但其大规模（或拓扑）的特性仍然是神秘的。这个集中的研究小组将探索4维空间的全球拓扑，其目的是了解哪些类型的空间（称为4维流形）可以作为数学对象以及这种歧管的属性存在。特别感兴趣的将是符号结构的存在和独特性的问题，以及确定给定歧管的对称性的问题。该小组将通过将不同歧管的碎片粘合在一起来研究歧管平滑结构的细微变化。通过结合几个学科的专业知识，包括从数学物理学规定得出的强大技术来检测到这种变化。