FRG: Collaborative Research in Gauge Theory

了解我们所处的四维宇宙的结构是现代数学和物理学研究的一个重要课题。规范理论是研究数学结构的关键工具，它为物理世界的标准模型提供了背景。该项目支持的研究将开发新的数学工具和理论，有助于测试这些模型，并促进我们对四维空间的理解。一个关键的数学思想是三维理论（即花理论）与四维空间不变量研究之间的相互作用。此外，该项目还为研究生提供了研究培训机会。该项目将发展和扩展不变量，包括阿贝尔规范理论模型，由Morse函数的临界值引起的协变不变量，如Daemi的Gamma不变量，等变扩展，如等变奇异瞬时结同调，以及结合Konno配合式复形与族的同伦不变量的新参数化不变量。它还将一些规范理论包中发现的结构转化为其他包，例如将l空间概念从Heegaard Floer理论转化为实例情况下的类似物，或将Seiberg-Witten Floer理论中的等变结构转化为Heegaard Floer理论。拓扑应用将包括对结调和、同调协、4流形的微分同构群和3流形群的SU(2)表示的新见解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Gauge theory is a crucial tool for the study of the mathematical structures that provide the context for standard models of the physical world. The research supported by this project will develop new mathematical tools and theories that will help test such models and advance our understanding of four-dimensional spaces. A key mathematical idea is the interaction between three-dimensional theories known as Floer theories and the study of invariants of four-dimensional spaces. In addition the project provides research training opportunities for graduate students.This project will develop and extend invariants including abelian gauge theoretic models, cobordism invariants arising from the critical values of Morse like functions such as Daemi's Gamma invariant, equivariant extensions such as equivariant singular instanton knot homology, and new parameterized invariants that combine Konno adjunction-style complexes with homotopy invariants of families. It will also translate structures uncovered in some gauge theory packages to other packages, for example translating L-space notions from Heegaard Floer theory to analogues in the instanton case, or equivariant structures from the Seiberg-Witten Floer theory to the Heegaard Floer theory. Topological applications will include new insights into knot concordance, homology cobordism, diffeomorphism groups of 4-manifolds, and SU(2) representations of 3-manifold groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.