FRG: Collaborative Research in Gauge Theory

理解我们所生活的四维宇宙的结构是现代数学和物理学研究的一个关键课题。规范理论是研究为物理世界的标准模型提供背景的数学结构的重要工具。该项目支持的研究将开发新的数学工具和理论，这将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被称为弗洛尔理论的三维理论和四维空间不变量的研究之间的相互作用。此外，该项目还为研究生提供研究培训机会。该项目将开发和扩展不变量，包括阿贝尔规范理论模型，从莫尔斯样函数的临界值产生的协边不变量，如Daemi的Gamma不变量，等变扩展，如等变奇异瞬子结同源，以及新的参数化不变量，联合收割机Konno复形与同伦不变量的家庭。它还将把一些规范理论包中发现的结构转换到其他包中，例如将Heegaard Floer理论中的L空间概念转换为瞬子情况下的类似物，或者将Seiberg-Witten Floer理论中的等变结构转换为Heegaard Floer理论。拓扑应用将包括新的见解结一致性，同源协边，4-流形的同构群，和SU（2）表示的3-流形groups.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.