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FRG: Collaborative Research in Gauge Theory

FRG：规范理论的合作研究

基本信息

批准号：
1952755
负责人：
David Auckly
金额：
$ 31.29万
依托单位：
Kansas State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952755&HistoricalAwards=false
关键词：
FRG Collaborative Research Gauge Theory

项目摘要

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.

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。规范理论是研究数学结构的重要工具，这些结构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，有助于测试这种模型，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研究之间的相互作用。此外，该项目还将为研究生提供研究培训机会。本项目将开发和扩展不变量，包括阿贝尔规范理论模型，由类Morse函数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调，以及将Konno附加型复形与族的同伦不变量相结合的新的参数化不变量。它还将把一些规范理论包中未涉及的结构翻译到其他包中，例如将L空间概念从Heegaard Floer理论翻译成瞬子情况下的类比，或者将Seiberg-Witten Floer理论的同变结构翻译成Heegaard Floer理论。拓扑学应用将包括对纽结调和、同调共边、4-流形的微分同胚群和3-流形群的SU(2)表示的新见解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。