FRG: Obstructions to Local-Global Principles and Applications to Algebraic Structures

FRG：局部全局原理的障碍以及代数结构的应用

• 批准号: 1463882
• 负责人: Parimala Raman
• 金额: $ 45.13万
• 依托单位: Emory University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1463882&HistoricalAwards=false
• 关键词: FRG; Obstructions; Local; Global; Principles

The interplay between number theory and algebraic geometry has been a source of inspiration in modern mathematics. Having led to the solution of a number of outstanding conjectures, such as Fermat's Last Theorem and the Mordell Conjecture, it continues to give rise to deep and important problems in algebra. Local-global principles are a central theme in this interplay of subjects, and many important outstanding problems can be expressed in terms of such principles. This project has the objective of understanding local-global principles and their obstructions, in contexts that are broader than those considered in number theory. The project will also support and enhance the training of graduate students and postdoctoral researchers through seminars, conferences and workshops, and mentoring activities.The Focused Research Group will focus on local-global principles for algebraic structures defined over function fields of curves over base fields such as p-adic fields, with a longer term goal of treating the case of function fields of curves over global fields. The obstructions to such local-global principles can often be formulated in terms of cohomology. Our project aims to study the finiteness of these obstructions and determine criteria for them to vanish. The resulting understanding will be applied to proving conjectures and solving open problems concerning algebraic structures such as quadratic forms and associative algebras. This will include situations that have been studied by many researchers but where solutions had previously seemed out of reach. Research methods will include field patching, cohomological methods including residues and duality, and approaches from geometry.

数论和代数几何之间的相互作用一直是现代数学的灵感来源。它导致了许多杰出猜想的解决，例如费马大定理和莫德尔猜想，并继续引发代数中深刻而重要的问题。局部-全球原则是这些主题相互作用的中心主题，许多重要的突出问题可以用这些原则来表达。该项目的目标是在比数论中考虑的范围更广泛的背景下理解局部全局原则及其障碍。该项目还将通过研讨会、会议和讲习班以及指导活动来支持和加强研究生和博士后研究人员的培训。重点研究小组将重点研究在基域上的曲线函数域上定义的代数结构的局部全局原理，例如作为 p 进场，长期目标是处理全局场上的曲线函数场的情况。这种局部-全局原理的障碍通常可以用上同调来表述。我们的项目旨在研究这些障碍物的有限性并确定它们消失的标准。由此产生的理解将应用于证明猜想并解决有关代数结构（例如二次形式和结合代数）的开放问题。这将包括许多研究人员已经研究过但以前似乎无法解决的情况。研究方法将包括现场修补、上同调方法（包括留数和对偶性）以及几何方法。