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: 1463901
• 负责人: Daniel Krashen
• 金额: $ 17.32万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1463901&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.

数字理论与代数几何形状之间的相互作用一直是现代数学中灵感的来源。导致了许多出色的猜想解决方案，例如Fermat的Last Therorem和Mordell猜想，它继续引起代数中的深层和重要问题。本地全球原则是该主题相互作用的中心主题，许多重要的问题可以从此类原则上表达出来。该项目的目的是理解本地全球原则及其障碍，而在数字理论中比被认为的情况更广泛。该项目还将通过研讨会，会议和研讨会以及指导活动来支持和增强研究生和博士后研究人员的培训。专注的研究小组将专注于本地全球全球范围的原则，用于定义的代数结构，这些原理定义了基础领域的曲线功能领域，作为P-ADIC字段，其长期目标是治疗全球字段上曲线功能场的情况。这种局部全球原则的障碍通常可以从同时学方面提出。我们的项目旨在研究这些障碍的有限性，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和解决有关代数结构（例如二次形式和联想代数）的开放问题。这将包括许多研究人员所研究的情况，但是解决方案以前似乎遥不可及。研究方法将包括现场弥补，包括残基和二元性在内的共同体学方法以及几何形状的方法。

项目成果

Local-Global Principles for Constant Reductive Groups over Semi-Global Fields

半全局域上常约简群的局部全局原理

• DOI: 10.1307/mmj/20217219
• 发表时间: 2022
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Colliot-Thélène, Jean-Louis;Harbater, David;Hartmann, Julia;Krashen, Daniel;Parimala, R.;Suresh, V.
• 通讯作者: Suresh, V.