FRG: Collaborative Research: Lifting Problems and Galois Theory

FRG：协作研究：提升问题和伽罗瓦理论

• 批准号: 1265290
• 负责人: Ted Chinburg
• 金额: $ 116万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265290&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Lifting; Problems

The object of this proposal is to pursue new approaches to some fundamental problems in Galois theory, following recent breakthroughs in this subject. The focus of research will be on lifting finite (possibly ramified) Galois covers of smooth projective curves over a commutative ring and on finding small rings of definition for such lifts. These problems pertain to the classical Inverse Galois Problem as well as to the question of finding the relations between etale fundamental groups of various kinds.This proposal deals with fundamental questions about the solutions of algebraic equations. The study of the symmetries of such equations has been the motivation for the development of much of abstract algebra. Advances in abstract algebra have been essential to the development of modern technology. Some particular examples of applications of abstract algebra include the compression and transmission of data, cryptography, search algorithms and solving optimization problems in the physical sciences, engineering and economics.

这一建议的目的是追求新的方法来解决伽罗瓦理论中的一些基本问题，在这个问题上最近的突破。 研究的重点将是提升有限（可能分歧）伽罗瓦覆盖光滑的射影曲线在一个交换环，并找到小环的定义，这样的升降机。 这些问题既涉及到经典的Galois逆问题，也涉及到寻找各种基本群之间的关系问题，这个建议涉及到关于代数方程解的基本问题。 对此类方程对称性的研究一直是大部分抽象代数发展的动力。 抽象代数的进步对现代技术的发展至关重要。 抽象代数应用的一些具体例子包括数据的压缩和传输，密码学，搜索算法和解决物理科学，工程和经济学中的优化问题。