FRG: Collaborative Research: Lifting Problems and Galois Theory

这一建议的目的是在伽罗瓦理论最近取得突破之后，寻求新的方法来解决该学科的一些基本问题。研究的重点将是在交换环上提升光滑投影曲线的有限（可能是分支的）伽罗瓦覆盖，并为这种提升找到定义的小环。这些问题既属于经典的伽罗瓦反问题，也属于寻找各种基本群之间关系的问题。这个建议处理关于代数方程解的基本问题。对这些方程的对称性的研究一直是许多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特殊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。

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.