FRG: Collaborative Research: Lifting Problems and Galois Theory

这一建议的目的是在伽罗瓦理论中的一些基本问题上寻求新的方法，在这一主题最近的突破之后。研究的重点将是提升交换环上光滑投射曲线的有限(可能是分支的)Galois覆盖，以及寻找此类提升的定义的小环。这些问题涉及经典的伽罗华逆问题，以及寻找各种基本群之间的关系的问题。这个建议涉及关于代数方程的解的基本问题。对这类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、搜索算法以及解决物理科学、工程和经济中的最优化问题。

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.