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Arithmetic Moduli Spaces and Gauge Theory

算术模空间和规范理论

基本信息

批准号：
EP/V046888/1
负责人：
Minhyong Kim
金额：
$ 25.87万
依托单位：
University of Edinburgh
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV046888%2F1
关键词：
Arithmetic Moduli Spaces Gauge Theory

项目摘要

This proposal is concerned with the theory of Diophantine equations, that is, the study of rational or integral solutions to algebraic equations. This is one of the oldest subjects in mathematics, going back possibly to the ancient Babylonians and systematised by Diophantus of Alexandria around the 3rd century. Nevertheless, it is still the source of some of the most difficult problems and wide-ranging programmes in mathematics, such as Fermat's Last Theorem or the conjectures of Birch and Swinnerton-Dyer. In spite of much progress over the last 100 years or so using the modern methods of arithmetic geometry, the major problems remain unsolved, especially when it comes to algorithmic methods that can find solutions to equations on a computer. (This is furthermore hampered by certain impossibility theorems of mathematical logic.) This research proposes to apply new ideas inspired by high energy physics to the study of Diophantine equations in two variables based on an analogy between the solution space to Euler-Lagrange equations in physics and the non-Archimedean geometry of 'arithmetic gauge fields' constructed by the PI. The main goal is to generalise to higher degree the methodology surrounding the conjecture of Birch and Swinnerton-Dyer, which is concerned with equations of degree 3. Eventually, this research should lead to substantial progress on the problem of devising a computer algorithm that will find all rational solutions to equations of degree at least 4 and two unknowns.

这个建议是关于丢番图方程的理论，即研究代数方程的有理解或积分解。这是数学中最古老的学科之一，可能可以追溯到古巴比伦人，并由亚历山大的丢番图在世纪左右系统化。尽管如此，它仍然是数学中一些最困难的问题和广泛的程序的来源，如费马大定理或伯奇和斯温纳顿-戴尔的著作。尽管在过去100年左右的时间里，使用算术几何的现代方法取得了很大的进步，但主要问题仍然没有解决，特别是当涉及到可以在计算机上找到方程解的算法方法时。(This此外，还受到数理逻辑中某些不可能性定理的阻碍。）本研究将高能物理学的新思想应用于双变量丢番图方程的研究，基于物理学中欧拉-拉格朗日方程的解空间与PI构建的“算术规范场”的非阿基米德几何之间的类比。主要目标是推广到更高程度的方法论周围的猜想的伯奇和Swinnerton-Dyer，这是有关方程的程度3。最终，这项研究将导致在设计一种计算机算法的问题上取得实质性进展，该算法将找到至少4次方程和两个未知数的所有合理解。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Topological nature of dislocation networks in two-dimensional moiré materials

二维莫尔材料中位错网络的拓扑性质

DOI：
10.1103/physrevb.107.125413
发表时间：
2023
期刊：
Physical Review B
影响因子：
3.7
作者：
Engelke, Rebecca;Yoo, Hyobin;Carr, Stephen;Xu, Kevin;Cazeaux, Paul;Allen, Richard;Valdivia, Andres Mier;Luskin, Mitchell;Kaxiras, Efthimios;Kim, Minhyong
通讯作者：
Kim, Minhyong

Linking emergent phenomena and broken symmetries through one-dimensional objects and their dot/cross products.

通过一维对象及其点/叉积将涌现现象和对称性破缺联系起来。