Non-commutative fundamental groups in Diophantine geometry

丢番图几何中的非交换基本群

• 批准号: EP/G024979/2
• 负责人: Minhyong Kim
• 金额: $ 15.06万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG024979%2F2
• 关键词: Non; commutative; fundamental; groups; Diophantine

This proposal is concerned primarily with Diophantine equations in two variables, i.e., polynomial relations with integers coefficients for which one seeks to understand the collection of integer solutions. The history of such investigations reaches back to the tradition of Greek mathematics, while the twentieth century has seen spectacular applications of abstract modern machinery to the resolution of difficult old questions, such as Wiles' proof of Fermat's last theorem. The investigator proposes a new approach to studying these classical problems by incorporating fundamental ideas of topology and geometry that go beyond the principal developments of the twentieth century in that the relevant structures are, in the main, non-commutative and non-linear. An eventual goal is to construct methods for effectively resolving Diophantine equations in two-variables.

该提案主要涉及两个变量的丢番图方程，即具有整数系数的多项式关系，人们试图理解整数解的集合。此类研究的历史可以追溯到希腊数学的传统，而二十世纪已经见证了抽象现代机械在解决困难的老问题上的惊人应用，例如怀尔斯对费马大定理的证明。研究者提出了一种新的方法来研究这些经典问题，通过结合拓扑和几何的基本思想，这些思想超越了二十世纪的主要发展，因为相关结构主要是非交换和非线性的。最终目标是构建有效求解二变量丢番图方程的方法。

项目成果

Tangential localization for Selmer varieties

Selmer 品种的切向定位

• DOI: http://dx.10.1215/00127094-1507332
• 发表时间: 2012
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Kim M

Number Theory, Analysis and Geometry

数论、分析和几何

• DOI: http://dx.10.1007/978-1-4614-1260-1_16
• 发表时间: 2012
• 作者: Kim M

The main conjecture of Iwasawa theory for totally real fields

岩泽理论关于全实域的主要猜想

• DOI: http://dx.10.1007/s00222-012-0436-x
• 发表时间: 2012
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Kakde M

Appendix and erratum to "Massey products for elliptic curves of rank 1"

“1 阶椭圆曲线的 Massey 产品”的附录和勘误表

• DOI: http://dx.10.1090/s0894-0347-2010-00675-3
• 发表时间: 2010
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Balakrishnan J