Motivic fundamental groups, multiple polylogarithms, and Diophantine geometry

动机基本群、多重多对数和丢番图几何

• 批准号: 0753012
• 负责人: Minhyong Kim
• 金额: $ 3.39万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0753012&HistoricalAwards=false
• 关键词: Motivic; fundamental; groups; multiple; polylogarithms

The theory of mixed motives provides one of the most fertile grounds for investigations in number theory via its connection to the theory of L-functions of algebraic varieties. Although the study of motives is essentially homological in nature, Deligne has defined a motivic fundamental group attached to varieties over number fields. Coordinate functions on such fundamental groups are related to special functions like multiple polylogarithms and, thereby, also to special values of L-functions. On the other hand, the proposer has discovered a direct connection betweenmotivic fundamental groups and Diophantine geometry, somewhat along the lines suggested by Gronthendieck's `anabelian' philosophy. The technical tools involve p-adic integration, p-adic Hodge theory, and the global study of Galois cohomology. He proposes to continue research into this connection, aiming towards homotopy-theoretic proofs of well-known theorems, such as those of Faltings or Wiles, and eventually new higher-dimensional results in the line of the conjectures of Serge Lang on hyperbolic varieties.The deep relationship between geometry and arithmetic is a venerable topic of study going back at least to ruler and compass constructions of special numbers in ancient Greece. The modern manifestation of this tradition is the subject of arithmetic geometry, an area of mathematics that has yielded some of the most profound mathematical results of the previous century.This proposal describes several related ideas for obtaining new results in the theory of algebraicequations using ideas at the interface of linear and non-linear geometry.

混合动机的理论通过与代数品种的L功能理论的联系来为研究数理论的研究提供了最肥沃的理由之一。尽管对动机的研究本质上是同源性的，但Deligne定义了一个与数字领域相关的动机基本群体。此类基本组上的协调功能与特殊功能（如多个多聚类阶层）有关，因此也与L功能的特殊值有关。另一方面，提议者发现了动态基本群体与二世的几何形状之间的直接联系，这在某种程度上符合Gronthendieck的“ Anabelian”哲学所建议的界限。技术工具涉及P-ADIC整合，P-ADIC HODGE理论和Galois协同学的全球研究。 He proposes to continue research into this connection, aiming towards homotopy-theoretic proofs of well-known theorems, such as those of Faltings or Wiles, and eventually new higher-dimensional results in the line of the conjectures of Serge Lang on hyperbolic varieties.The deep relationship between geometry and arithmetic is a venerable topic of study going back at least to ruler and compass constructions of special numbers in ancient Greece.这种传统的现代表现是算术几何形状的主题，这是一个数学领域，它产生了上一个世纪的一些最深刻的数学结果。该提案描述了使用线性界面和非线性微音界面上的思想理论中获得代数理论的几个相关思想。