Asymptotic algebraic geometry and Diophantine applications

渐近代数几何和丢番图应用

• 批准号: 261900-2013
• 负责人: Roth, Michael
• 金额: $ 1.38万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=547908
• 关键词: Asymptotic; algebraic; geometry; Diophantine; applications

Algebraic geometry is a rich branch of mathematics which unifies algebra, topology, number theory, and differential geometry. It allows one to reformulate questions which seem to be about algebra or numbers into geometrical questions about shape, or vice versa. Its origins lie in the realization that finding solutions to algebraic equations or understanding the properties of the solutions are intimately tied to the intrinsic geometric nature of the algebraic varieties those equations describe. The Lang-Vojta conjectures are one of the most spectacular realizations of this philosophy. In general terms they connect the global behaviour of rational solutions of equations to the global behaviour of the curvature on the corresponding algebraic variety.

代数几何是数学的一个丰富的分支，它统一了代数、拓扑学、数论和微分几何。它允许人们将看似关于代数或数字的问题重新表述为关于形状的几何问题，反之亦然。它的起源在于认识到寻找代数方程的解或理解解的性质与这些方程所描述的代数变体的内在几何性质密切相关。Lang-Vojta猜想是这一哲学最引人注目的实现之一。一般来说，它们将方程的有理解的整体行为与相应代数变量上曲率的整体行为联系起来。