Topics in Arithmetic Geometry and K-theory

算术几何和 K 理论专题

• 批准号: 0100587
• 负责人: Henri Gillet
• 金额: $ 10.18万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100587&HistoricalAwards=false
• 关键词: Topics; Arithmetic; Geometry; theory

This proposal concerns questions in higher dimensional Arakelov theory andalgebraic K-theory. The PI intends to extend results of previous work withSoule on motivic weight complexes of algebraic varities to the arithmetic case.In collaboration with J. Hu, he intends to use the deformation to the normalcone techniquess for Arakelov theory (developed by J. Hu in his thesis) to thecase of stacks. He also intends to study questions in arithmetic geometryrelated to the Mordell conjecture using methods from differential algebra andArakelov theory. This proposal deals with arithmetic geometry and algebraic K-theory.Arithmetic geometry is the study of the properties of equations withcoefficients that are whole numbers, and using methods both from numbertheory (the study of properties of whole numbers) and algebraic geometry, whichstudies geometric figures that can be defined by the simplest of equations,namely polynomials. Algebraic K-theory studies properties of linear equationsusing methods from geometry. The questions and phenomena which arise fromcombining number theory and geometry serve as driving forces in much ofcontemporary mathematics research. Moreover, arithemetic geometry hascontributed to many applications including cryptography and theoreticalcomputer science.

这一建议涉及高维Arakelov理论和代数K-理论中的问题。PI打算将Soule以前关于代数变体的Motivic Weight复形的结果推广到算术情形，并与J.Hu合作，打算将对Arakelov理论(由J.Hu在他的论文中发展)的正规锥技巧的变形用于堆栈的情形。他还打算用微分代数和阿拉克洛夫理论中的方法来研究与莫德尔猜想有关的算术几何问题。这项建议涉及算术几何和代数K理论。算术几何是研究系数为整数的方程的性质，并使用数论(研究整数的性质)和代数几何的方法，它研究可以由最简单的方程定义的几何图形，即多项式。代数K-理论利用几何方法研究线性方程的性质。数论和几何相结合所产生的问题和现象是当代许多数学研究的驱动力。此外，算术几何还促进了许多应用，包括密码学和理论计算机科学。