Topics in Arithmetic Geometry and K-Theory

算术几何和 K 理论专题

• 批准号: 9801219
• 负责人: Henri Gillet
• 金额: $ 23.17万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-06-15 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801219&HistoricalAwards=false
• 关键词: Topics; Arithmetic; Geometry; Theory

Henri GilletDMS-98 01219Professor Gillet will work on problems involving the interaction between K-theory and arithmetic geometry. He will study algebraic analogs to Arakelov theory for varieties over discretely valued fields. He hopes to establish an arithmetic Riemann-Roch theorem. It is hoped that these researches will lead to improved insights into the structure of sets of rational points on arithmetic varieties. In addition to these algebraic problems, Professor Gillet will study intersections of analytic cycles from a purely analytic perspective.Analysis, geometry and algebra are three of the major attacks used to investigate modern mathematical problems. All three have been used repeatedly on the ancient problem of solving equations using fractions alone. Geometrically this problem amounts to finding rational points (points described by fractions) on curves (equations viewed as graphs.) Giving such solutions a geometric interpretation makes them more concrete. But having done this, the modern approach to geometry constructs abstract algebraic objects that behave like familiar geometric objects. At the same time analytic techniques that use Calculus can help understand the same geometric objects in a different way. In this project Professor Gillet will study certain collections of rational solutions of equations, varieties, using geometry, algebra, and analysis in tandem. The combination of several modern ideas will lead to new insights into the structure of the rational solutions of rational equations.

Henri GilletDMS-98 01219 Gillet教授将致力于涉及K理论和算术几何之间相互作用的问题。他将研究离散值域上簇的阿拉克洛夫理论的代数类比。他希望建立算术Riemann-Roch定理。希望这些研究将有助于对算术变种上的有理点集的结构有更深入的了解。除了这些代数问题，吉尔特教授还将从纯分析的角度研究分析圈的交集。分析、几何和代数是研究现代数学问题的三种主要方法。在仅用分数解方程的古老问题上，这三种方法都被反复使用。从几何上讲，这个问题相当于在曲线(方程式被视为图形)上寻找有理点(用分数描述的点)。对这些解进行几何解释会使它们更加具体。但在这样做之后，现代几何方法构建了抽象的代数对象，其行为类似于熟悉的几何对象。同时，使用微积分的分析技术可以帮助以不同的方式理解相同的几何对象。在这个项目中，Gillet教授将同时使用几何、代数和分析来研究方程、变种的某些有理解集。几种现代思想的结合将对有理方程有理解的结构产生新的见解。