CAREER: Rational points on varieties and non-abelian Galois groups

职业：簇上的有理点和非阿贝尔伽罗瓦群

• 批准号: 0448750
• 负责人: Jordan Ellenberg
• 金额: --
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0448750&HistoricalAwards=false
• 关键词: CAREER; Rational; points; varieties; non

ABSTRACT for CAREER proposal DMS-0448750: Rational points onvarieties and non-abelian Galois groupsPrincipal investigator: Jordan S. EllenbergInstitution: University of WisconsinThe investigator proposes to address an ensemble of problems inarithmetic geometry, centered on the problem of describing thedistribution of rational points on varieties over number fields. Thestudy of rational points is, on the one hand, passed down to us fromthe very beginning of number theory; on the other hand, the presentstate of the art in this extremely active area involves ideas from awide variety of subjects, including etale cohomology, methods ofanalytic number theory, and the complex algebraic geometry of modulispaces of morphisms. In the first project, the investigator and hiscolleagues will continue their investigation of well-known conjectureson distribution of rational points and distribution of discriminantsof number fields. In the second project, the investigator will studythe arithmetic of towers of curves; rational points in such towersseem to be intimately related with objects of non-abelian Iwasawatheory, a subject which has enjoyed an explosion of interest in thelast five years. The proposed projects will involve a great deal ofcollaboration with researchers from every part of number theory andalgebraic geometry, and should provide many opportunities for graduatestudents and postdoctoral fellows.The foundational problem of number theory is this: find all solutionsin rational numbers to an equation. For instance, one can ask forsolutions of the Pythagorean equation: for which rational numbers x,y,and z is it the case that the square of x plus the square of y equalsthe square of z? The Pythagorean equation has been well-understoodfor centuries; but many others, like Fermat's equation, present -- tosay the least! -- more serious difficulties, and most equations arepresently beyond our current ability to analyze. In recent years, thesubject of "rational points", or the analysis of solutions ofequations, has been taking shape as a subject of its own, drawingideas and techniques from many different parts of mathematics. Theinvestigator proposes several research projects in the area ofrational points. Because this area of research combines classical,easy-to-state problems with advanced methods and opportunities forexperimentation, it is very well-suited for fledgling mathematicians.The investigator will launch a research program for students atMadison Memorial High School, and will expand the scope of theWisconsin Talent Search; the aim will be to develop in Wisconsin aninfrastructure for mathematical exploration, enrichment and researchin secondary school, which will have enough momentum to continuebeyond the duration of the present award.

职业提案摘要 DMS-0448750：簇上有理点和非阿贝尔伽罗瓦群主要研究者：Jordan S. Ellenberg 机构：威斯康星大学研究者建议解决算术几何中的一系列问题，重点是描述簇上有理点的分布问题超过数字字段。 一方面，有理点的研究是从数论诞生之初就传承下来的；另一方面，另一方面，这个极其活跃的领域目前的最新技术涉及来自广泛学科的思想，包括 etale 上同调、解析数论方法以及态射模空间的复杂代数几何。 在第一个项目中，研究者和他的同事将继续研究有关有理点分布和数域判别式分布的著名猜想。 在第二个项目中，研究者将研究曲线塔的算法；这些塔中的理性点似乎与非阿贝尔岩泽理论的对象密切相关，这一主题在过去五年中引起了人们的极大兴趣。 拟议的项目将涉及与数论和代数几何各个领域的研究人员的大量合作，并为研究生和博士后提供许多机会。数论的基本问题是：找到方程有理数的所有解。 例如，我们可以求毕达哥拉斯方程的解：对于哪些有理数 x、y 和 z，x 的平方加 y 的平方等于 z 的平方？ 毕达哥拉斯方程几个世纪以来一直为人们所理解。但至少可以说，还存在许多其他方程，例如费马方程！ ——更严重的困难，大多数方程目前超出了我们目前的分析能力。 近年来，“有理点”学科，即方程解的分析，已经成为一门独立的学科，它借鉴了数学许多不同部分的思想和技术。 研究者在理性点领域提出了几个研究项目。 由于这一研究领域将经典的、易于表述的问题与先进的方法和实验机会结合起来，因此非常适合初出茅庐的数学家。研究者将为麦迪逊纪念高中的学生启动一项研究计划，并将扩大研究范围威斯康星州人才搜索；其目标是在威斯康星州开发中学数学探索、丰富和研究的基础设施，该基础设施将有足够的动力在当前奖项的期限后继续下去。