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

摘要职业建议DMS-0448750：品种和非阿贝尔伽罗瓦群的合理点主要研究者：Jordan S。Ellenberg机构：威斯康星大学研究者提出解决算术几何中的一系列问题，集中在描述数域上的变量上的有理点的分布问题上。 理性点的研究是，一方面，传递给我们从一开始的数论;另一方面，目前的艺术状态在这一极其活跃的领域涉及的想法，从各种各样的主题，包括etale上同调，方法ofanalytic数论，和复杂的代数几何modulispaces的态射。 在第一个项目中，研究者和他的同事们将继续他们对有理数点的分布和数域的判别式的分布的著名定理的研究。 在第二个项目中，研究者将研究曲线塔的算法;在这样的塔中的合理点似乎与非阿贝尔岩泽理论的对象密切相关，这是一个在过去五年中引起人们极大兴趣的主题。 这些项目将涉及大量的合作研究人员从每一个部分的数论和代数几何，并应提供许多机会，为研究生和博士后研究员数论的基本问题是：找到所有的解决方案，在有理数方程。 例如，人们可以问毕达哥拉斯方程的解：对于哪个有理数x，y和z，x的平方加上y的平方等于z的平方？ 几个世纪以来，人们对毕达哥拉斯方程的理解已经很好了;但至少可以说，还有许多其他的方程，比如费马方程。--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，“合理点”或方程解的分析这一主题已形成为一门独立的学科，它从数学的许多不同部分汲取了思想和技术。 研究者提出了分数领域的几个研究项目。 由于这一研究领域结合了经典的、易于表述的问题和先进的方法以及实验机会，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发起一项研究计划，并将扩大威斯康星州人才搜索的范围;目标是在威斯康星州发展一个基础设施，用于中学数学探索、充实和研究，which哪一个will have enough足够momentum势头to continue继续beyond超过the duration期of the present当前award奖项.

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.