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

摘要：求职申请：DMS-0448750：有理点的多样性和非阿贝尔伽罗瓦群。负责人：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，重点是描述有理点的多样性在数域上的分布问题。对有理点的研究，一方面是从数论一开始就传给我们的；另一方面，在这个极其活跃的领域中，目前的技术状况涉及到各种各样的主题，包括表上同调，解析数论的方法，以及模态空间的复代数几何。在第一个项目中，研究者和他的同事将继续他们关于有理点分布和数域判别分布的著名猜想的研究。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理论的对象密切相关，这一主题在过去五年中受到了极大的关注。这些项目将涉及数论和代数几何各个领域的研究人员的大量合作，并将为研究生和博士后提供许多机会。数论的基本问题是：找到一个方程的有理数的所有解。例如，人们可以问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的平方吗？几个世纪以来，毕达哥拉斯方程一直为人们所熟知；但至少可以这么说，还有许多其他的方程，比如费马方程！——更严重的困难，而且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”这门学科，或者说方程解的分析，已经形成了自己的一门学科，从数学的许多不同部分汲取思想和技巧。研究者在理性点领域提出了几个研究项目。因为这一领域的研究结合了经典的，易于陈述的问题与先进的方法和实验的机会，它非常适合初出茅台的数学家。研究者将为麦迪逊纪念高中的学生启动一项研究计划，并将扩大威斯康星人才搜索的范围；其目标是在威斯康辛州发展一个中学数学探索、丰富和研究的基础设施，该基础设施将有足够的动力在本奖项的持续时间之后继续发展。

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.