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

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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.