ABSTRACT for CAREER proposal DMS-0448750: Rational points on varieties and non-abelian Galois groups
Principal investigator: Jordan S. Ellenberg
Institution: University of Wisconsin
The investigator proposes to address an ensemble of problems in arithmetic geometry, centered on the problem of describing the distribution of rational points on varieties over number fields. The study of rational points is, on the one hand, passed down to us from the very beginning of number theory; on the other hand, the present state of the art in this extremely active area involves ideas from a wide variety of subjects, including etale cohomology, methods of analytic number theory, and the complex algebraic geometry of moduli spaces of morphisms. In the first project, the investigator and his colleagues will continue their investigation of well-known conjectures on distribution of rational points and distribution of discriminants of number fields. In the second project, the investigator will study the arithmetic of towers of curves; rational points in such towers seem to be intimately related with objects of non-abelian Iwasawa theory, a subject which has enjoyed an explosion of interest in the last five years. The proposed projects will involve a great deal of collaboration with researchers from every part of number theory and algebraic geometry, and should provide many opportunities for graduate students and postdoctoral fellows.
The foundational problem of number theory is this: find all solutions in rational numbers to an equation. For instance, one can ask for solutions 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 equals the square of z? The Pythagorean equation has been well-understood for centuries; but many others, like Fermat's equation, present -- to say the least! -- more serious difficulties, and most equations are presently beyond our current ability to analyze. In recent years, the subject of "rational points", or the analysis of solutions of equations, has been taking shape as a subject of its own, drawing ideas and techniques from many different parts of mathematics. The investigator proposes several research projects in the area of rational points. Because this area of research combines classical, easy-to-state problems with advanced methods and opportunities for experimentation, it is very well-suited for fledgling mathematicians. The investigator will launch a research program for students at Madison Memorial High School, and will expand the scope of the Wisconsin Talent Search; the aim will be to develop in Wisconsin an infrastructure for mathematical exploration, enrichment and research in secondary school, which will have enough momentum to continue beyond the duration of the present award.
