Quantitative Aspects of Number Theory

The aim of this project is to investigate problems in number theory that have a quantitative component. Typically, the problems would involve some algebraic objects, such as algebraic number fields or rational points on algebraic varieties. The questions that we ask are, however, of analytic nature. For example, how many number fields are there of a given type and with given properties, such that their discriminant is bounded by a (large) number B? The answer that we are seeking would then be an asymptotic formula or, if this is too hard, bounds for this number as B tends to infinity.

Quantitative results of this kind are of interest by themselves, but moreover they are useful in other proofs, often to establish the existence of objects with certain properties. This holds, for example, for many applications of the Hardy-Littlewood circle method from analytic number theory.

A candidate for this project would have some background in analytic or algebraic number theory, as well as an interest in learning the other field. An interest in algebraic geometry could be useful, but is by no means required. The concrete problems to work on will be chosen to match the student’s interests.