Rings of Low Rank

The goal of the workshop is to better understand the amazing work of Manjul Bhargava on rings of low rank, and to explore its impact on computational number theory.

Bhargava finished his dissertation in 2001 in Princeton under the supervision of Andrew Wiles, who is himself famous for proving Fermat's Last Theorem. The individual chapters of his thesis have been appearing as separate articles in the acclaimed journal Annals of Mathematics. He wil be the http://www.stieltjes.org Visiting Professor 2006 in Leiden.

Bhargava's papers address very classical problems, and his results can be viewed as a direct extension of work of Gauss on quadratic forms. It seems fair to say that the mathematical community has not fully understood these developments, and it is our hope that the workshop will contribute in this respect.  Bhargava's work is a break-through in the subject of classifying and counting rings whose underlying abelian group is free of a fixed low rank. His results on quartic and quintic rings took all experts by surprise.  He discovered an as yet mysterious link with exceptional Lie groups.  Many themes remain to be explored, such as counting the number of subrings in a given ring of a given index. It is also hoped that Bhargava's approach will shed some light on results of Nakagawa from 1996 which suggest a functional equation for zeta-functions associated to counting quartic rings.

Applications of Bhargava's results are both of a theoretical and of a computational nature: the counting techniques can both be applied in an asymptotic setting, where we estimate growth behaviour, and in a computer algebra setting where we ask for an exact number of, say, all quintic rings with discriminant of at most 10 digits.