Principal Investigator: Matilde Marcolli
This proposal aims at using techniques from noncommutative geometry, especially analytic techniques based on functional analysis, operator algebras, global analysis and index theory, to approach questions of relevance to arithmetic geometry. The main themes covered in this proposal include: the theory of noncommutative tori with real multiplication as a parallel for real quadratic fields to the theory of elliptic curves with real multiplication; the use of quantum statistical mechanical systems associated to number fields and function fields as an approach to explicit class field theory; an application of techniques from functional analysis and quantum statistical mechanics to the study of the asymptotic problem for error correcting codes; the construction of invariants of curves with p-adic uniformization from noncommutative geometry, in terms of index theorems and spectral triples; investigating the relation between motives and noncommutative spaces, in relation to L-functions of varieties and motives; relating the noncommutative geometry approach to the Riemann zeta function to arithmetic topology, matrix models, and foliated spaces; understanding manifestations of modularity on the noncommutative boundary of modular curves.
The project is interdisciplinary in scope, as it is aimed at exploring the interactions between tools and techniques from noncommutative geometry and motivating problems from number theory and arithmetic. This presents a novel approach to classical questions of modern mathematics, where we plan on using techniques and ideas derived from the world of quantum mechanics, statistical mechanics, and quantum field theory. Noncommutative geometry is especially suitable as a tool, since it was developed precisely a a geometry adapted to quantum physics: as such, it provides a way to combine quantum mechanical ideas and techniques with more classical algebro-geometric and number-theoretic methods. We expect that a continuing investigation of this approach will lead to further advances and developments within the field of noncommutative geometry itself, by the challenge of fine tuning the available theory to fit specific number theoretic problems, while at the same time it will provide new possible approaches and different viewpoints on questions of number theoretic relevance. This proposal has a strong educational components, with six graduate students involved in various parts of the project. A network of international contacts and collaborations will also be involved.
