Total positivity, quantised coordinate rings and Poisson geometry

Matrices are central objects in mathematics, but also in other sciences. In particular, totally nonnegative matrices, that is, matrices whose minors are all nonnegative, have been recently used in areas as diverse as computer science, chemistry, physics and economics.In the 90's Lusztig generalises this notion and defines the space of totally nonnegative elements in a real flag variety---a very beautiful geometric object from algebraic Lie theory. As often, putting things in a more general context has led to many ground breaking developments such as for instance the theory of cluster algebras by Fomin and Zelevinsky. Recently, a connection between total positivity and quantised coordinate rings was observed by the applicant and his collaborators. More precisely, it was observed that in recent publications the same combinatorial object has appeared as a device to classify objects in combinatorics (total positivity), noncommutative algebra (quantised coordinate rings) and Poisson geometry. This very exciting connection was then studied by Goodearl, Lenagan and the applicant in the matrix case. Building up on this success, the main aim of this proposal is to investigate this new and unexpected similarity in the more general framework of flag varieties. In particular, we aim to create a bridge between these three rich branches of mathematics, and to use it to solve problems in number theory and combinatorics. Our approach through algorithmic methods should lead to rapid progress in all three areas. As often, unifying different theories should lead not only to ground breaking results, but also to new and exciting developments.

There are four main activities that will be carried forward during the project in order to ensure it reaches its full potential. 1. Academic dissemination. In order for the outputs of the project to be widely known, the PI will continue his strategy of publishing his results in a wide range of journals. This contribution to several subject areas on the archive is a way to ensure wide readership. 2. Linking Research and Teaching. It is something the PI has tried to develop since he got a Lectureship at the University of Kent. This has been done so far via projects, and in particular projects for first year students. The PI will continue this strategy. He will also seek funding in order to organise an intensive course for PhD students on the topics of this proposal. (Note that the PI already organised such a course in May 2009.) 3. Communication and Engagement. The PI has always been involved in outreach activity. In 2009/2010, he ran two well-attended masterclasses for high school pupils on his research topics. During the project, the PI is planning to run at least four masterclasses for high school pupils. They will take place at the University of Kent with the collaboration of the Royal Institution, and will be based on the outcomes of this project. 4. Link with mathematicians closer to the end users. Recently, the combinatorial aspects of this proposal have been used in mathematical physics and statistical physics. The PI has recently organised a mini-workshop in Oberwolfach where mathematicians involved in some of these applications were present, hence making them aware of his results. The PI with the help of the PDRA will continue organising meetings in order to facilitate/initiate such links. During the project, the PI will apply for funding in order to organise either a workshop at the University of Kent, or in a Mathematical Centre such as Luminy (France). This will ensure that the outputs of this project are available not only to pure mathematicians but also to mathematicians who are a priori closer to potential users.