Total positivity, quantised coordinate rings and Poisson geometry

点击翻译按钮获取中文摘要

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.