Total positivity, quantised coordinate rings and Poisson geometry

矩阵是数学的中心对象，也是其他科学的中心对象。特别是，完全非负矩阵，即矩阵的子矩阵都是非负的，最近已经在计算机科学，化学，物理学和经济学等领域得到了广泛的应用。在90年代，Lusztig推广了这个概念，并定义了空间的完全非负元素在一个真实的标志品种--一个非常美丽的几何对象，从代数李理论。通常，把事情在一个更一般的情况下，导致了许多突破性的发展，例如理论的集群代数的福明和Zelevinsky。最近，申请人和他的合作者观察到总正性和量子化坐标环之间的联系。更确切地说，有人指出，在最近的出版物中，同样的组合对象已经出现作为一种设备来分类对象在组合（全正性），非交换代数（量子化坐标环）和泊松几何。然后，Goodearl、Lenagan和申请人在矩阵案例中研究了这种非常令人兴奋的联系。建立在这一成功的基础上，本建议的主要目的是调查这种新的和意想不到的相似性，在更一般的框架旗品种。特别是，我们的目标是在这三个丰富的数学分支之间建立一座桥梁，并用它来解决数论和组合数学中的问题。我们通过算法方法的方法应该会导致所有三个领域的快速进展。通常，统一不同的理论不仅会导致开创性的结果，而且会导致新的和令人兴奋的发展。

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.