Total nonnegativity, quantum algebras and growth of algebras

这是一个广泛的项目，涉及三个领域的noncommutativealgebra，泊松代数几何和线性代数。此外，解决方案涉及表示论和组合数学。此外，该项目将考虑有关代数增长的问题。量子代数理论的发展是由1980年代以来的物理问题所推动的。完全非负矩阵已经涉及到的问题，在不同的领域，如机械系统，出生和死亡的过程，平面电阻网络，计算机辅助几何设计，杂耍，等有关结果的增长代数已经获得从20世纪60年代起，但这个问题是在一个静止的状态，直到2000年取得了重大进展。在过去的五年里，人们发现并调查了第一段中提到的三个领域之间令人惊讶的联系。对这些联系，特别是矩阵的坐标代数的特殊情况，已经有了部分的了解。本项目的目的是通过深化矩阵的情况下的知识，并通过扩大知识的范围，包括代数，如grassmannians，部分旗品种和德Concini-Kac-Procesi代数，以促进这种理解。该项目的增长部分将集中在两种特定类型的增长：二次增长/Gelfand-Kirillov维数2，和中间增长（超多项式，但次指数）。

This is wide ranging project that involves the three areas of noncommutativealgebra, Poisson algebraic geometry and linear algebra. Also, the solutionsoften involve representation theory and combinatorics. In addition, theproject will consider problems concerning growth of algebras. The development of the theory of quantum algebras was motivated by problems inPhysics from the 1980s onwards. Totally nonnegative matrices have beeninvolved in problems in such diverse areas as mechanical systems, birth anddeath processes, planar resistor networks, computer aided geometric design,juggling, etc. Results concerning growth of algebras have been obtained fromthe 1960s onwards, but the subject was in a quiescent state until the 2000swhen significant advances have been made. In the past five years, surprising links between the three areas mentioned inthe first paragraph have been discovered and investigated. A partialunderstanding of these connections has been gained, especially in the particularcase of coordinate algebras of matrices. The present project aims to furtherthis understanding by deepening the knowledge of the matrix case and byexpanding the scope of the knowledge to include algebras such asgrassmannians, partial flag varieties and De Concini-Kac-Procesi algebras. The growth part of the project will concentrate on two specific types ofgrowth: quadratic growth/Gelfand-Kirillov dimension two, and intermediategrowth (super polynomial, but subexponential).