Some questions in total positivity and cluster algebras

该项目的重点是集群代数和全积极性。 福明和泽列文斯基在他们的工作中多次遇到一种新的组合学，他们将其公理化为所谓的簇代数。 近年来，簇代数在数学的许多领域都有应用，例如代数表示，非交换几何，离散可积系统，Teichmuller理论甚至弦理论。在与Sergey Fomin的联合工作中，PI将根据张量图探索某些集群代数的模型。PI还将与托马斯林共同研究一个密切相关的主题。代数群的全正部分与簇代数和Lusztig的标准基密切相关。仿射全正性研究是仿射全正性研究的一个新方向，该研究项目源于两个数学领域：表示论和组合数学。表示论是一个重要的领域，是量子力学和粒子物理研究的基础。簇代数为表示论和相关学科提供了一种新的研究方法。PI的研究重点是簇代数，它超出了我们目前的理解范围（所谓的无限突变类型）。全正性的概念最早出现在上个世纪初的文献中，它与簇代数密切相关。它是理解机械系统振荡的主要工具。PI的工作集中在将该理论扩展到仿射（或无限维）情况。

The proposed project focuses on cluster algebras and total positivity. Fomin and Zelevinsky have repeatedly encountered in their work a new kind of combinatorics, which they axiomatized as so-called cluster algebras. Cluster algebras have found applications in numerous areas of mathematics in recent years, such as quiver representations, non-commutative geometry, discrete integrable systems, Teichmuller theory and even string theory. In joint work with Sergey Fomin, the PI will explore a model for certain cluster algebras in terms of tensor diagrams. The PI will also study a closely related topic of total positivity in joint work with Thomas Lam. Totally positive parts of algebraic groups are closely related to cluster algebras and Lusztig's canonical bases. The proposed research continues the study of affine total positivity, which is a new and fruitful direction in this area.The proposed project is rooted in two areas of mathematics: representation theory and combinatorics. Representation theory is an important area that is fundamental for the study of quantum mechanics and particle physics. Cluster algebras provide a new approach to representation theory and related disciplines. The PI's research focuses on cluster algebras that lie just beyond the current frontier of our understanding (the so called infinite mutation type). The notion, closely related to cluster algebras, of total positivity first appeared in the literature at the beginning of previous century. It is a major tool for understanding oscillations in mechanical systems. The PI's work concentrates on extending this theory to the affine (or infinite dimensional) case.