Some questions in total positivity and cluster algebras

提出的项目重点是簇代数和总正性。Fomin和Zelevinsky在他们的工作中反复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，簇代数在许多数学领域得到了应用，如颤振表示、非交换几何、离散可积系统、Teichmuller理论甚至弦理论。在与Sergey Fomin的合作中，PI将根据张量图探索某些簇代数的模型。私家侦探亦会与林兆贤共同研究一个密切相关的课题，即“整体积极性”。代数群的全正部分与聚类代数和Lusztig正则基密切相关。本研究是对仿射总正性研究的延续，是该领域一个富有成果的新方向。提议的项目植根于数学的两个领域：表示理论和组合学。表征理论是量子力学和粒子物理研究的一个重要领域。聚类代数为表征理论和相关学科的研究提供了新的途径。PI的研究重点是超越我们目前理解的前沿的簇代数（所谓的无限突变型）。与簇代数密切相关的全正性概念最早出现在上个世纪初的文献中。它是理解机械系统振荡的主要工具。PI的工作集中于将这一理论扩展到仿射（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.