Collaborative Research: Generalized Cluster Structures on Poisson Varieties and Applications

这个项目位于代数的一个领域，它正在发展，以期在数学和理论物理中的应用。一个特别的焦点是泊松-李群和簇代数的理论。长期以来，前者一直是研究经典力学和量子力学中重要的精确可解模型的自然框架。后者是由弗明和泽列文斯基在2001年发现的，后来被证明与广泛的数学学科有许多令人兴奋的联系，包括组合学、表示理论、代数和泊松几何，以及镜像对称、统计和高能物理。pi将在之前合作的基础上，继续系统地研究在代数几何、表示理论和数学物理中具有重要意义的坐标环中的多个簇结构，并研究相应簇代数之间的相互作用。这项研究将与本科和研究生课程和研究项目的发展联系起来。计划开展协同活动，以促进机构间和部门间的合作，吸引来自代表性不足和教育背景不同的群体的研究生，并通过社区外联使高中生接触数学研究。pi将研究泊松几何在聚类代数理论中的应用。具体而言，项目的主要目标包括：1)构建和研究泊松-李群和泊松齐次变异上的广义团簇结构，包括带Belavin-Drinfeld托架的泊松-李群、简单泊松-李群的Drinfeld双元和泊松-李偶元，以及三维N = 4 SUSY规范理论的k理论库仑分支；2)泊松变簇上的广义簇结构在经典非交换离散可积系统中的应用，这些系统由簇变换序列、量子群在单位根的表示和更高的Teichmuller理论产生。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project lies in an area of algebra that is being developed with a view toward applications in Mathematical and Theoretical Physics. A particular focus is on the theory of Poisson-Lie groups and cluster algebras. The former has long served as a natural framework in which important exactly solvable models of classical and quantum mechanics can be studied. The latter, discovered by Fomin and Zelevinsky in 2001, have since been shown to have numerous exciting connections with a wide range of mathematical subjects, including combinatorics, representation theory, algebraic and Poisson geometry, as well as mirror symmetry and statistical and high energy physics. The PIs will build upon their previous collaborations to continue a systematic study of multiple cluster structures in coordinate rings of a number of varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. This research will be linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through community outreach, to expose high school students to mathematical research.The PIs will work on applications of Poisson geometry to the theory of cluster algebras. In more detail, the main goals of the project include: 1) construction and study of generalized cluster structures on Poisson-Lie groups and Poisson homogeneous varieties including Poisson-Lie groups equipped with Belavin-Drinfeld brackets, Drinfeld doubles and Poisson-Lie duals of simple Poisson-Lie groups,and K-theoretic Coulomb branches of 3d N = 4 SUSY gauge theories; and 2) applications of generalized cluster structures on Poisson varieties to classical and non-commutative discrete, integrable systems that arise as sequences of cluster transformations, representations of quantum groups at roots of unity, and higher Teichmuller theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.