Collaborative Research: Generalized Cluster Structures on Poisson Varieties and Applications

这个项目位于一个正在开发的代数领域，着眼于在数学和理论物理中的应用。一个特别的焦点是泊松-李群和簇代数理论。前者长期以来一直是一个自然的框架，可以在其中研究重要的经典和量子力学的精确可解模型。后者由Fomin和Zlevinsky在2001年发现，此后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、表示论、代数和泊松几何，以及镜像对称性和统计和高能物理。PI将在他们以前的合作的基础上，继续系统地研究在代数几何、表示理论和数学物理中具有多种重要性的坐标环中的多个团簇结构，并研究相应的团簇代数之间的相互作用。这项研究将与本科生和研究生课程和研究项目的发展相联系。计划的协同活动旨在促进机构间和部门间的合作，吸引来自代表性不足群体和不同教育背景的研究生，并通过社区推广活动，让高中生接触数学研究。个人投资促进机构将致力于泊松几何在簇代数理论中的应用。更详细地说，该项目的主要目标包括：1)建立和研究泊松-李群和泊松齐次簇上的广义团簇结构，包括带有Belavin-Drinfeld括号的泊松-李群、简单泊松-李群的Drinfeld对偶和Poisson-Lie对偶，以及3DN=4SUSY规范理论的K-理论库仑分支；2)Poisson簇上的广义簇结构应用于经典和非交换离散、可积系统，这些系统作为簇变换序列、单位根处量子群的表示和更高的TeichMuller理论产生。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.