Collaborative Research: Generalized Cluster Structures on Poisson Varieties and Applications

合作研究：泊松簇的广义簇结构及其应用

• 批准号: 2100791
• 负责人: Michael Shapiro
• 金额: $ 36.5万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2100791&HistoricalAwards=false
• 关键词: Collaborative; Research; Generalized; Cluster; Structures

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.

该项目属于正在开发的代数领域，旨在数学和理论物理中的应用。特别关注泊松李群和簇代数的理论。前者长期以来一直作为一个自然框架，可以在其中研究经典和量子力学的重要的精确可解模型。后者由 Fomin 和 Zelevinsky 于 2001 年发现，已被证明与广泛的数学学科有许多令人兴奋的联系，包括组合数学、表示论、代数和泊松几何，以及镜像对称、统计和高能物理。 PI将在之前的合作基础上继续系统地研究代数几何、表示论和数学物理中多种重要的坐标环中的多簇结构，并研究相应的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及研究项目的开发联系起来。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群体和具有不同教育背景的研究生，并通过社区外展，让高中生接触数学研究。PI 将致力于泊松几何到簇代数理论。更详细地说，该项目的主要目标包括：1）构建和研究泊松李群和泊松同质簇的广义簇结构，包括配备Belavin-Drinfeld括号的泊松李群、Drinfeld双组和Poisson-Lie对偶组。简单的泊松李群，以及 3d N = 4 SUSY 规范理论的 K 理论库仑分支； 2）泊松簇上的广义簇结构在经典和非交换离散可积系统中的应用，这些系统作为簇变换序列、统一根处量子群的表示以及更高的 Teichmuller 理论而出现。该奖项反映了 NSF 的法定使命和通过使用基金会的智力优点和更广泛的影响审查标准进行评估，该项目被认为值得支持。

项目成果

ROOTS OF CHARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID

辛群曲面特征方程的根

• 发表时间: 2022
• 期刊: Uspehi matematičeskih nauk
• 作者: Chekhov, L.;Shapiro, M.;Shibo, H.
• 通讯作者: Shibo, H.

Log-canonical coordinates for symplectic groupoid and cluster algebras

辛群群和簇代数的对数正则坐标

• DOI: 10.1093/imrn/rnac101
• 发表时间: 2022
• 期刊: International mathematics research notices
• 影响因子: 1
• 作者: Chekhov, L.;Shapiro, M.
• 通讯作者: Shapiro, M.

Introducing isodynamic points for binary forms and their ratios

介绍二元形式的等力点及其比率

• DOI: 10.1007/s40627-022-00112-4
• 发表时间: 2023
• 期刊: Complex Analysis and its Synergies
• 作者: Hägg, Christian;Shapiro, Boris;Shapiro, Michael
• 通讯作者: Shapiro, Michael