Representation Theory, Quantum Groups and Piecewise-Linear Combinatorics

表示论、量子群和分段线性组合学

• 批准号: 0102382
• 负责人: Arkady Berenstein
• 金额: $ 9.05万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102382&HistoricalAwards=false
• 关键词: Representation; Theory; Quantum; Groups; Piecewise

Arkady Berenstein investigates the area lying at thecrossroads of the Representation Theory of Lie Groups, Quantum Groupsand Piecewise-Linear Combinatorics. He studies canonical bases andcrystal bases, the multiplicities for the representations of reductive groups,and the totally positive varieties. The main tools he uses for this study arethe valuations of the corresponding (quantum) algebras, the method ofinvolutions, convex polyhedra, rational maps and geometric crystals.The subjects of Arkady Berenstein's research are combinatorialstructures in the area of Mathematics known asRepresentation Theory of Lie algebras. These structuresnaturally emerge in classical enumeration problems that arisefrom physics, chemistry and other basic sciences in additionto mathematics. Quite surprisingly, these purelydiscrete structures have continuous counterparts. Understanding the relationship between these enumerativecombinatorial structures and the geometric structure of theircontinuous counterparts is a question of the greatestimportance. This relationship proved to be a useful tool in thestudy of Langlands Correspondence -- the most mysteriousand inspiring correspondence between Algebra andGeometry of the 20th century Mathematics.

Arkady Berenstein研究了李群表示理论、量子群和分段线性组合的交叉领域。他研究正则碱基和晶体碱基，还原群表示的多样性，以及完全正的变化。他在这项研究中使用的主要工具是相应（量子）代数的赋值、卷积方法、凸多面体、有理映射和几何晶体。Arkady Berenstein的研究主题是数学领域的组合结构，即李代数的表示理论。这些结构自然地出现在经典的枚举问题中，这些问题产生于物理、化学和数学以外的其他基础科学。令人惊讶的是，这些纯粹离散的结构有连续的对应物。理解这些数列组合结构和它们的连续对应物的几何结构之间的关系是一个非常重要的问题。这种关系被证明是研究朗兰兹对应性的一个有用的工具——朗兰兹对应性是20世纪数学中代数和几何之间最神秘、最鼓舞人心的对应性。