Representation Theory, Quantum Groups and Piecewise-Linear Combinatorics

Arkady Berenstein研究了李群表示论、量子群和分段线性组合学的交叉点。 他研究规范基地andcrystal基地，多重代表性的还原群体，以及完全积极的品种。 主要工具，他用于这项研究的估值相应的（量子）代数，方法ofinvolutions，凸多面体，合理的地图和几何crystals.The主题的Arkady Berenstein的研究是combinatorialstructures在该地区的数学被称为表示理论的李代数。这些结构自然地出现在经典的枚举问题中，这些问题除了数学之外，还来自物理学、化学和其他基础科学。令人惊讶的是，这些纯粹的离散结构有连续的对应物。理解这些枚举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。这一关系被证明是研究20世纪世纪数学中代数与几何之间最有启发性的对应关系--朗兰兹对应关系的一个有用工具。

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.