Representation Theory, Quantum Groups and Piecewise-Linear Combinatorics

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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.