Representation Theory, Quantum Groups and Piecewise-Linear Combinatorics

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Arkady Berenstein investigates the area lying at the crossroads of the Representation Theory of Lie Groups, Quantum Groups and Piecewise-Linear Combinatorics. He studies canonical bases and crystal bases, the multiplicities for the representations of reductive groups, and the totally positive varieties. The main tools he uses for this study are the valuations of the corresponding (quantum) algebras, the method of involutions, convex polyhedra, rational maps and geometric crystals. The subjects of Arkady Berenstein's research are combinatorial structures in the area of Mathematics known as Representation Theory of Lie algebras. These structures naturally emerge in classical enumeration problems that arise from physics, chemistry and other basic sciences in addition to mathematics. Quite surprisingly, these purely discrete structures have continuous counterparts. Understanding the relationship between these enumerative combinatorial structures and the geometric structure of their continuous counterparts is a question of the greatest importance. This relationship proved to be a useful tool in the study of Langlands Correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.