Representation Theory, Quantum Groups, and Birational Algebraic Geometry

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This project is devoted to investigation of the area lying at the crossroads of the representation theory of Lie groups, quantum groups, birational algebraic geometry, and piecewise-linear combinatorics. A new approach to the study of Lusztig's canonical bases and Kashiwara'scrystal bases is proposed, based on quantum cluster algebras and geometric crystals. New information resulting from this study will be applied to computing the multiplicities for the representations of reductive groups and for constructing new totally positive varieties. The results of this study will also be used for solving problems emerging in the representations of discrete subgroups of reductive algebraic groups as well as for explication and elaboration of related combinatorial and geometric structures. Representation theory of Lie algebras and quantum groups is one of the most dynamically developing fields of modern Mathematics. This theory has a large impact on other fields of Mathematics and generates numerous applications in other NaturalSciences. In their turn, the concepts of canonical and crystal bases are of great importance for the representation theory: a mere establishing of their existence has helped in solving classical enumeration problems (e.g., the problem of computing multiplicities of irreducible representations or decomposing tensor products of irreducible representations). Therefore, any information on canonical or crystal bases would be very beneficial for the representation theory. Understanding the relationship between the discrete (i.e., combinatorial) and continuous (i.e., geometric) structures of the canonical bases is one of the main priorities of this project. This relationship has proved to be a useful tool in the study of a famous Langlands correspondence -- the most mysterious and inspiring correspondence between Algebra and Geometry of the 20th century Mathematics.