Representation Theory of Quantum Groups and Integrable Systems

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We proved the existence of the crystal bases in the sense of Kashiwara for KR modules of the quantum groups of nonexceptional affine type, and determined their crystal structure. We then settled affirmatively the conjecture on the perfectness of KR crystals. By examining the combinatorial structure of the KR crystal, we solved the X=K conjecture by Shimozono-Zabrocki. On the other hand, we showed that a similar relation holds for the generating functions of rigged configurations, thereby solving the X=M conjecture for all nonexceptional affine types when the rank is sufficiently large.