Representation theory of affine Hecke algebras and quanum groups via geometry, category and combinatorics

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We mainly study the structure of Johnson cokernels for the mapping class group of surfaces via several representation theoretic approaches. The notion of Johnson cokernels gives an approximation of the mapping class group of surfaces using a certain graded Lie algebras. We have no series in the Johnson cokernels except for "the Morita obstruction" in 1990's.In this research, we introduce a new class in Johnson cokernels for the automorphism groups of the free groups and the mapping class groups. Moreover, the class is combinatorially computable by using some representation theoretic method for the general linear groups and the symplectic groups. Moreover, we give a new series "anti-Morita obstructions" in the Johnson cokernels. This is a joint work with Takao Staoh.