Mathematical Sciences: Geometric Aspects of Representations of Quantum Groups and Kac-Moody Lie Algebras

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This research is concerned with the study of representations and cohomology of Kac-Moody Lie algebras and quantum groups and their relationship with the geometry of configuration spaces and infinite flag varieties, and some related questions. The principal investigator will study the connection between representations of quantum groups and perverse sheaves in configuration spaces; cohomology of affine Lie algebras and cohomology of configuration spaces; vertex operators and cohomology of affine lie algebras; connections with the geometry of infinite flag spaces; Selberg integrals, the structure constants of operator algebras in conformal field theory, and their analogues over finite fields; and higher dimensional generalizations. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.