Infinite-dimensional Lie algebras and their applications

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Lie theory, named after a Norwegian mathematician Sophus Lie, is a vibrant area of modern mathematics with applications ranging from string theory to satellite control and quantum computing.******We propose to develop representation theory of Lie algebras of vector fields on affine algebraic varieties. This theory will combine the features of commutative and non-commutative algebra. Building on the recent advances we made in representation theory of vector fields on a torus, we would like to prove a classification theorem in the setting of an arbitrary smooth algebraic variety.******A part of this proposal is of a more applied nature. We propose to solve several optimal control problems on unitary groups. Solving these problems will have ramifications for quantum control and quantum computing.**