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James Gilbert Glimm (born 24 March 1934) is an American mathematical physicist, and Professor at the State University of New York at Stony Brook.

He received his Ph.D. in mathematics from Columbia University in 1959; his advisor was Richard V. Kadison.

He has been noted for contributions to C*-algebras, quantum field theory, partial differential equations, fluid dynamics, scientific computing, the modeling of petroleum reservoirs, and geometric models for structural biology. Together with Arthur Jaffe, he has founded a subject called constructive quantum field theory.

James Glimm was elected to the National Academy of Sciences in 1984. He won the National Medal of Science in 2002. Starting January 1, 2007, he is serving a 2-year terms as President of the American Mathematical Society.

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