Differential operators on von Neumann algebras

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Differential operators on von Neumann algebrasNikolai WeaverProject SummaryDifferential operators on operator algebras have been studiedextensively as a model for the infinitesimal time developmentof a quantum-mechanical system, either isolated or interactingwith a macroscopic system. The proposed research involves astudy of differential operators on von Neumann algebras buildingon the investigator's prior work on the special case of first-orderdifferential operators.Operators on a Hilbert space are infinite analogs of finitematrices. They have applications throughout mathematics, butthe proposed research relates specifically to mathematicalphysics. Von Neumann algebras of operators have been used to modelcertain quantum mechanical systems, particularly those arising inquantum statistical mechanics. The proposal involves using ideasof Fields medalist Alain Connes to study the time-evolution ofthese algebras, with the double aim of clarifying the generaltheory and giving insight into specific examples.