Mathematical Science: Cohomology, Multivariable Operator Theory and Perturbation of Differential Operator

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9400766 Xia The project involves work in three directions. First work is proposed on finding complete unitary invariants for operator tuples. The operators in these tuples commute and are chosen from classes of non-normal operators with trace class self-commutators. The goal is to derive trace formulae for products of commutators and to establish some K-theory related to the cyclic cohomology associated with almost commuting operators. The second problem is to study the operators of the form P+Q where P is a partial differential operator. The project here is to derive results on the cyclic cohomology associated with a product of twisted commutators for this type of perturbation and to study the related spectral analysis. The final project is to study the Gelfand-Fuks cohomology of a Poisson algebra and its quantized version. This is a project on operator theory. The roots of this project can be traced back to mathematical physics and the study of the electron. The behavior of the electron was modeled by a pair of mathematical objects called operators. The states of the electron are determined by the "spectra" of the operators. Here the project involves a study of tuples of operators that are similar to the pair studied in quantum mechanics. The results will have applications to quantum mechanics and the mathematical areas of pseudo-differential operators and algebraic geometry. ***