Mathematical Sciences: Cyclic Cohomology, Local Index and N-Tuple of Operators

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Professor Xia's project will focus on situations in which operators on Hilbert space almost satisfy various algebraic relationships, where almost means up to perturbation by compact, trace class, or other suitably small operators. He is particularly interested in almost Lie algebras of operators, and in n-tuples of operators almost satisfying identities of Schrodinger type. Formulas explicitly giving the trace of operators known in such contexts to be trace class will be sought, and powerful generalizations of trace formulas involving the pairing of cyclic cohomology with K-theory will be explored. This is mathematical research in the theory of operators on Hilbert space. Operators may be thought of as enriched numbers, obeying the same laws of arithmetic as ordinary numbers with two notable exceptions: multiplication of operators depends upon the order in which the factors are taken, and not every non-zero operator has an inverse. For finite-dimensional operators, i.e. square matrices of numbers, these phenomena account for the richness of the subject of linear algebra. In the infinite- dimensional setting of Hilbert space, there is furthermore the phenomenon of operators obeying the laws of arithmetic for numbers except for error terms that are small in an appropriate sense and that contain important information about whatever situation gave rise to the operators. In a broad sense, Professor Xia's project is concerned with how one can extract this information efficiently.