Mathematical Sciences: Connes Invariant X(M) and Related Invariants

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9501515 Chen This research involves the study of an invariant (called chi invariant) of Alain Connes on von Neumann algebras. The connection between this invariant and the cohomology of groups will be explored. Of special interest is the cohomology theory of perfect groups. He will also use cohomology theory of continuous groups. One objective is to clarify the relations between the algebraic invariants associated with the chi invariant. Another objective is to realize the invariants for appropriate von Neumann algebras. Finally, a study will be made of the generalization of the chi invariant to bimodules. This invariant also plays an important role in the theory of subfactors. The basis of the theory underlying this project is algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics. ***