Professor Natsume's project has to do with functorial invariants, called K-theory and cyclic cohomology, for algebras of operators. The problems to be tackled concern noncommutative ramifications of ideas from geometry. One such problem is to exhibit geometrically an explicit nontrivial generator for K- nought of certain operator algebras associated to the fundamental group of a closed Riemann surface, and to find as well a cyclic cocycle on an appropriate dense subalgebra that pairs nontrivially with this generator. Another facet of the project involves foliated manifolds and the operator algebras to which they give rise. The fundamental insight of the area of mathematical research in which Professor Natsume works is the following: complete information about the structure of a space (a surface, say, or some higher dimensional analogue) is stored in an appropriate algebra of functions on the space. (Functions assign number values to points in the space.) Everything one would want to say about the geometry of a compact manifold, for instance, can be stated in terms of the algebra of infinitely differentiable complex functions on the manifold. This algebra, like so many interesting objects in mathematics, can be represented usefully as an algebra of operators on Hilbert space. Quite often, geometrical or topological notions that make sense for algebras of smooth or continuous functions also make sense for more or less arbitrary operator algebras. Working algebraically has the twofold merit of clarifying ideas, and also of permitting the use of machinery such as K-theory that works much better in the context of algebras than of spaces. Professor Natsume's project will consider several specific problems in pursuit of this agenda of looking at geometric phenomena through an operator-algebraic lens.
