Mathematical Sciences: Cyclic Homology, Algebra, Topology and Geometry

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Burghelea will study some aspects and implications of cyclic homology in algebra, topology and geometry. In algebra, he plans to study the cyclic homology of algebraic varieties and crossed product algebras associated to actions of groups on algebraic varieties; he will also study the Chern character for group rings. In topology he will study the homology type of B Diff(M) in the stability range, and at the prime 2. In geometry he will study the relationship between the Selberg trace formula, cyclic homology and closed geodesics. Exploring the uses of cyclic homology will render a powerful technique more versatile. Ultimately applications to theoretical physics and other consumers of higher mathematics should profit. **8 8701643 Bergman Three important properties of universal algebras will be studied: the amalgamation property, deductiveness, and congruence modular varieties. The research on the amalgamation property centers on two problems. 1. Determination if the amalgamation property plus residual smallness implies the congruence extension property. 2. Characterization of the amalgamation bases of the variety. The plans for deductiveness are less specific. The general issue is to attempt to classify those varieties that are deductive. A third avenue to be pursued is the representation problem for the commutator in congruence modular varieties. All of these topics concern the foundations of algebra and are likely to be applied outside of algebra only to the extent that they condition a mode of thought about algebraic questions.