Mathematical Sciences: The Joint Torsion of Koszul Complexes

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9501387 Pincus The investigator is studying operator algebras. More specifically, he is studying what are known as Steinberg symbols, a generalization of multiplicative commutators in such algebras. Steinberg symbols are factored by means of a Koszul complex construction, and this gives rise to a new invariant, the joint torsion. This invariant is related to a local or maximal ideal index in the setting of commutative operator algebras, and it provides new understanding of the principal function in the non- commutative case. Similar constructions involving higher algebraic K-groups will also be investigated. The study of Wiener-Hopf integral equations began more than sixty years ago in connection with physical problems such as the diffraction of electromagnetic or sound waves. A natural operator W arises. The investigator's contributions began with the introduction of a naturally paired operator U for which a combination of the two operators known as the determinant took on a particularly simple and useful integral form. Using this determinant, he found that the study of the Wiener-Hopf operator W could be deepened, that new solutions could be found to the original integral equations, and that the study of functions h(W,U) of the pair of operators could be put into a new geometric context. This was an early step in development of the hugely successful new area of modern mathematics sometimes known as "non-commutative geometry." Now non-commutative geometry has drawn together several diverse areas of mathematics, and "algebraic K-theory" figures prominently among them. In the context of algebraic K-theory, the further analysis of the structure of the determinants mentioned above has led to a new object called the joint torsion of the pair {W,U}. This object in turn is linked to geometry in still another way and occurs in the study of differential equations and in the study of dynamical systems. ***