9704891 Lodder A major topic of research in the 1970's was the interplay between foliations and Lie algebra cohomology. A foliation represents the solution of a partial differential equation on a space, and the cohomology classes that detect a given foliation provide a concise numerical invariant measuring the (global) twisting of the solution set of the equation. The first cohomology class to be discovered in this way was the Godbillon-Vey invariant, a single class in dimension three that is an element of the classical de Rham cohomology group of the underlying space. (For codimension one foliations there are no other invariants.) This project is based on a fundamentally new method of computing cohomology that does not require the classical symmetries needed for de Rham cohomology. The ideas for this construction arise from work of Jean-Louis Loday of Strasbourg, France, and are referred to as Leibniz cohomology. The results of preliminary computations are striking. In the codimension one case alone, there are infinite families of invariants for foliations. They begin with the Godbillon-Vey invariant and then continue in dimensions 4n and 4n+3 (n any positive integer). Although the new classes result from a one-parameter variation of a foliation, it remains an open question to interpret their meaning in terms of physical properties. A more general topic of research in mathematics remains to classify space according to the criteria set forth by Riemann nearly 150 years ago. Much progress has been made in the past century, but the classification is not complete. As an unexpected and highly non-trivial invariant of space, Leibniz cohomology will be a new tool in this field. To aid in the interpretation of Leibniz cohomology, a construction for this invariant in terms of more familiar spaces would be helpful. It is auspicious that the formalism used to define the Leibniz groups appears quite naturally in quantum field theory. By work of V. Kac, the elements of a field theory form a classical Lie algebra---but only after a certain number of them are set to zero. If these elements are not set to zero, the original elements form a Leibniz algebra lacking the symmetries of a Lie algebra. The cohomology groups of this Leibniz algebra will be new invariants of quantum field theory. ***
