9310433 McLaughlin In his work on algebraic K-theory, Beilinson introduced characteristic classes for holomorphic bundles, which refine the usual Chern classes. At present, they cannot be constructed directly from a given bundle and are poorly understood. McLaughlin will attempt to remedy this by finding explicit formulae for the Beilinson classes. This work will lead to a new interpretation of polylogarithms in terms of multicategories. He also expects to shed new light on Bloch's conjecture concerning characteristic classes of flat bundles. His methods are inspired by the degree three non-abelian cohomology recently defined by Breen. This geometric technique will also be used to attack some "classical" problems in topology, e.g., lifting group actions in bundles with non-abelian structure group. This research project has important implications for several areas of mathematics and physics. In number theory, there is the two hundred-year-old problem of finding the values of Riemann's zeta function at odd whole numbers. Some of these unknown values are related to the Beilinson classes, and this project will lead to a better understanding of the subject. Furthermore, the research provides a framework for understanding the emerging area of non-abelian geometry. This encompasses such diverse areas as the classification of three-dimensional spaces (a problem known as the Poincare conjecture) and even quantum physics (in particular, string theory). The project's unifying approach has already yielded new insights, and there is the promise of more to come. ***
