Mathematical Sciences: Geometry of Characteristic Classes and Non-Abelian Cohomology

9310433 McLaughlin在代数k理论中，Beilinson引入了全纯束的特征类，改进了通常的Chern类。目前，它们不能直接从给定的包中构造，而且很难理解。McLaughlin将试图通过为Beilinson类找到明确的公式来弥补这一点。这项工作将导致在多范畴方面对多对数的新解释。他还希望对布洛赫关于平束特征类的猜想提供新的启示。他的方法受到Breen最近定义的三次非阿贝尔上同的启发。这种几何技术还将用于解决拓扑中的一些“经典”问题，例如，非阿贝尔结构群束中的群作用提升问题。这个研究项目对数学和物理的几个领域具有重要意义。在数论中，有一个存在了两百年的难题：如何在奇数整数处找到黎曼ζ函数的值。其中一些未知的值与Beilinson类有关，这个项目将使我们更好地了解这个主题。此外，该研究为理解非阿贝尔几何的新兴领域提供了一个框架。这包含了许多不同的领域，如三维空间的分类（一个被称为庞加莱猜想的问题），甚至量子物理学（特别是弦理论）。该项目的统一方法已经产生了新的见解，并且有望产生更多的见解。＊＊＊

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. ***