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

9310433 McLaughlin在他关于代数K-理论的工作中，Beilinson引入了全纯丛的特征类，从而提炼了通常的陈氏类。目前，它们不能直接从给定的捆绑包构造出来，而且人们对它们的理解很少。麦克劳克林将试图通过寻找贝林森类的显式公式来纠正这一点。这项工作将导致多范畴对多对数的一种新的解释。他还希望对Bloch关于平面丛的特征类的猜想有新的解释。他的方法是受Breen最近定义的三度非交换上同调的启发。这种几何技巧也将被用来解决拓扑学中的一些“经典”问题，例如具有非交换结构群的丛中的提升群作用。这项研究项目对数学和物理的几个领域具有重要影响。在数论中，有一个两百年前的问题：求Riemann‘s Zeta函数在奇数上的值。其中一些未知值与贝林森班级有关，这个项目将导致对这一主题的更好理解。此外，这项研究还为理解非阿贝尔几何这一新兴领域提供了一个框架。这包括不同的领域，如三维空间的分类(这个问题被称为庞加莱猜想)，甚至量子物理(特别是弦理论)。该项目的统一方法已经产生了新的见解，而且还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. ***