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

小行星9310433 在他的工作代数K-理论，贝林森介绍了特征类的全纯丛，完善了通常的陈类。 目前，它们不能直接从一个给定的捆绑包中构造出来，而且人们对它们的理解也很少。 McLaughlin将试图通过为Beilinson类找到显式公式来弥补这一点。 这项工作将导致一个新的解释多元的多元范畴。 他还希望能阐明新的光布洛赫猜想有关的特征类的单位束。 他的方法的灵感来自度三非交换上同调最近定义的布林。 这种几何技术也将用于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。 这个研究项目对数学和物理的几个领域有重要的影响。 在数论中，有200年历史的问题，寻找黎曼zeta函数的值在奇数整数。 这些未知值中的一些与Beilinson类有关，这个项目将导致对该主题的更好理解。 此外，研究提供了一个框架，了解新兴领域的非阿贝尔几何。 这包括了三维空间的分类（庞加莱猜想），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，并且有希望获得更多。 ***

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. ***