Mathematical Sciences: Characteristic Classes, the Space of Knots and Groups of Diffeomorphisms

Brylinski打算继续他关于特征类的几何构造和它们的显式整数值切赫环的工作。他期望发展出对规范群和微分同态群构造上同调类的一般方法。他还将继续研究光滑流形中结点的几何和拓扑，重点研究非模微分同态群的作用。他将使用模空间上的线束严格分析一些拓扑量子场论。这些部分的细节各不相同，但都与将几何信息简化为计算主题有关。所涉及的几何信息的性质是难点的关键。虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但这与几何对象的拓扑特性大不相同。这些属性包括连通性（整体）、打结性、无孔等等。所有对这些性质的系统研究，例如，如何判断两个几何物体在其中一个性质上是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。

Brylinski intends to continue his work on the geometric construction of characteristic classes and on explicit integer- valued Cech cocycles for them. He expects to develop general methods to construct cohomology classes for the gauge groups and groups of diffeomorphisms. He will also continue his study of the geometry and topology of knots in a smooth manifold, with emphasis on the action of the group of unimodular diffeomorphisms. He will analyze rigorously some topological quantum field theories, using line bundles on moduli spaces. The details of these parts vary, but all are concerned with reducing geometric information to a subject for calculation. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation.