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

Brylinski打算继续他在特征类的几何构造以及它们的显式整值Cech上循环方面的工作。他期望发展出一般的方法来构造规范群和微分同胚群的上同调类。他还将继续研究光滑流形中纽结的几何和拓扑，重点是单模微分同态群的作用。他将使用模空间上的线丛，严格分析一些拓扑量子场论。这些部分的细节各不相同，但都与将几何信息归结为一个计算对象有关。涉及的几何信息的性质是困难的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有对这些性质的系统研究，例如，如何区分两个几何物体在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些只有在归结为计算的问题时才被真正理解和掌握。

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.