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

布林斯基打算继续他的工作， 构造特征类和显式整数- Cech cocycles为他们。 他预计， 构造规范群上同调类的方法， 群的超同态。 他还将继续研究 几何和拓扑的节点在一个光滑的流形，重点 关于幺模同态群的作用 他将 严格分析一些拓扑量子场论，使用 模空间上的线丛 这些部分的细节各不相同，但都涉及到 将几何信息还原为计算对象。 的 所涉及的几何信息的性质是关键的 困难 而长度、面积、角度、体积， 等几乎哭了，以减少计算，它是 与我们所知的 几何物体 这些属性比如连通性 （都在一块），打结，没有洞，等等 向前。 所有系统的研究，例如，如何 来判断两个几何对象是否真的在 这些属性之一，或者只是表面上的不同， 对可能发生的各种差异进行分类，所有这些 只有当他们真正理解和掌握， 减少到计算的问题。

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.