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.
