Mathematical Sciences: Research in Topology

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The investigator will be carrying out research projects on several basic problems in topology and geometry of high and low dimensional manifolds and of stratified spaces and on topological and algebraic-geometric invariants of varieties. One set of problems to be studied concerns the role, calculation, and relationships among fundamental invariants of varieties and stratified spaces, such as the characteristic classes. Another group of problems concerns the classification of fixed points of transformation groups and their relationships to these basic invariants. Still a third set of projects to be investigated concerns deep new invariants of three-dimensional manifolds and their relationship to the representation theory of the fundamental group. This work will entail a study of relevant analytical problems concerning spectral flow and symplectic geometrical questions concerning the Maslow index. 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.