Mathematical Sciences: Research in Topology

研究人员将开展研究项目， 拓扑几何中的几个基本问题 维流形和分层空间以及拓扑 和代数几何不变量的品种。 一组 要研究的问题涉及的作用，计算， 变种的基本不变量之间的关系 分层空间，如特征类。 另一 一组问题涉及不动点的分类， 变换群及其与这些基本 不变量 仍有第三组项目有待调查 涉及三维流形的深层新不变量， 它们与基本表象理论的关系 组 这项工作将需要研究相关的分析 谱流和辛几何问题 关于Maslow Index的问题 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。

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.