Mathematical Sciences: Research in Topology

数学科学：拓扑研究

• 批准号: 9215134
• 负责人: Sylvain Cappell
• 金额: $ 21.9万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-12-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9215134&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Topology

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.

研究员将针对高维和低维流形和分层空间的拓扑和几何中的几个基本问​​题以及簇的拓扑和代数几何不变量开展研究项目。 要研究的一组问题涉及品种​​和分层空间（例如特征类）的基本不变量之间的作用、计算和关系。 另一组问题涉及变换群不动点的分类及其与这些基本不变量的关系。 还有第三组有待研究的项目涉及三维流形的深层新不变量及其与基本群表示论的关系。 这项工作将需要研究有关谱流的相关分析问题和有关马斯洛指数的辛几何问题。 这些部分的细节各不相同，但都涉及将几何信息简化为计算对象。 所涉及的几何信息的性质是困难的关键。 虽然关于长度、面积、角度、体积等的问题实际上迫切需要简化为计算，但它与所谓的几何对象的拓扑性质有很大不同。 这些属性包括连通性（全部为一体）、打结性、无孔等。 所有对这些性质的系统研究，例如，如何判断两个几何对象在这些性质之一上是否确实不同或只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在简化为计算问题时才真正被理解和掌握。