Mathematical Sciences: Algebraic and Geometric Topology

数学科学：代数和几何拓扑

• 批准号: 9505024
• 负责人: William Dwyer
• 金额: $ 29.18万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 2000-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505024&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Geometric; Topology

9605024 Dwyer The project touches on several areas in topology. One of its goals is to express some classical index theorems, especially those involving the Euler and signature operators, as homotopy theoretic statements, and then to study the higher-order information provided by this reinterpretation. The researchers are also investigating geometric questions about surfaces in four-dimensional manifolds, as well as algebraic issues surrounding the action of permutation groups on the homology of configuration spaces. Finally, they are working towards a classification of p-compact groups (i.e. "homotopy Lie groups") as well as towards a more complete understanding of how topology and number theory are mixed together by the constructions of algebraic K-theory. The general theme of the project is the study of some remarkable and exciting connections between geometry (the science of topological spaces, that is, of higher-dimensional shapes) and algebra (the science of equations and numbers). The connections go both ways. For instance, deciding whether or not it is possible to use the methods of calculus on a particular topological space turns out to involve solving some fairly subtle equations. On the other hand, the collection of all solutions to an equation can often be assembled in one way or another into a topological space. The investigators are looking at some particular aspects of the interplay between geometry and algebra, and they expect that taking a dual point of view will lead to major advances in both areas. ***

小行星9605024 该项目涉及拓扑学的几个领域。 它的目标之一是表达一些经典的指标定理，特别是那些涉及欧拉和签名算子，同伦理论的陈述，然后研究高阶信息提供了这种重新解释。 研究人员还在研究四维流形中曲面的几何问题，以及围绕置换群对配置空间同调作用的代数问题。 最后，他们正致力于对p-紧群（即“同伦李群”）进行分类，以及更完整地理解拓扑学和数论如何通过代数K-理论的构造而混合在一起。 该项目的总主题是研究几何（拓扑空间的科学，即高维形状）和代数（方程和数字的科学）之间的一些显着和令人兴奋的联系。 这种联系是双向的。 例如，决定是否可以在特定的拓扑空间上使用微积分方法，结果涉及到求解一些相当微妙的方程。 另一方面，一个方程的所有解的集合通常可以以这样或那样的方式组合成一个拓扑空间。 研究人员正在研究几何和代数之间相互作用的一些特定方面，他们希望采取双重观点将导致这两个领域的重大进展。 ***