Mathematical Sciences: Undergraduate Research in Nilpotent Spaces

该奖项将于1988年夏季在福特汉姆大学为4名学生设立一个REU项目。研究问题出现在代数拓扑，特别是同伦理论，一个通常被认为远远超出了本科课程的纯数学领域。这个问题在大学生的范围内，因为它可以简化为对某些类型矩阵的计算机搜索。在代数拓扑中，关于拓扑空间的问题被转化为关于代数对象的问题，通常是关于阿贝尔群的问题。这些阿贝尔群，以及它们之间的同态，可以依次表示为矩阵和矩阵运算。这允许计算机表示和操作。本科生将设计算法并在计算机上实现，以搜索某种拓扑对象，称为幂零复合体。通过算法解决这个问题，本科生将开创一种解决这些空间问题的新方法。具体来说，该项目试图回答一个尚未解决的问题，即有限三维幂零复合体的存在。

This award is for an REU site for 4 students to take place during the summer of 1988 at Fordham University. The research problem arises in algebraic topology, specifically homotopy theory, a field of pure mathematics usually considered to be well beyond the undergraduate curriculum. This question is within range of undergraduates because it can be reduced to a computer search for certain kinds of matrices. In algebraic topology, questions about a topological space are translated into questions about algebraic objects, usually Abelian groups. These Abelian groups, and homomorphisms between them, can in turn be represented as matrices and matrix operations. This allows for computer representation and manipulation. The undergraduates will design algorithms and implement them on a computer to search for a certain kind of topological object, called a nilpotent complex. By attacking the problem algorithmically, the undergraduates will pioneer a new approach to questions about these spaces. Specifically, the project seeks to answer an unsolved question about the existence of finite three-dimensional nilpotent complexes.