Mathematical Sciences: Undergraduate Research in Nilpotent Spaces

由REU网站资助的六名学生将探索拓扑学领域中数学问题的几个研究领域。基本问题出现在代数拓扑学中，特别是同伦理论，这是一个通常被认为超出本科课程范围的纯数学领域。这个问题在学生的能力范围内，因为它可以归结为计算机搜索特定类型的矩阵。在代数拓扑学中，关于拓扑空间的问题被转化为关于代数对象的问题，通常是阿贝尔群。这些阿贝尔群以及它们之间的同态又可以用矩阵和矩阵运算来表示。这允许计算机表示和操作。本科生将设计算法，并在计算机上实现它们，以搜索特定类型的空间，称为幂零复合体。计算机方法在这里很有价值，因为尽管人们在很多方面都理解了幂零复形，但很难用几何方法来描绘。利用这些想法，已经开发了计算机程序，并发现了新的幂零复合体。然而，主要的生存问题仍然没有答案。这个项目的一个具体目标是回答一个尚未解决的问题，即有限的三维幂零复形的存在。一个次要但非常重要的目标是改进去年REU项目期间制定的方案之一。搜索更复杂的组是很耗时的。必须创造有效、更快的方法来克服这一障碍。

The six students supported by this REU site grant will explore several areas of research on mathematical problems in the field of topology. The basic question arises in algebraic topology, specifically homotopy theory, a field of pure mathematics generally considered beyond the undergraduate curriculum. The question is within the students' capabilities 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 by matrices and matrix operations. This allows for computer representation and manipulation. The undergraduates will design algorithms and implement them on computers to search for certain kinds of space, called a nilpotent complex. The computer approach is valuable here because, though understood in many ways, nilpotent complexes are hard to picture geometrically. Using these ideas, computer programs have been developed and new nilpotent complexes discovered. However, major existence questions are still unanswered. A specific goal of this project is to answer an unsolved question about the existence of finite three-dimensional nilpotent complexes. A secondary, but very important, goal is one of improving programs developed during the past year's REU project. Searching more complex groups is time consuming. Efficient, faster methods must be created to overcome this obstacle.