Mathematical Sciences: Exploration in Algebraic Topology

该奖项支持一个面向本科生的研究体验计划，该计划专注于代数拓扑学数学理论中出现的问题。六名学生将与首席研究员一起参加为期九周的活动，旨在开发和实施算法来搜索一种称为细胞复合体的特定拓扑结构。单元复合体产生于对拓扑空间的研究中，这些拓扑空间是球面、环面、多面体等的高维版本。人们通过以规定的方式将单元组合在一起来从基本的构建块构造单元复合体(也称为CW-复合体)。关于细胞复合体的问题可以转化为关于代数对象、阿贝尔群的问题，这些问题可以表示为矩阵。矩阵是表示矩阵及其之间运算的理想工具。感兴趣的复形是那些被称为幂零的复形。很难看出如何将细胞组合在一起得到幂零复合体，但只要编程得当，计算机就可以构造出许多例子。这个特殊的项目试图回答一个基本的悬而未决的问题，即有限的三维幂零复形的存在。该课程首先向学生介绍在本科课程中通常不会遇到的拓扑概念。然后，利用学生在以前的REU项目中发现的结果，该团队将寻求改进现有的算法和代码，以提高它们的速度和范围。最后的任务将是找到新的、更复杂的例子。要么出现一个具有有限基本群的群，要么提出一个证明它们不存在的想法。无论是哪种情况，该计划都将为学生提供全面的研究体验感。

This award supports a program of Research Experiences for Undergraduates focusing on problems arising in the mathematical theory of algebraic topology. Six students will join with the principal investigator in a nine-week activity designed to develop and implement algorithms to search for a certain kind of topological construct called a cell complex. Cell complexes arise in the study of topological spaces that are higher- dimensional versions of the sphere, torus, polyhedra, etc. One constructs cell complexes (also known as CW-complexes) from elementary building blocks by combining cells together in prescribed ways. Questions about cell complexes translate into questions about algebraic objects, Abelian groups, which can be represented as matrices. Matrices are ideal tools for representing matrices and the operations between them. The complexes of interest are those called nilpotent. It is difficult to see how to put cells together to get nilpotent complexes, but computers - properly programmed - can construct multitudes of examples. This particular project seeks to answer a basic unsolved problem about the existence of finite three- dimensional nilpotent complexes. The program first introduces the students to topological concepts not normally encountered in undergraduate courses. Then, exploiting results of students discovered on prior REU projects, the team will seek to improve existing algorithms and codes to increase their speed and scope. The final tasks will be to find new and more complex examples. Either a group with a finite fundamental group will turn up or else, an idea for a proof of their non-existence will be suggested. In either event, the program will provide of full measure of sense of a research experience to the students.