Mathematical Sciences: Exploration in Algebraic Topology

该奖项支持本科生研究经验项目，重点关注代数拓扑数学理论中出现的问题。六名学生将与首席研究员一起进行为期九周的活动，旨在开发和实现算法，以搜索某种称为细胞复合体的拓扑结构。细胞复合体出现在对球面、环面、多面体等高维拓扑空间的研究中。人们通过将细胞以规定的方式组合在一起，从基本的构建块构建细胞复合物（也称为cw复合物）。关于细胞复合体的问题转化为关于代数对象的问题，阿贝尔群，它可以表示为矩阵。矩阵是表示矩阵和它们之间的运算的理想工具。我们感兴趣的复合体被称为幂零复合体。很难看出如何将细胞组合在一起以得到零势复合体，但是计算机——适当地编程——可以构建大量的例子。这个特殊的项目试图回答一个基本的未解决的问题，即有限三维幂零复合体的存在性。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，团队将寻求改进现有的算法和代码，以提高其速度和范围。最后的任务将是找到新的和更复杂的例子。要么会出现一个具有有限基本群的群，要么会提出一个证明它们不存在的想法。无论哪种情况，该项目都将为学生提供充分的研究体验感。

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.