Mathematical Sciences: Exploration in Algebraic Topology

该奖项支持一项研究经验计划， 本科生专注于数学中出现的问题 代数拓扑理论 六名学生将参加 一项为期九周的活动的首席研究员， 开发和实现算法来搜索某种 一个叫做细胞复合体的拓扑结构。 细胞复合物 出现在拓扑空间的研究中， 三维版本的球体，环面，多面体等。 构建细胞复合物（也称为CW复合物）， 通过将细胞组合在一起， 规定的方式。 关于细胞复合体的问题 关于代数对象的问题，阿贝尔群，它可以是 表示为矩阵。 矩阵是理想的工具， 表示矩阵和它们之间的运算。 的 感兴趣的复形是那些称为幂零的复形。 是 很难看出如何把细胞放在一起， 复杂，但计算机-适当编程-可以构建 很多例子。 这个特别的项目试图回答 一个基本的未解决的问题，存在有限的三个- 维幂零复形 该计划首先向学生介绍拓扑 在本科课程中通常不会遇到的概念。 然后，利用先验REU发现的学生结果， 项目，该团队将寻求改进现有的算法， 代码以提高其速度和范围。 最后的任务是 寻找新的更复杂的例子 要么是一个 有限基本群会出现，否则， 将提出它们不存在的证据。 无论哪种情况， 该计划将提供充分措施意义上的研究 经验给学生。

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.