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.
