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.
