Algebra in Stone-Cech Compactifications and its Combinatorial Applications

Howard University, Washington DC

Investigators

Abstract

The principal investigator proposes to study questions dealing with the algebraic structure of the Stone-Cech compactification of a discrete semigroup and the combinatorial applications of that structure. Some of the questions, while easy to state, are notoriously difficult. For example, is there a nontrivial continuous homomorphism from the Stone-Cech compactification of the positive integers to its remainder? Equivalently, is there a finite subsemigroup of the remainder whose elements are not all idempotents? Among the applications of this algebraic structure have been results establishing that certain infinite matrices are image partition regular. It is hoped that progress will be made on the difficult question of characterizing which infinite matrices are image partition regular. The study of the algebraic structure of the Stone-Cech compactification of a discrete semigroup is important for two main reasons. First, many of the questions that remain open are very simple to state and quite natural. Secondly, experience has shown that information about this algebraic structure has significant applications to the field of mathematics known as Ramsey Theory. The first such application was the algebraic proof in 1975 by Fred Galvin and Steven Glazer of the "Finite Sums Theorem", which asserts that whenever the positive integers are divided into finitely many classes, one of these classes contains a sequence and all of its sums of finitely many distinct terms. After this initial application many other applications have been found. For example very easy proofs of van der Waerden's Theorem (which asserts that whenever the positive integers are divided into finitely many classes, one of these must contain arbitrarily long arithmetic progressions) and some of its extensions have been found. The principal investigator intends, with some of his students, to continue the investigation of this algebraic structure and its applications.