Algebra in Stone-Cech Compactifications and its Combinatorial Applications

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The principal investigator proposes to study questionsdealing with the algebraic structure of the Stone-Cechcompactification of a discrete semigroup and the combinatorialapplications of that structure. Some of the questions,while easy to state, are notoriously difficult. For example, is there a nontrivial continuous homomorphismfrom the Stone-Cech compactification of the positiveintegers to its remainder? Equivalently, is there afinite subsemigroup of the remainder whose elementsare not all idempotents? Among the applicationsof this algebraic structure have been results establishingthat certain infinite matrices are image partition regular.It is hoped that progress will be made on the difficultquestion of characterizing which infinite matrices areimage partition regular. The study of the algebraic structure of the Stone-Cechcompactification of a discrete semigroup is importantfor two main reasons. First, many of the questions thatremain open are very simple to state and quite natural.Secondly, experience has shown that information aboutthis algebraic structure has significant applicationsto 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 SumsTheorem", which asserts that whenever the positiveintegers are divided into finitely many classes, oneof these classes contains a sequence and all of itssums of finitely many distinct terms. After thisinitial application many other applications have been found. For examplevery easy proofs of van der Waerden's Theorem (which asserts thatwhenever the positive integers are divided into finitelymany classes, one of these must contain arbitrarilylong arithmetic progressions) and some of itsextensions have been found. The principal investigatorintends, with some of his students, to continuethe investigation of this algebraic structureand its applications.