Algebra in Stone-Cech Compactifications and its Combinatorial Applications

主要研究离散半群的石-切紧化代数结构及其组合应用问题。有些问题虽然很容易回答，但却是出了名的难。例如，是否存在从正整数的stone - ech紧化到其余数的非平凡连续同态？同样地，是否存在余数的有限子半群，其元素不都是幂等的？在这种代数结构的应用中，得到了某些无限矩阵是象分割规则的结果。希望在确定无穷矩阵是象分割正则的这一难题上取得进展。研究离散半群的stone - cech紧化的代数结构具有重要的意义，主要有两个原因。首先，许多悬而未决的问题都很简单，也很自然。其次，经验表明，关于这种代数结构的信息在数学领域有重要的应用，即拉姆齐理论。第一个这样的应用是1975年由Fred Galvin和Steven Glazer对“有限和定理”的代数证明，该定理断言，每当正整数被分成有限多个类时，其中一个类包含一个序列及其所有有限多个不同项的和。在这个初始应用程序之后，发现了许多其他应用程序。例如，范德瓦尔登定理（该定理断言，每当正整数被分成有限多类时，其中一类必须包含任意长的等差数列）的非常简单的证明及其一些扩展已经被发现。首席研究员打算和他的一些学生继续研究这个代数结构及其应用。

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.