Algebra in Stone-Cech Compactifications and its Combinatorial Applications

主要研究者建议研究离散半群的Stone-Cech紧化的代数结构以及该结构的组合应用。 有些问题虽然容易陈述，但却出了名的难。 例如，从正整数的Stone-Cech紧化到它的余项是否存在非平凡连续同态？等价地，是否存在元素不全是幂等元的剩余半群的有限子半群？ 在这种代数结构的应用中，已经有结果证明某些无限矩阵是象分划正则的，希望在刻画哪些无限矩阵是象分划正则的这一难题上能取得进展。 研究离散半群的Stone-Cech紧化的代数结构有两个重要原因。 首先，许多尚未解决的问题陈述起来非常简单，而且非常自然。其次，经验表明，有关这种代数结构的信息在数学领域有着重要的应用，即拉姆齐理论。第一个这样的应用是弗雷德·加尔文（Fred Galvin）和史蒂文·格雷泽（Steven Glazer）在1975年对“有限和定理”的代数证明，它断言，无论何时正整数被分成1000个类，这些类中的一个包含一个序列及其所有1000个不同项的和。在这最初的应用程序许多其他的应用程序已被发现。例如，非常容易证明货车德瓦尔登定理（它断言，每当正整数分为有限多类，其中一个必须包含任意长的算术级数）和它的一些扩展已被发现。校长打算，与他的一些学生，继续调查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.