Mathematical Sciences: Problem in Set-Theoretic Topology

9322613尼科斯这个项目针对的是拓扑学家和集合论者之间的互动似乎可能导致这两个领域取得重大进展的一些问题领域。在过去的二十年里，集合论拓扑学在这种相互作用下得到了爆炸性的发展，也激发了集合论的大量研究成果。然而，在过去的几年里，结果的流动已经显示出减弱的迹象，可能需要发展新的集合论公理和技术来保持这一拓扑学领域的良好健康。将特别强调的问题领域是遗传正规性；可数紧空间的结构和乘积；以及不可度量流形理论。集合论拓扑学比其他拓扑学更是一门基础性学科，主要涉及非常抽象的空间以及集合论公理变体的采用对拓扑学的影响。其次，它不仅在集合论和拓扑学的其他方面有应用，而且在分析和代数，特别是布尔代数理论中也有应用。尼科斯的研究强调了所有这些方面。例如，他对可以施加在长直线上的微分结构(即光滑化)及其相关的2维切向流形进行了广泛的研究，为那些希望了解更熟悉但更高维的流形的非平凡光滑化的人提供了有价值的启发式方法。(这是今天一个非常重要的课题，特别是在与理论物理相互作用很强的4个维度上。)长线的一些平滑需要集合论中通常公理之外的公理。自从十多年前，尼科斯发现关于摩尔空间的一个著名的未经证明的猜想实际上独立于通常的集合论基本公理S以来，他一直是集合论拓扑学中使用这些公理的领导者。此外，这一猜想被证明与康托连续统假说不一致，康托连续统假说是一个更熟悉的命题，如果必须做出选择，大多数在职数学家很可能会选择。这一结果暗示了最佳集合论拓扑的味道。***

9322613 Nyikos This project is aimed at some problem areas where interaction between topologists and set theorists seems likely to lead to significant progress in both areas. Set-theoretic topology has grown explosively in the last two decades as a result of such interaction, stimulating much fruitful research in set theory as well. In the last few years, however, the flow of results has shown signs of abating, and the development of new set-theoretic axioms and techniques will probably be needed to keep this area of topology in good health. The problem areas on which particular stress will be laid are hereditary normality; the structure and products of countably compact spaces; and the theory of nonmetrizable manifolds. Set-theoretic topology is more of a foundational subject than the rest of topology, primarily concerning itself with very abstract spaces and with the impact on topology of the adoption of variants of the axioms of set theory. Secondarily, it has applications not only to set theory and the rest of topology, but also to analysis and algebra, particularly the theory of Boolean algebras. Nyikos's research has emphasized all these aspects. For example, he has done an extensive study of the differential structures (i.e. smoothings) that can be imposed on the long line, and of the associated 2-dimensional manifold of its tangent vectors, providing valuable heuristics for those wishing to understand nontrivial smoothings of more familiar but higher-dimensional manifolds. (This is a subject of major importance today, especially in 4 dimensions, where the interplay with theoretical physics is strong.) Some of the smoothings of the long line required axioms outside the usual axioms of set theory. Nyikos has been a leader in the use of these axioms in set-theoretic topology ever since his involvement over a decade ago in the discovery that a famous unproved conjecture concerning Moore spaces was actually independent of the usual basic axiom s of set theory. Moreover, this conjecture was shown to be inconsistent with the Cantor Continuum Hypothesis, a far more familiar proposition that most working mathematicians would be likely to choose if they had to make a choice. This result suggests the flavor of the best set-theoretic topology. ***