Mathematical Sciences: Problem in Set-Theoretic Topology

小行星9322613 这个项目的目的是在一些问题领域，拓扑学家和集理论家之间的相互作用似乎有可能导致这两个领域的重大进展。 由于这种相互作用，集合论拓扑在过去的二十年里得到了爆炸性的发展，也激发了集合论中许多富有成果的研究。 在过去的几年里，然而，流动的结果已经显示出减弱的迹象，和新的集合理论公理和技术的发展可能需要保持这一领域的拓扑结构健康。 问题领域，特别强调将奠定遗传正常;结构和产品的可数紧空间;和理论的nonmetrizable流形。 集合论拓扑学比拓扑学的其他学科更像是一个基础学科，主要涉及非常抽象的空间，以及集合论公理的变体对拓扑学的影响。 其次，它不仅应用于集合论和拓扑学的其余部分，而且还应用于分析和代数，特别是布尔代数理论。 尼科斯的研究强调了所有这些方面。 例如，他做了广泛的研究微分结构（即平滑），可以施加在长的线，以及相关的2维流形的切向量，提供有价值的数学为那些希望了解非平凡平滑更熟悉，但高维流形。(This是一个非常重要的课题，特别是在4维空间中，与理论物理学的相互作用很强。 这条长长的线的某些平滑需要集合论通常公理之外的公理。 Nyikos一直是一个领导者在使用这些公理集理论拓扑自从他参与了十多年前的发现，一个著名的未经证明的猜想有关摩尔空间实际上是独立的通常基本公理集理论。 此外，这个猜想被证明与康托连续统假设不一致，康托连续统假设是一个更为熟悉的命题，如果必须做出选择，大多数数学家可能会选择。 这个结果表明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. ***