Cardinal Invariants and Sets of Reals

这些研究项目涉及集合论的真实的线，一部分描述集理论。 主要研究真实的数集合的各种理想。连续统的基数特征和相关的小集合族是本研究的核心。 具体而言，他侧重于问题的一致性，各种推广的博雷尔猜想，声明断言，家庭的问题集完全由可数集。 他还研究了两个经典的小概念之间的二重性的程度：测量零和第一类。这个问题涉及到在测度概念如强测度零和泛测度零与它们的第一类类似物之间寻找相似之处。对真实的数的结构的研究标志着集合论的起源，并且自上个世纪开始以来一直是系统研究的对象。 测度和范畴的概念已经被严格地研究了一百多年，并在现代数学的许多领域得到了成功的应用。 Bartoszynski在这一领域中研究了几个问题，这些问题可能会产生数学定理的肯定答案，也可能会被证明与集合论的标准公理无关。 肯定的答案涉及有限和无限组合学的新结果，并具有超越传统集合论到实分析、测度论和拓扑学的应用。 另一方面，独立的结果，特别是那些使用强迫，有应用集理论本身。

These research projects concern the set theory of the real line,a part of descriptive set theory. The principal investigatorstudies various ideals of sets of real numbers. Cardinalcharacteristics of the continuum and the associated families ofsmall sets are central to this research. Specifically, hefocuses on problems concerning the consistency of variousgeneralizations of the Borel Conjecture, a statement assertingthat the families of sets in question consist entirely ofcountable sets. He also studies the extent of the dualitybetween the two classical notions of smallness: measure zero andthe first category. This problem concerns finding parallelsbetween the measure concepts such as strong measure zero anduniversal measure zero, and their first category analogs.The study of the structure of the real numbers marks the originsof set theory and has been the object of systematic researchsince the beginning of the last century. Concepts of measure andcategory have been studied rigorously for about one hundredyears, and have been successfully used in many areas of modernmathematics. Bartoszynski pursues several problems in this area.These problems may yield positive answers that are theorems ofmathematics or they may turn out to be independent from thestandard axioms of set theory. The positive answers involve newresults in both finite and infinite combinatorics and haveapplications reaching beyond traditional set theory to realanalysis, measure theory, and topology. On the other hand, theindependence results, particularly ones using forcing, haveapplications within set theory itself.