Mathematical Sciences: Determinancy, Descriptive Set Theoretic Methods in Topology, Random Homeomorphisms

数学科学：决定性、拓扑中的描述集理论方法、随机同胚

• 批准号: 9207707
• 负责人: Stephen Jackson
• 金额: $ 4.4万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1995-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9207707&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Determinancy; Descriptive; Set

The investigators are engaged in ongoing research in set theory, descriptive set theory, topology, and analysis. Part of this research will be conducted by the investigators separately and part will be conducted jointly. A main line of research for Jackson continues the development of the structural theory for the model L(R) assuming determinacy. This includes pushing the current theory further, as well as refining the theory at the projective levels. Jackson will also work on problems in other areas of set theory. The investigators will work jointly on problems in descriptive set theory, the theory of equivalence relations, and problems both set-theoretic and topological in nature. Some examples include the determinacy transfer problem at the odd levels, questions about metrizable continua with sigma-finite linear Hausdorff measure, and the existence of natural norms on K(X), the n-dimensional kernel of a complex metric space. Mauldin will also investigate some problems in the theory of random homeomorphisms, for example: for one natural method for producing circle homeomorphisms, do almost all homeomorphisms of the unit circle have periodic orbits? Are there natural methods which produce homeomorphisms with irrational rotation numbers? Considering that the real numbers can be thought of as the ordinary number line of grade school arithmetic, it is surprising how much structure can be imposed upon them and how intricate the questions that can be asked about this structure. Descriptive set theory is the theory that addresses these questions with all the machinery of modern mathematical logic. For example, a standard construct of this theory is the Borel hierarchy of sets, consisting of two infinite sequences of families of sets, the Pi sets and the Sigma sets, defined inductively, and hence of increasing complexity. One of the obvious applications of the theory is to locate precisely in this hierarchy a particular set which arises in analysis, say the set of points at which a function is differentiable. In fact, it is a typical and classical theorem of descriptive set theory (due to Mazurkiewicz) that any such set of differentiability belongs to the family Pi-one-one in the Borel hierarchy. A central concern of the investigators are questions of this character, i.e. questions which make logic relevant to the wider world of mathematics.

研究人员从事集合理论，描述性集理论，拓扑和分析的持续研究。 这项研究的一部分将由研究人员分别进行，部分将共同进行。 杰克逊的主要研究延续了假设确定性的模型L（R）的结构理论的发展。 这包括进一步推动当前的理论，以及在投影级别上完善理论。 杰克逊还将在设定理论的其他领域中处理问题。 研究人员将共同研究描述性集理论，等价关系理论以及本质上的理论和拓扑问题的问题。 一些示例包括在奇数级别上的确定性转移问题，有关Sigma-Finite线性Hausdorff测量的可衡量连续性问题，以及K（x）上的自然规范的存在，这是复杂度量空间的n维内核。 毛丁还将研究随机同态理论中的一些问题，例如：对于产生圆的同态同态的一种自然方法，几乎​​所有单位圆的同态都具有定期轨道吗？ 是否有自然方法产生具有非理性旋转数字的同态形态？ 考虑到实数可以被认为是小学算术的普通数字线，这令人惊讶的是，可以将多少结构施加到它们上，以及有关该结构的问题。 描述性集理论是通过现代数学逻辑的所有机械来解决这些问题的理论。 例如，该理论的标准构建体是集合的Borel层次结构，由两个无限的集合家族，PI集和Sigma集序列组成，这些序列是归纳定义的，因此复杂性的增加。 该理论的明显应用之一是精确地定位在该层次结构中，该集合在分析中出现的特定集合，例如函数可区分的点集。 实际上，这是描述性集理论的典型和古典定理（由于Mazurkiewicz的原因），任何此类可区分性都属于Borel层次结构中的家族。 调查人员的主要关注点是这个角色的问题，即使逻辑与更广泛的数学世界相关的问题。