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.

研究人员正在进行一系列的研究。 理论，描述集合论，拓扑学和分析。 的一部分 本研究将由研究者单独进行， 部分将联合进行。 一条研究主线， 杰克逊继续发展结构理论， 模型L（R）假设确定性。 这包括推动电流 理论进一步，以及完善理论在投影 程度. 杰克逊还将致力于解决其他领域的问题 理论 调查人员将共同努力解决 描述性集合论、等价关系理论和 问题集理论和拓扑性质。 一些 例子包括确定性转移问题在奇数 水平，关于可度量化连续统的问题 线性Hausdorff测度，以及上自然范数的存在性 K（X），复度量空间的n维核。 莫尔丁 本文还将探讨随机理论中的一些问题 同胚，例如：对于一个自然的方法， 圆同胚，几乎所有的单位同胚 圆有周期轨道吗 有没有自然的方法， 产生具有无理旋转数的同胚 考虑到真实的数可以被认为是 普通数行小学算术，令人吃惊 有多少结构可以强加给他们，以及如何复杂的， 关于这个结构的问题。 描述集合 理论是用所有方法解决这些问题的理论 现代数理逻辑的机器。 例如标准 这个理论的结构是集合的Borel层次，包括 两个集合族的无限序列，Pi集合和 Sigma集合，归纳定义，因此 复杂性 该理论的一个明显的应用是 在这个层次中精确地定位一个特定的集合， 分析，比如说一个函数在其上的一组点， 可微的 事实上，它是一个典型的和经典的定理， 描述性集合论（由于Mazurkiewicz），任何这样的集合， 可微性属于Borel方程中的Pi-one-one族 等级制度。 调查人员的一个主要关切是 这个特征，即使逻辑与问题相关的问题。 更广阔的数学世界