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

研究人员正在进行集论、描述集论、拓扑学和分析方面的研究。部分研究将由研究人员单独进行，部分将共同进行。Jackson的研究主线是继续发展假设确定性的L(R)模型的结构理论。这包括进一步推动当前的理论，以及在投影层面上完善理论。Jackson还将研究集合论其他领域的问题。研究人员将共同研究描述性集合论、等价关系理论以及集合论和拓扑性质的问题。一些例子包括奇阶上的确定性转移问题，关于具有有限线性Hausdorff测度的可度量连续体的问题，以及复度量空间的n维核K(X)上自然范数的存在性。Mauldin还将研究随机同胚理论中的一些问题，例如：对于产生圆同胚的一种自然方法，是否几乎所有单位圆的同胚都有周期轨道？是否存在产生无理数的同胚的自然方法？考虑到实数可以被认为是小学算术中的普通数轴，那么令人惊讶的是，它们可以被赋予如此多的结构，以及关于这种结构可以提出的问题是多么复杂。描述性集合理论是用现代数理逻辑的所有机制来解决这些问题的理论。例如，这个理论的一个标准构造是Borel集合层次，由两个集合族的无限序列组成，Pi集合和Sigma集合，归纳定义，因此越来越复杂。这个理论的一个明显的应用是精确地在这个层次中找到分析中出现的一个特定的集合，比如一个函数可微的点的集合。事实上，它是描述集合论（由Mazurkiewicz）的一个典型的经典定理，即任何这样的可微集合都属于Borel层次中的pi - 1 - 1族。研究者关注的中心问题是这种性质的问题，即使逻辑与更广阔的数学世界相关的问题。

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.