Mathematical Sciences: Infinitary Combinatorics and Social Choice

泰勒的调查将全部集中在 组合集合论，但要考虑的问题 有着不同的灵感 也就是说， 第一个被讨论的群体来自于人们认为是 传统的无穷组合，而那些第二 数学政治学分支中的一个群体 由诺贝尔奖获得者， 肯尼斯·埃罗 传统无穷大中的考虑 组合学将开始与具体的问题， 在真实的直线上的可定义图形。 其结果是希望 已经证明，对于闭集， 由M。福尔曼和调查员;期望的扩展 可以同时概括出两种优雅的 已知的结果。 更一般的调查将在 分区理论的领域，主要涉及一些希望 最近与A. Blass和P. Erdos。 阿罗的不可能性定理可以被看作是一个 超积构造的优美刻画 适用于线性（或预线性）排序。 （自 有限集上的超滤子是主要的，这产生了 “独裁者”与阿罗定理联系在一起）。 本 社会选择理论的研究将集中在 研究人员最近工作推广阿罗定理， 有限语境和超积语境。

Taylor's investigations will all be in the area of combinatorial set theory, but the questions to be considered have rather different inspirations. That is, those of the first group to be discussed arise from what one thinks of as traditional infinitary combinatorics, while those of the second group arise from the subarea of mathematical political science launched into prominence by the Nobel Prize winning work of Kenneth Arrow. The considerations within traditional infinitary combinatorics will begin with a specific question about definable graphs on the real line. The result it is hoped to establish for all Borel sets has been shown to hold for closed sets by M. Foreman and the investigator; the desired extension would provide a simultaneous generalization of two elegant known results. The more general investigations will be in the area of partition theory and primarily involve some hoped for extensions of recent joint work with A. Blass and P. Erdos. Arrow's Impossibility Theorem can be regarded as an elegant characterization of the ultraproduct construction applied to linear (or pre-linear) orderings. (Since ultrafilters on a finite set are principal, this yields the "dictator" one associates with Arrow's theorem). The instant investigations in social choice theory will focus on the investigator's recent work generalizing Arrow's theorem in both the finite context and that of ultra-products.