Mathematical Sciences: Infinitary Combinatorics and Social Choice

泰勒的研究都将在组合集理论的领域，但要考虑的问题有相当不同的启发。也就是说，我们要讨论的第一组问题源于人们所认为的传统无穷组合，而第二组问题则源于因诺贝尔奖得主肯尼斯·阿罗（Kenneth Arrow）的工作而变得突出的数学政治学分支领域。传统的无穷大组合学将从一个关于实线上可定义图的特定问题开始。希望对所有Borel集建立的结果已被M. Foreman和研究者证明对闭集成立；期望的扩展将提供两个优雅的已知结果的同时泛化。更一般的研究将在分割理论领域，主要涉及与A. Blass和P. Erdos最近的联合工作的一些希望的扩展。阿罗不可能定理可以看作是应用于线性（或预线性）排序的超积构造的一个优雅的表征。（由于有限集合上的超过滤器是主要的，这就产生了与阿罗定理相关的“独裁者”）。社会选择理论的即时调查将集中在研究者最近在有限和超产品的情况下推广阿罗定理的工作。

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.