Mathematical Sciences: Games, partitions and harmless forcing extensions

同一类别的数学对象通常有两种：规则的，易于描述或分类，或具有一些“好”的性质；以及不规则的，更难以捉摸的。一个很好的例子是C(？0，1！)上的同态。转化为一些Banach代数：人们会认为连续的是规则的，不连续的是不规则的。就像上面的例子一样，在给定的范畴中，不规则对象的存在既不能被证明，也不能被反驳，至少是借助于普遍接受的数学工具(如公理集合论中形式化的)。这个项目的研究集中在三种方法上，以一种允许消除某些不规则对象的方式来扩展数学工具包：-确定性公理，它说没有任何不规则的(即，未确定的)游戏，-开放着色公理，它说某些不规则的划分不存在，-假设数学发生在可构建的宇宙的某些“无害的强迫扩展”中。这些方法的相对强度有待确定，并寻求它们的可能改进和新的应用。

Mathematical objects of the same category often come in two varieties: regular ones, which are easy to describe or classify, or have some "nice" properties; and the irregular, more elusive ones. A good example are homomorphisms on C(?0, 1!) into some Banach algebra: one would think of the continuous ones as regular, the discontinuous ones as irregular. It happens (as in the above example) that the existence of irregular objects in a given category can neither be proved nor disproved, at least with the help of only universally accepted mathematical tools (as formalized in axiomatic set theory). The research of this project focuses on three methods to extend the toolkit of mathematics in a way that allows elimination of certain irregular objects: - the Axiom of Determinacy, which says that there are no irregular (i.e., undetermined) games of sorts, - the Open Coloring Axiom, which says that certain irregular partitions do not exist, - assumptions that mathematics takes place in certain "harmless forcing extensions" of the constructible universe. The relative strength of these methods is to be determined and possible improvements and new applications of them sought.