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，容易描述或分类，或者有一些“好”的属性;不规则的，更难以捉摸的。一个很好的例子是C（？0，1！）在某些Banach代数中：人们会认为连续的是正则的，不连续的是不规则的。正如上面的例子一样，在一个给定的范畴中，不规则对象的存在既不能被证明，也不能被反驳，至少只能借助于普遍接受的数学工具（如公理集合论中的形式化）。该项目的研究重点是三种方法，以允许消除某些不规则对象的方式扩展数学工具包：-确定性公理，它说没有不规则（即，不确定的）游戏的排序，-开放着色公理，其中说，某些不规则的分区不存在，-假设数学发生在某些“无害的强制扩展”的可构造的宇宙。这些方法的相对强度有待确定，并寻求可能的改进和新的应用。

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.