Mathematical Sciences: Games, partitions and harmless forcing extensions

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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.