Descriptive set theory is a classical topic that dates back to the 19th century. It studies simply definable sets and functions on complete metric spaces, for instance Borel, analytic and projective sets. A typical question is whether sets of a certain complexity are Lebesgue measurable. There are subtle logical issues here, for instance, strong axioms of infinity beyond the axioms of set theory are necessary to show that all projective sets are Lebesgue measurable. In the context of descriptive set theory, it is useful to understand reals, or elements of some other complete metric space, as infinite words (of length N) whose entries are natural numbers. Measurability, and other similar properties, can then be formulated via infinite games, where each move is a natural number. This approach was used extensively by Kechris and others.
The objective of this project is to study the descriptive set theory of uncountable words from the viewpoints of set theory and higher computability. Spaces of such words are called generalised Baire spaces. The study of these spaces was initiated by Väänänen and has become a major research theme only in the last decade. Major challenges appear because of fundamental differences to the countable setting, caused by combinatorics of uncountable cardinals, for example the existence of Kurepa trees.
One approach links set theory with higher computability. We study uncountable words that can be detected by an algorithm. An analysis of countable recognisable words was carried out in work of Carl, Schlicht and Welch. Hamkins, Leahy and Groszek proved results on the related notion of implicitly definable sets. We aim to understand the precise nature of uncountable recognisable sets, using tools in inner model theory and large cardinals.
Another approach lies in applications of games of uncountable length to definable subsets of generalised Baire spaces. Previous results show that it is consistent that some natural games of uncountable length are determined, for instance the perfect set game and the Banach-Mazur game. We aim to study games for analogues to classical dichotomies and other important games such as Lipschitz games in the uncountable setting.
