Long Games and Strengthenings of Determinacy

In a perfect information two-player game without draws, one of the players al- ways has a winning strategy; in this case, the game is said to be determined. This project is motivated by a natural yet profound question: which games of infinite length are determined? This question is one of the fundamental questions in set theory the mathematical study of infinity. This research explores the determinacy infinite length games from a set theoretic perspective to deepen our understanding of its connection with large cardinals. We focus on two central themes: (I) the determinacy of games of variable countable length, where the total length is always countable but depends on the players moves; and (II) models of strengthening of the Axiom of Determinacy with struc- tures that cannot be coded into sets of reals, which naturally arise in the context of long games. Inner model theory, which investigates canonical models with large cardinals, will play a central role in this research, and part of our motivation is to gain deeper insights into its underlying machinery. Over the past 15 years, the theory of hod mice has emerged and significantly advanced the field. The combination of long games and hod mice offers a promising framework for discovering new, natural determinacy models.