Dynamical Systems Seminar

In this talk, we introduce Schmidt's game, which in recent years has been used extensively to prove theorems in number theory. We will contrast Schmidt's game with the Banach-Mazur game to motivate investigation into certain set theoretic properties of the game. The central concept of this talk will be the property of a game that one of the two players has a winning strategy (determinacy). Under certain strong hypotheses, we present a proof that the Banach-Mazur game is determined (which has implications for subsets of the real line, for instance). In light of this proof, we will examine how Schmidt's game differs and sketch a proof of a partial result of the determinacy of Schmidt's game.