Abstract: We shall survey some recent work on the existence of canonical inner models, also known as mice, satisfying large cardinal hypotheses. One key open problem here is the Mouse Set Conjecture: Assume AD+, and that there are no iteration strategies for mice with superstrong cardinals. Let x and y be reals such that x ε M.
Here AD+ is a certain technical strengthening of the Axiom of Determinacy. The conjecture is a basic test of our ability to construct mice which are correct at higher levels of logical complexity.
We shall describe some recent work related to the Mouse Set Conjecture. The author's work in the area leads to optimal consistency strength lower bounds for ADℝ (the axiom of determinacy for games on the reals), and related theories. The most significant recent progress is due to G. Sargsyan, and we shall attempt to describe it in general terms.