I will discuss some recent development in understanding connections between three major subjects in modern set theory: large cardinals, forcing axioms, and determinacy. Large cardinals form a linear hierarchy of axioms extending the standard axioms of Zermelo-Fraenkel with choice (ZFC) and every known, natural theory can be interpreted by one of the large cardinal axioms. Forcing axioms form another hierarchy generalizing the Baire Category Theorem and are widely used in many applications in other fields such as functional analysis, topology, and more recently $C^*$-algebra. The Axiom of Determinacy and its generalizations postulate that complicated sets of reals have nice regularity properties and are incompatible with the Axiom of Choice. These axioms have been studied extensively by descriptive set theorists and inner model theorists. I will particularly focus on recent works of Woodin, Steel, Sargsyan, and myself on constructing models of determinacy from large cardinals and from forcing axioms such as the proper forcing axiom (PFA) and Martin's maximum (MM). These works make significant progress on the Inner Model Problem, one of the most central problems in set theory.
The talk is mostly accessible to non-logicians; there will be plenty of backgrounds, historical contexts given to motivate the topics.