Connes Embedding Problem, Kirchberg's Conjecture and Tsirelson's Problem

The three problems referred to in the title originate in the theory of von Neumann algebras, C* algebras, and quantum information theory respectively. Each of them has been a deep long-standing open problem in its respective area. Surprisingly, the three problems are equivalent. Recently we gave a negative answer to all three problems by taking an unexpected path -- through complexity theory. The goal of these three lectures is to give a flavor of the resolution of these three problems, for a mathematical audience. In the first lecture I will introduce the problems described in the title and explain their equivalence. I will then formulate the approach, which goes through the careful design of families of linear functionals acting on "quantum correlation spaces" and derived from the theory of nonlocal games in quantum information theory.