Explore the fascinating world of operator algebras and quantum correlations in this 45-minute lecture by Thomas Vidick for the International Mathematical Union. Delve into the Connes embedding problem and its equivalent reformulations, including Tsirelson's problem. Discover the groundbreaking result MIP^{∗} = RE and its implications for quantum information theory. Learn about the basic approach of separating convex sets and the undecidability of quantum value. Examine key ingredients such as the rigidity of quantum correlations and the role of Probabilistically Checkable Proofs (PCPs) and Multi-prover Interactive Proofs (MIPs) in this negative resolution. Gain insights into the birth of operator algebras and their significance in modern mathematics and theoretical physics.