A countable group G is said to be W*-superrigid if G can be entirely recovered from its ambient group von Neumann algebra L(G). I will present a series of joint works with Milan Donvil in which we establish new degrees of W*-superrigidity: isomorphisms may be replaced by virtual isomorphisms expressed by finite index bimodules, the group von Neumann algebra may be twisted by a 2-cocycle, the group G might have infinite center, or we may enlarge the category of discrete groups to the broader class of discrete quantum groups.

A non-local game involves two non-communicating players who cooperatively play to give winning pairs of answers to questions posed by an external referee. Non-local games provide a convenient framework for exhibiting quantum supremacy in accomplishing certain tasks and have become increasingly useful in quantum information theory, mathematics, computer science, and physics in recent years. Within mathematics, non-local games have deep connections with the field of operator algebras, group theory, graph theory, and combinatorics. In this talk, I will provide an introduction to the theory of non-local games and quantum correlation classes and show their connections to different branches of mathematics. We will discuss how entanglement-assisted strategies for non-local games may be interpreted and studied using tools from operator algebras, group theory, and combinatorics. I will then present a general framework of non-local games involving quantum questions and answers.