Why was Connes' embedding conjecture refuted and there are still no known non-hyperlinear groups?

Michael Chapman examines the gap between the refutation of Connes’ embedding conjecture by MIP*=RE and the construction of non-hyperlinear groups. The talk introduces non-local games, their values and Tsirelson’s problem, then associates a finitely presented *-algebra to a synchronous game. It explains how representations of this algebra relate to different game values, and why the relevant von Neumann closure cannot embed into an ultraproduct of the hyperfinite II₁ factor. A route to a non-hyperlinear group would require an appropriate counterexample algebra to be a complex group ring. Chapman introduces linear constraint system games, their associated game groups, and explains how the MIP*=RE construction differs from such a game.