  The Connes Embedding Problem (CEP) is a problem in the theory of tracial von
Neumann algebras and asks whether or not every tracial von Neumann algebra
embeds into an ultrapower of the hyperfinite II$_1$ factor. The CEP has had
interactions with a wide variety of areas of mathematics, including C*-algebra
theory, geometric group theory, free probability, and noncommutative real
algebraic geometry (to name a few). After remaining open for over 40 years, a
negative solution was recently obtained as a corollary of a landmark result in
quantum complexity theory known as $\operatorname{MIP}^*=\operatorname{RE}$. In
these notes, we introduce all of the background material necessary to
understand the proof of the negative solution of the CEP from
$\operatorname{MIP}^*=\operatorname{RE}$. In fact, we outline two such proofs,
one following the "traditional" route that goes via Kirchberg's QWEP problem in
C*-algebra theory and Tsirelson's problem in quantum information theory and a
second that uses basic ideas from logic.
