Connes embedding conjecture

The Connes embedding conjecture is a major open question in mathematics and operator algebras. It asks whether every infinite-dimensional, type II₁ factor (a kind of mathematical structure describing complex systems of operators) can be approximated arbitrarily well by finite-dimensional matrix algebras. In simpler terms, it questions if these complex infinite structures can be represented as limits of simpler, well-understood finite systems. A positive answer would have far-reaching implications across mathematics and quantum physics, linking areas like quantum information theory, free probability, and the fundamental understanding of infinite versus finite structures.