Operator Ideal membership of Quantised derivatives

Abstract: 

In 1994 Alain Connes introduced a ``quantised" calculus, based on operator theoretic expressions which play roles analogous to derivatives and differentials in ordinary calculus. Quantised calculus in one variable relies on results of V. V. Peller from the 1980s about operator ideal membership of commutators of multiplication operators with the Hilbert transform. I will discuss the proofs Peller's results.