The invention of stable anti-Yetter-Drinfeld modules as coefficients of Hopf-cyclic cohomology by Hajac, Khalkhali, Rangipour, and Sommerhaeuser. The article "Hopf-cyclic homology and cohomology with coefficients" opened up a new area of research and introduced the term "Hopf-cyclic" that is now used as standard. Its main theorem links the anti-Yetter-Drinfeld property with the existence of the cyclic operator.
The discovery of a pairing between super-Lie-Rinehart homology with coefficients in traces and periodic cyclic homology, extending a previously known special case defined by means of an invariant trace (Connes-Moscovici) generalized sufficiently to express celebrated index pairing as an above homological pairing (On a pairing between super-Lie-Rinehart and periodic cyclic homology, Maszczyk, Comm. Math. Phys., 2006).
