There is interplay among the three topics of "Hopf Algebras, Renormalization and Noncommutative Geometry" by Connes and Kreimer (1998), which are of continuing interest as illustrated by Kreimer and Yeats' "Diffeomorphisms of quantum fields" (2017), Yeats' "A Combinatorial Perspective on Quantum Field Theory" (2017), and Balduf's "Perturbation theory of transformed quantum fields" (2019). (Trees and vector fields play integral roles.)

There is interplay among the three topics of "Hopf Algebras, Renormalization and Noncommutative Geometry" by Connes and Kreimer (1998), which are of continuing interest as illustrated by Kreimer and Yeats' "Diffeomorphisms of quantum fields" (2017), Yeats' "A Combinatorial Perspective on Quantum Field Theory" (2017), and Balduf's "Perturbation theory of transformed quantum fields" (2019). (Trees and vector fields play integral roles.)

Edit Dec. 18, 2021:

Interesting applications of NC geometry to investigations of Monstrous Moonshine can be found via the workshop synopsis "Novel approaches to the finite simple groups" by John McKay and Roland Friedrich (2012):

As an effect of the early preparation of all attendees, several conjectures previously made, in particular by J. McKay, could in fact be turned into concrete research plans during the time in Banff, one example being the application of non-commutative geometry and the Bost-Connes system to Monstrous Moonshine and replicability.

(Anyone have a copy or link to Jorge Plazas' roadmap mentioned in the synopsis?)

There are connections among noncommutative free probability theory, enumerative combinatorics (e.g. noncrossing partitions), random matrix models, NC geometry, and Monstrous Moonshine as well. See, e.g., the He and Jejalla ref in OEIS A134264.