Hazewinkel wrote this article in 2005. Perhaps it's time for an update.
For example, updating item
34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of Lie-Butcher numerical methods for solving autonomous and non-autonomous ODE's (evolution equations) by , e.g., Ebrahimi-Fard, Hans Munthe-Kaas and their colleagues. See, e.g.,
i) "Butcher series: A story of rooted trees and numerical methods for evolution equations" by McLachlan, Modin, Munthe-Kaas, and Verdier
ii) "Lie-Butcher series, Geometry, Algebra and Computation" by Munthe-Kaas and K. Føllesdal
This is closely allied to item
81: Quantum theory
through the HA (combinatorial Faa di Bruno bialgebra, trees, Feynman diagrams) of renormalization to which Alain Connes, Christian Brouder, David Broadhurst, Dirk Kreimer, Kurush Ebrahimi-Fard, Hector Figueroa, Jose Gracia-Bondia, Hans Munthe-Kass, Loic Foissy, Karen Yeats, Paul-Hermann Balduf, et al. have contributed.
52: Convex polytopes (relabeled)
Hopf monoids have been introduced to explain the association of permutohedra and associahedra with compositional and multiplicative inversion and optimization:
"Hopf monoids and generalized permutahedra" by Marcelo Aguiar and Federico Ardila.
It would be motivational and useful in pursuing research in these topics if others would list some of their favorite interests under appropriate items and give associated authors and/or papers, particularly of an introductory nature.
Please feel free to note your own work (with no false modesty--if you are taking the time and effort to publish, you must feel it could be of interest to others, otherwise ...).