The bicycle problem that nearly broke mathematics in Nature News & Comment is a badly titled (click-bait) article that talks about one person who contributed to the development of the differential equations that accurately describe bicycle balancing (which has been incorrectly or incompletely described many times in the physics and engineering literature).
The one-line summary of the article is pretty accurate:
Jim Papadopoulos has spent a lifetime pondering the maths of bikes in motion. Now his work has found fresh momentum.
There is nothing in the article giving any indication that the equations Papadopoulos derived provided any stress to mathematics. The problem, as in many physics problems, is all in deciding what needs to be included in the model to get the best compromise between the tractability of the model and its accuracy. So far as I can tell from the vague descriptions in the article, the equations themselves are pretty much standard PDEs.
Unfortunately, the article does not give the equations themselves, so this article is particularly disappointing. It is People article, not a science article.
The article did give one prediction from the equations that showed their worth: it is possible to design a rideable bike with no gyroscopic balancing and negative trail, which would be inherently unstable in previous, simpler models. The trick is to move the center of gravity far enough forward to be ahead of the steering axis. Supposedly, such a bike has been built [Kooijman, J. D., G. Meijaard, J. P., Papadopoulos, J. M., Ruina, A., Schwab, A. L. A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects Science 3(32), 339–342 (2011) http://dx.doi.org/10.1126/science.1201959], but that article is hidden behind the Science paywall, so you’ll need to go to a university library to access it.
The supplementary material for the Science article is where the equations are presented and explained.