Kolmogorov–Arnold–Moser theorem

Water confined between two cylinders flows in smooth layers when the inner one rotates. But once the speed crosses a certain threshold, the layers crumple into toroidal rolls, like a stack of bagels. Physicists call these Taylor vortices: the flow loses stability and a regular pattern emerges.

In a strong magnetic field, electrons in a thin metal film move along circular orbits. When the film thickness is comparable to the size of such an orbit, the resistance starts to oscillate: these are Sondheimer oscillations, known since the mid-20th century as a purely geometric effect.

A quantum computer is like an orchestra where each musician plays their own part. If the conductor slips up, the music falls apart. In digital quantum simulations, the conductor's role is played by a sequence of simple operations, and errors accumulate like wrong notes.

Fourteen rubidium atoms lined up like beads on a string. Each atom is a tiny magnet that can point up or down. Together they follow strict rules, similar to a train schedule: everything is predictable, everything repeats. But introduce a barely noticeable change, and the schedule goes off the rails.

The orchestra plays the symphony of climate. Each violin, each wind instrument carries its own part: these are the Kolmogorov modes, the system's natural rhythms. Suddenly, a thunderclap bursts into the hall: a sharp jump, like a gust of wind or a downpour. Previously, we could only listen to the smooth sound, but we couldn't decipher such chords. In the new work, mathematicians have completed the score.