The Navier-Stokes equations are sophisticated nonlinear equations that are challenging to solve, particularly in three dimensions where singularities can form.

The Navier-Stokes equations govern fluid flow for incompressible fluids like water, and the Navier-Stokes regularity problem asks whether a smooth velocity field can ever concentrate to the point of becoming infinite, which would indicate a singularity.

The phenomenon of supercriticality versus criticality and subcriticality is a key qualitative feature that distinguishes equations that are predictable from those that are not, such as in fluid dynamics.

Grigori Perelman transformed the supercritical Navier-Stokes problem into a critical problem by introducing concepts like reduced volume and permanence entropy, which simplified the analysis of singularities.

Mathematicians care about whether something holds true 100% of the time, unlike most fields where 99.99% suffices. This is crucial in fluid dynamics, where we need to understand if there are special initial states that could lead to blow-up scenarios.

Maxwell's equations are conformally invariant, meaning they depend on the space-time structure independent of scaling.

Through engineering, we can create clever constructions that lead to interesting behaviors in fluid dynamics, similar to how glider guns and gates operate in the Game of Life.

The Clay Foundation offers a million-dollar prize for solving one of the seven Millennium Prize Problems, one of which is the Navier-Stokes problem, and only one of these problems has been solved so far.

The Game of Life, a cellular automaton, demonstrates how simple local rules can lead to complex emergent behavior, similar to how fluid dynamics can exhibit complex structures under certain conditions.