Terence Tao on Hard Problems, Navier-Stokes, and the Future of AI with Liquid Computers

Terence Tao, widely regarded as one of history's greatest mathematicians, delves into the nature of challenging research-level problems, beginning with the Kaya problem. This seemingly simple puzzle, concerning the minimal area required to turn a needle, serves as a gateway to understanding complex connections across partial differential equations, number theory, and geometry. Tao explains how the problem's evolution from 2D to 3D scenarios, such as rotating a telescope in minimal volume, illuminates fundamental concepts in wave propagation, including the potential for wave concentration and the formation of mathematical singularities in physical systems.

The discussion then transitions to the formidable Navier-Stokes regularity problem, one of the Clay Millennium Prize problems. This challenge seeks to determine if smooth fluid flows can ever develop singularities (infinite velocities) in finite time. Tao emphasizes the unique rigor of mathematics, where absolute certainty (100% proof) is paramount, contrasting with the statistical likelihood often accepted in other scientific fields. He illustrates the profound difficulty of proving global regularity by invoking the concept of "Maxwell's demon," where extremely improbable but mathematically possible configurations could theoretically lead to blow-up, despite not being observed in everyday fluid dynamics.

Tao elaborates on his innovative approach to this problem, where he engineered a finite-time blow-up in an "averaged" three-dimensional Navier-Stokes equation by strategically modifying the laws of physics. This work, while not a direct solution to the original problem, provides a crucial "obstruction," ruling out certain classes of proof techniques for the true Navier-Stokes equations and guiding future research efforts. A central insight from this line of inquiry is the distinction of "supercriticality," a condition where nonlinear transport terms overpower linear dissipation terms at small scales, rendering systems like fluid dynamics (and weather prediction) inherently unpredictable beyond short time horizons, unlike more stable "critical" or "subcritical" equations.

Perhaps the most imaginative aspect of Tao's work involves the conceptualization of a "liquid computer" or "water punk" Turing machine. He describes a theoretical framework where, through carefully programmed delays and nonlinear interactions, one could construct a self-replicating fluid machine. This machine would transfer its energy from a larger configuration to a smaller, faster-operating copy, thereby demonstrating a finite-time blow-up. While acknowledged as a theoretical "pipe dream," this thought experiment powerfully connects fluid dynamics to foundational concepts in computation theory, such as the halting problem and Von Neumann machines, and highlights mathematics' unparalleled ability to uncover deep, unifying connections between seemingly disparate scientific and conceptual domains, akin to the emergent complexity seen in cellular automata like Conway's Game of Life.

What's really interesting are the problems just at the on the boundary between what we can do relatively easily and what are hopeless.

So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, comics.

There's a famous unsolved problem called the Navia Stokes regularity problem.

This is what distinguishes mathematicians from pretty much everybody else like it If something holds 99.99% of the time, um that's good enough for most, you know, uh for for most things, but mathematicians are one of the few people who really care about whether every like 100% really 100% of all um situations are covered by by um yeah, so most fluid most of the time um water that does not blow up.

Short answer is Maxwell's demon.

So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation...

So this in principle would create a blow up uh for the actual Navia Stokes and this is what I managed to accomplish for this average Navia Stokes.

The thing about mathematics is that it's really good at um spotting connections between what you think of what you might think of as completely different um problems.

But if the mathematical form is the same you you can you you can you can draw a connection.

I mean that's a key qualitative feature that distinguishes some equations for being sort of nice and predictable and you know like like planetary motion and I mean there are certain equations that that you can predict for millions of years and or thousands at least. Again, it's not really a problem, but but there's a reason why we can't predict the weather past 2 weeks into the future because it's a super critical equation.