Science: After 125 years, math science makes a major advance in uniting math and physics

The few top notch math people on planet Earth, maybe 0.00001% of us (1 in 10 million), can see and think about things that most people, including me, cannot even articulate. I need experts to translate opaque math into American. SciAm reports about a major breakthrough in uniting math with physics (mildly edited):

Mathematicians Crack 125-Year-Old Problem, 

Unite Three Physics Theories

A breakthrough in Hilbert’s sixth problem is a major step in grounding physics in math

At the International Congress of Mathematicians at Sorbonne University in Paris, legendary mathematician David Hilbert presented 10 unsolved problems as ambitious guideposts for the 20th century. He later expanded his list to include 23 problems, and their influence on mathematical thought over the past 125 years cannot be overstated.

Hilbert’s sixth problem was one of the loftiest. He called for “axiomatizing” physics, or determining the bare minimum of mathematical assumptions behind all its theories. Broadly construed, it’s not clear that mathematical physicists could ever know if they had resolved this challenge.

In March mathematicians Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma of the University of Michigan posted a new paper to the preprint server arXiv.org that claims to have cracked one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in math and may open the door to analogous breakthroughs in other areas of physics.

In the paper, the researchers suggest they have figured out how to unify three physical theories that explain the motion of fluids. These theories govern a range of engineering applications from aircraft design to weather prediction—but until now, they rested on assumptions that hadn’t been rigorously proven. This breakthrough won’t change the theories themselves, but it mathematically justifies them and strengthens our confidence that the equations work in the way we think they do. 

Each theory differs in how much it zooms in on a flowing liquid or gas. At the microscopic level, fluids are composed of particles—little billiard balls bopping around and occasionally colliding—and Newton’s laws of motion work well to describe their trajectories.

But when you zoom out to consider the collective behavior of vast numbers of particles, the so-called mesoscopic level, it’s no longer convenient to model each one individually. In 1872 Austrian theoretical physicist Ludwig Boltzmann addressed this when he developed what became known as the Boltzmann equation. Instead of tracking the behavior of every particle, the equation considers the likely behavior of a typical particle.

Zoom out further, and you find yourself in the macroscopic world. Here we view fluids not as a collection of discrete particles but as a single continuous substance. At this level of analysis, a different suite of equations—the Euler and Navier-Stokes equations—accurately describe how fluids move and how their physical properties interrelate without recourse to particles at all.

This does not just pertain to the physics of fluid motion. It pertains to all of physics. All of it. The researchers put it like this in their arXiv.org manuscript:

Hilbert’s sixth problem. In his address to the International Congress of Mathematics in 1900, David Hilbert proposed a list of problems as challenges for the mathematics of the new century. Of those problems, the sixth problem asked for an axiomatic derivation of the laws of physics. In his description of the problem, Hilbert says: 

"The investigations on the foundations of geometry suggest the problem: To treat in the  same manner, by means of axioms, those physical sciences in which already today  mathematics plays an important part; in the first rank are the theory of probabilities and mechanics." 

Broadly interpreted, this problem can encompass all of modern mathematical physics.

The core of the math advance simplified into American 

In Fig. 1, math was applied independently to the atomic or microscopic scale of fluid flow on the left (Newtonian physics, or classical mechanics), and the next larger scale, mesoscopic (Boltzman's kinetics equations), in the middle to the macroscopic or large scale on the right (hydrodynamics equations). The three levels of analysis each describe the same underlying reality—how fluids flow. Three different sets of equations for three different sets of conditions. However, the math of those three sets of physics was not unified. 

What Deng, Hani and Ma did was find a way to unify all three sets of equations. Their tactic was simple. Well sort of, I guess. The three researchers built on decades of incremental progress with incomplete math subject to caveats or limitations. They got rid of the incomplete math with caveats//limitations and replaced it with free range, unrestricted math.

They did it in three steps. First, derive the macroscopic theory from the mesoscopic. Then derive the mesoscopic theory from the microscopic. Finally, they stitched the equations together in a single derivation of the macroscopic laws all the way from the microscopic ones.

Since this is a pure math breakthrough, it does not qualify for a Nobel Prize. But it might qualify for a top-level mathematics prize like the Fields Medal, Abel Prize, or Wolf Prize.

And not surprisingly, there are critics. Shan Gao, in a critical paper, argues that while the work is mathematically rigorous, it contains "two critical physical flaws" that prevent it from truly capturing the physics of dense fluids. Gao contends that "Hilbert's Sixth Problem remains open," suggesting the mathematical community has not reached consensus on the completeness of this particular solution.

Give it another ~6-12 months. Maybe by then, math geeks will come to a consensus or see the alleged (or other) flaws.

By Germaine: Not a mathematician