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Just as Richard's family began to dream about the future, the world of mathematics suddenly erupted in excitement!
The initial reason was that Tao Xuanzhi published a letter sent by Qiao Yu on his blog.

It's common for successful mathematicians to frequently communicate and discuss mathematical problems via email. This is especially true for the most accomplished mathematicians.

Moreover, from an outsider's perspective, the two should actually have something in common to some extent. For example, they were both child prodigies when they were young, and they didn't waste their talents as adults.

In particular, both of them have a wide range of knowledge in mathematics.

Not to mention that Tao Xuanzhi had a very high opinion of Qiao Yu, even before Qiao Yu was widely recognized by the world's mathematics community.

The fact that he has supported Qiao Yu on more than one occasion is proof enough that the two would have contact in private, which was expected by everyone.

The reason this letter caused such a stir in the mathematics community was because it explored the issue of turbulence and the Navier-Stokes equations!

Seven years later, Qiao Yu finally turned his attention back to mathematics.

The contents of this letter are as follows:

Mr. Tao Xuanzhi: I hope to see your letter and meet you in person.

A while ago, Mr. Yuan made a calculation and believed that I had the potential to solve the fundamental problems of turbulence. So, I have been thinking about turbulence and the smoothness and uniqueness of the Navier-Stokes equations.

I must say this is indeed a very interesting question. Coincidentally, while I was researching this question, I came across your paper published in the Proceedings of the American Mathematical Society in 2014, "Finite-time bursting of the average solution of the three-dimensional Navier-Stokes equations".

So I wrote this letter to discuss some of my recent thoughts on the three-dimensional Navier-Stokes equations.

The average version of the Euler bilinear operator you constructed in your paper proves that a turbulent system with an initial value of u0 will explode in a finite time.

I roughly understand it as a robot A spilling a bottle of cola, so it clones itself into robot B to clean up the mess, and robot B then clones itself into robot C to clean up the mess...

This process of replication continued until Robot X released explosive energy, cleaning up the spilled cola and eliminating all the robots.

I find this very interesting. Your research has effectively eliminated the possibility of proving one particular approach to the Navier-Stokes equations. It has also given me great inspiration—that is, the proof process must include a method to distinguish between the original operator and the averaging operator.

This also gave Joe's algebraic geometry a new purpose.

In the traditional analytical framework, the primitive operator and the averaging operator will form an irreconcilable contradiction in the Banach space, just like the explosion mechanism you revealed.

However, if we project each velocity field element u(x,t) into the modal space (α,β), through modal projection of N_α,β(u), we can construct a new bilinear form with the following properties:

B(u, v)=⊕_{γ∈Γ}[N_{α+γ, β}(u)_Ω_γ N_{α, β-γ}(v)]
Where Γ is the critical frequency range defined in your paper. Now, let's both temporarily forget the boundary between Riemannian surfaces and Euclidean space.

Come and appreciate the ingenuity of this structure!

As you may have noticed, when γ approaches the explosion threshold, the corresponding modal component N_{α+γ,β}(u) will automatically annihilate due to its self-conservation requirement—this essentially transforms the explosion of robot X that you observed into a conservation law in modal space.

Now let's recall the modal conservation theorem in Geometric algebraic geometry.

If the initial condition u0 is rewritten as N_α, β(u0) = ⊕[φ_kψ_l], where each φ_k satisfies the modal unit number stability condition ‖N_α, β(φ_k)‖ ≡ 1, then the energy transfer chain will inevitably experience a directional reversal of the parametric manifold M at the k+l≤dimM step.

To this end, I construct a special characteristic class on the modal manifold M and prove that any solution that leads to a finite-time singularity necessarily violates the modal unity theorem of N_α,β(1).

Of course, by now you've probably already spotted the problem!
My approach still has two fatal flaws that I cannot verify. One is how to embed the viscosity term Δu into the curvature tensor of the modal space; the other is that I still cannot explain the asymptotic behavior of the brute-force crack when the modal parameters (α, β) → (0, π/2).

In fact, I have already performed several singular vortex mode decompositions using a quantum simulation supercomputer. However, the current results clearly do not provide direct evidence that it possesses a smooth solution or uniqueness.

So there are definitely things I haven't thought of yet. If you're not busy, maybe we can discuss these two issues in more depth together.

If your team has the time, you can also access the computing resources. Let's work together to solve this mystery as soon as possible.

P.S.: Actually, I wanted to rest. But my teacher and Mr. Yuan think I've had enough rest! They have high hopes for me and don't want me to slack off.

So please help me think of a solution! And I have a feeling that when we fully understand the essence of turbulence, or rather, its mathematical essence, we will be able to open up a new track in the aerospace field, and our names will be on that track.

……

After Tao Xuanzhi published the letter on his blog, he also shared his own insights.

"Although Qiao Yu painted a very rosy picture for me, I found that with my limited knowledge, I was probably not able to complete the task he entrusted to me on my own."

So if anyone has better ideas, we can discuss them together. In particular, we can discuss how to embed the viscosity term Δu into the curvature tensor of the modal space.

Δu represents the diffusion effect of the velocity field. Its effect in space is usually related to the rate of change of the velocity field; intuitively, the viscosity term controls the smoothness of the velocity field.

However, within the framework of modal space, the viscosity term needs to consider not only the gradient of the velocity field, but also how it interacts with the modal structure.

This involves figuring out how to map transformations in these spaces to modal spaces and understanding how these transformations affect the properties of the solutions.

Furthermore, can we understand the mode space as a space projected onto the velocity field, where each mode corresponds to a specific basis function or frequency?

In this space, the complexity of the problem may be simplified because each component in the modal space can be viewed as a representation or decomposition of the solution.

However, the curvature tensor in modal space involves the geometric properties of the hydrodynamic equations, especially the changes and interactions of the velocity field in different directions.

Therefore, my initial idea is to treat the nonlinear terms of the fluid dynamics equations as a geometric object, similar to a manifold on a Lie group or a generalized mechanical system in variational methods.

Within this framework, the effect of the viscosity term can be described by the curvature tensor, capturing the diffusion behavior of the fluid in different modes.

However, it is conceivable that the curvature tensor in modal space represents the local geometric properties of the velocity field within that space, while the viscosity term may influence the propagation and changes of these geometric properties. Therefore, embedding the viscosity term into the framework of the curvature tensor may mean constructing a nonlinear geometric operator that can sensitively capture changes in the velocity field and its diffusion behavior.

This is obviously difficult! If you have better ideas, please leave a comment below the blog post, or email me or Qiao Yu directly! But clearly, it's not just as Qiao Yu said, or perhaps he's being too modest...

I believe that if this problem can be solved, we will not only make a name for ourselves in the future aerospace field, but in many other fields!

……

It can only be said that Tao Xuanzhi is truly adept at posing a problem and then gathering collective wisdom. But it's clearly far more difficult than the other problems he's publicly discussed!
Okay, that seems like a pointless statement!

If it weren't difficult, it wouldn't have been listed as one of the seven unsolved mathematical problems of the millennium.

To the astonishment of countless mathematicians worldwide, Qiao Yu simplified the Navier-Stokes problem with a single stroke.

Theoretically speaking, by following the method given by Qiao Yu and conducting deductions, it can be proven that the Navier-Stokes equations have a smooth and unique solution.

It was simply a matter of resolving two questions he had raised in his mind that could not yet be verified.

But as I said before, the greatest significance of solving these world-class problems is not actually the problem itself, but the way of thinking that solves them can provide people with new tools and perspectives to understand some of the essential aspects of this world.

Qiao Yu's ingenious integration of the Navier-Stokes equations into Qiao algebraic geometry and Qiao spaces has undoubtedly opened a window for mathematicians worldwide!

In layman's terms, Qiao Yu is carrying out a mathematical revolution, more specifically a modal revolution in topological analysis, which even involves the cognitive upgrading of mathematical ontology and a paradigm shift in instrumental rationality.

This undoubtedly dissolves the barriers between disciplines and even launches another dimensional reduction attack on computational mathematics!

All mathematicians who can understand this letter and Tao Xuanzhi's analysis probably share this feeling.

The essence of Qiao Yu's proposed method can be understood as directly translating the differential structure of physical space into the topological invariants of modal space.

When mathematicians realized that the nonlinear terms of the Navier-Stokes equations could be characterized as fiber bundle sections on a parametric manifold M, this effectively built a quantum bridge between partial differential equations and algebraic geometry.

Just as Michael Atiyah and Isador Singer pioneered the Atiyah-Singer index theorem to unify analysis and topology, Qiao Yu's spatial methodology is creating a deep duality between dynamics and geometry.

It is important to know that in traditional analysis, turbulent singularities are often regarded as catastrophic, but in the N_α,β modal framework, these burst points are precisely the critical sources for generating conformal mappings in the modal space.

Well, when non-Euclidean geometry first emerged, it was essentially a reinterpretation of the parallel postulate. The situation is quite similar now.

Mathematicians no longer need to fight a life-or-death battle with ubiquitous singularities, but can directly transform them into new dimensional regulators by adjusting the (α, β) parameters.

The originally chaotic turbulent energy spectrum was deconstructed into coherent resonances of countable modal layers. Even more astonishingly, when someone followed this line of thought to verify it, this method provided a topological interpretation of the Kolmogorov scale law—the inertial region corresponds to the geodesic dense region of the parametric manifold M, while the dissipative region is the Riemannian folds of its curvature burst…

And that's not all...

Vortex structures are equivalent to special dividers on complex surfaces; the existence of Leray weak solutions corresponds to the mirror symmetry of the Calabi-Yau manifold; turbulent fluctuations are discretized as ⊕-superpositions of modal feature layers; smoothness is redefined as the connectivity of parametric manifolds…

Therefore, the proof of its existence can be understood as the turbulent trajectory necessarily passing through a three-dimensional slice...

Yes, Qiao Yu only sent a letter to Tao Xuanzhi, and a month later the entire mathematics community was in an uproar!
Yes, it wasn't just lively, it wasn't just discussion, it was boiling over! Various in-depth discussions even spread directly into the field of philosophy.

After all, the method proposed by Qiao Yu, which uses manifold curvature to encode viscous dissipation, directly points to the transcendental isomorphism between mathematics and physics.

In other words, humanity may never know whether mathematics was discovered or simply defined and reconstructed by human understanding of civilization.

The reason why it took a month was mainly because initially very few people could understand what the two were talking about.

Over the past month, many truly influential figures have stepped forward to provide explanations and verifications, simplifying the extremely information-dense content of the letter into something that everyone can understand.

For example, "the energy transfer chain will inevitably experience a directional reversal of the parametric manifold M at the k+l≤dimM step..."

Qiao Yu only mentioned it briefly in the letter, but it actually involved the infinite compression of the volume of a fluid element within a finite time.

And after the Qiao Yu method intervenes, it directly manipulates the object, transforming the singularity in the original physical space into a smooth pole on the M-manifold...

Mathematically, the detonation condition of the original Navier-Stokes equations, ‖u(t)‖→∞, is transformed into: ∫_{M} N_α，β(u) dσ= 0……

When zero flux occurs at the boundary of a parametric manifold, physical space explosions will inevitably be prevented.

This part alone could easily be written into a nearly 100-page mathematical paper!
This also explains why real ocean currents don't suddenly burst apart without warning just because of a small turbulence; the accumulated energy will eventually be released through some channel...

There are so many things like these that require mathematical explanation! Without these experts patiently publishing articles to explain, many mathematicians wouldn't understand what Qiao Yu was talking about with Tao Xuanzhi.

Some people have even summarized the explanations given by leading figures in the mathematics community and created a corresponding table.

For example, vortex stretching in turbulence is roughly equivalent to the symplectic deformation of a complex structure in mathematics, and the corresponding modal equation interpretation fragment is _tω= N_α，β(ω)v.

For example, viscous dissipation corresponds to anisotropic diffusion of Ricci curvature, and the modal equation fragment is νΔu Ric(g_{αβ})...

As people gradually uncovered that Qiao Yu was attempting to explain physical phenomena directly using mathematics, the entire mathematics community naturally experienced a gradual uproar...

There's no way around it, this is truly a mathematical theory that can drive the entire applied mathematics community crazy!
If Qiao Yuzhen has structured the Navier-Stokes equations using this method, it means that in the future, applied mathematicians may even be able to skip physics to some extent and directly structure and restore nature...

So Tao Xuanzhi is right about this!
If this problem can be solved, the impact will extend far beyond the aerospace field! (End of Chapter)