When Richard's family began to look forward to the future, the world's mathematics community suddenly became excited without warning!

The initial reason was that Tao Xuanzhi posted a letter sent by Qiao Yu on his blog.

It is common for successful mathematicians to communicate and discuss mathematical problems frequently with emails. The better the mathematician, the more this is true.

And to the outside world, the two should actually have a common language to some extent. For example, they were both child prodigy when they were young, and they did not waste their lives when they grew up.

Especially the two of them have a wide range of knowledge on the mathematical level.

Not to mention that before Qiao Yu was widely recognized by the world's mathematics community, Tao Xuanzhi had a high evaluation of this rising star.

It is proof that the two have a private connection, which is what everyone expected to cause a sensation in the mathematics community because the problems discussed in this letter are turbulence and N-S equations!

After seven years, Qiao Yu finally started math again.

The content of this letter is as follows:

Mr. Tao Xuanzhi: See you literally.

A few days ago, Mr. Yuan calculated and thought that I had the potential to solve the essential problem of turbulence, so during this period I have been thinking about the smoothness and uniqueness of turbulence and N-S equations.

I have to say that this is indeed a very interesting question. Coincidentally, when I was studying this problem, I happened to see the paper you published in the journal of the American Mathematical Society in 2014, "Finite Time Blasting of the Average Solution of the Three-Dimensional N-S Equation".

So I wrote this letter to explore some of my recent thoughts on three-dimensional N-S equations.

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

I probably understood it as a robot A sprinkled a bottle of Coke, so he copied his own robot B to clean up the mess, and robot B copied robot C to clean up.

It just kept copying until the robot × directly released explosive energy, and the spilled coke was cleaned up.

All robots no longer exist.

I think it's very interesting. Your research has made a research idea for the N-S equation cut off the possibility of proof from then on.

It also gave me great inspiration, namely, the proof process must have a method to distinguish between the original operator and the average operator.

This also gave Qiao algebraic geometry a place to work once again.

Under the traditional analysis framework, the original operator and the averaging operator will form an irreconcilable contradiction in the Banach space, just like the blasting mechanism you have revealed.

But if we project each velocity field unit u(,t) into the modal space (α,β), through N_α,β(u)

The modal projection of the method can create a new bilinear type with the following characteristics:

B(u,v)=_{∈"}[N_{α+Y,β}(u)β_QV_N_{α,β-}(v)]

Among them, "is the critical frequency interval defined in your paper. Now, please forget the boundary between the Riemann surface and Euclidean space for the time being.

Come and appreciate the exquisiteness of this structure!

I believe you have also discovered that when approaching the explosion threshold, the corresponding modal component N_{α+Y,β}(u) will automatically annihilate due to its self-conservation requirements. This essentially converts the explosion of the robot × you discovered into the conservation law in modal space.

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

If the initial condition u0 is rewritten to N_α, β(u0)=[Φ_k_I], where each Φ_k satisfies the modal unit stability conditions IlN_α, β(Φ_k)Il=1, then the energy transfer chain will inevitably have a directional inversion of the parameter manifold M when the k+I≤dimM step.

For this reason, I constructed a special exemplary class on the modal manifold M7 and proved that any solution that leads to a finite time singularity must violate the modal unity theorem of N_α, β(1).

Of course, I believe you have found the problem after seeing this!

There are two fatal loopholes in my thinking that cannot be verified. One is how to embed the sticky term △u into the curvature tensor of the modal space; the other is that I cannot explain the asymptotic behavior of blasting when modal parameters (α,β)→(0,π/2).

In fact, I have borrowed quantum simulation supercomputers to perform several singular vortex mode decompositions. But obviously, the current results do not directly prove that they have smooth solution and uniqueness.

So there must be something I didn't expect. If you are not busy, maybe we can discuss these two issues in a more in-depth way.

If your team has time, you can also access computing. Let us work together to solve this unsolved mystery as soon as possible.

Also: Actually, I want to take a break. But my teacher and Mr. Yuan think I have been resting for a long time! They have high hopes for me, which makes it inconvenient for me to be lazy.

So please be sure to help me think of a solution! And I have a premonition that when we fully realize the nature of turbulence, or the nature of mathematics, will be able to open up another new track in the field of aerospace, and our names will be included on the track.

After Tao Xuanzhi made the letter public on his blog, he later sent his own opinions.

"Although Qiao Yu drew me a big cake, I found that with my shallow knowledge reserves, I might not be able to complete the tasks he entrusted to me independently.

So if anyone has a better idea, maybe we can discuss it together. Especially the issue of how to embed the viscosity term △u into the curvature tensor of the modal space.

Au represents the diffusion effect of the velocity field. Its role 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, in the framework of modal space, the viscosity term not only needs to consider the gradient of the velocity field, but also how it interacts with the modal structure.

This involves how to map transforms in these spaces to modal spaces and understand how these transforms affect the properties of solutions.

In addition, can we understand modal space as a space after projecting the velocity field, where each modal corresponds to a specific basis function or frequency?

Then in this space, the complexity of the problem may be simplified, because the various components in the modal space can be regarded as a representation or decomposition of the solution.

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

So my initial idea is to regard the nonlinear terms of the fluid dynamics equation as a geometric object, similar to the generalized mechanics system in manifold or variational method on Li Group.

Then under this framework, the role of the viscosity term can be described by curvature tensors, capturing the diffusion behavior of the fluid under different modes.

But it can be imagined that the curvature tensor in modal space is the local geometric properties of the velocity field in that space, and the viscosity term may affect the propagation and change of these geometric properties.

Therefore, embedding the viscosity term into the framework of curvature tensors may mean the need to construct a nonlinear geometric operator that needs to keenly capture the changes in the velocity field and its diffusion behavior.

Obviously this is difficult! If you have a better idea, please leave a message on the blog, or send an email to me or Qiao Yu directly! But obviously, this is not just like Qiao Yu said, or he is still too modest. I believe that if this problem can be solved, it will definitely not only be able to give a name in the future track in the aerospace field, but also in many tracks!"

I can only say that Tao Xuanzhi is really good at throwing out a problem and then brainstorming. But obviously it is actually far more difficult than other problems he has made public!

OK, this seems like nonsense!

If it is not difficult, it will not be listed as one of the seven major mathematical problems of the millennium.

What makes countless mathematicians in the world speechless is that Qiao Yu simplified the N-S problem as soon as he takes action.

Theoretically, according to the method given by Qiao Yu, deduction can be proved that the N-S equation has the only solution with smooth solution roots.

It is nothing more than solving the two problems he mentioned in his mind that cannot be verified for the time being.

But the same thing is true. The greatest significance of solving these world-class problems is not actually solving the problem itself, but the idea of ​​solving the problem can provide people with new tools and perspectives for understanding some of the essential things of the world.

Qiao Yu's clever way of integrating N-S equations into Qiao algebraic geometry and Qiao space undoubtedly opened a window for mathematicians around the world!

In terms of language that the public can understand, Qiao Yu is undergoing a mathematical revolution, more specifically a modal revolution in topological analysis, and even involves the cognitive dimension upgrade of mathematical ontology and the paradigm transition of instrumental rationality.

This is undoubtedly a dissolving of discipline barriers, and even another dimensionality reduction blow to computational mathematics!

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

Because the essence of this method proposed by Qiao Yu can actually be understood as directly translating the differential structure of physical space into topological invariants of modal space.

When mathematicians realized that the nonlinear term of the N-S equation could be characterized as the cross section of the fibers on the parametric manifold M, this actually set up a quantum bridge between partial differential equations and algebraic geometry.

Just as the Atiyah-Singer index theorem developed by Michael Atiya and Isador Singer back then, Qiao Yu's spatial methodology is creating a deep dualism between dynamics and geometry.

It should be noted that in traditional analysis, turbulent singularities are often regarded as disaster, but under the N_α and β modal framework, these blasting points become the critical sources for the formation of conformal mapping in modal space.

How to say it, when non-European geometry emerged, it was a reinterpretation of parallel axioms. The situation at this time was actually similar.

Mathematicians no longer need to fight the ubiquitous singularity, but instead directly convert it into a new dimension regulator by adjusting (α,β) parameters.

The original chaotic turbulence energy spectrum is deconstructed into coherent resonances that can be listed in modal layers. What is even more amazing is that when someone follows this idea to verify, this method can give the topological interpretation of the Kolmogorov scale law a geodesic dense region corresponding to the parametric manifold M, while the dissipation region is the Riemann fold of its curvature burst, and even more than this vortex structure is equivalent to the special denominator on the complex surface; the existence of Leray weak solution corresponds to the mirror symmetry of the Calabi-Yau manifold; the turbulent pulsation discretes as the -superposition of the modal feature layer; the smoothness is redefined as the connectivity of the parametric manifold·—.

Therefore, the proof of existence can be understood as the turbulent trajectory must be sliced ​​in three-dimensionally. Yes, Qiao Yu just sent a letter to Tao Xuan, which made the entire mathematics community completely boil in a month!

Yes, it’s not lively, it’s not discussion, it’s boiling! Various in-depth discussions even spread directly to the field of philosophy.

After all, Qiao Yu proposed that the method of encoding viscous dissipation using manifold curvature directly points to the transcendent isomorphism between mathematics and physics.

In other words, humans may never know whether mathematics is discovered, or that mathematics is just artificially defined and reconstructed the cognition of human civilization.

The reason why there was one month in this was mainly that at first few people could understand what the two of them were talking about.

This month, many real big shots jumped out to explain and verify various explanations and verifications, and then simplified the extremely information-intensive content in the letters into content that everyone can understand in step.

For example, "The energy transfer chain will inevitably have a directional inversion of the parameter manifold M when the k+≤dimM step—"

Qiao Yu just wrote it in a simple sentence in the letter, but in fact the content involved includes infinite compression of the volume of fluid micronumerals in a limited time.

And after Qiao Yu's method intervenes, it will be operated directly, and then the singularity of the original physical space is converted into smooth pole on the M manifold. The mathematical representation is: the bursting condition of the original N-S equation Ilu(t)Il→∞o is converted to: J_{M}N_α,β(u)

do=O....

When zero flux appears in the boundary of the parameter manifold, physical space blasting will inevitably be prevented.

And just this part of the content can even be written directly with a math paper of nearly a hundred pages!

This also explains that the real ocean current will definitely not explode directly because of a small turbulent current without warning. The accumulated energy will eventually pour through a certain channel, such as too many things that require someone to explain mathematically! Without these big guys patiently posting articles to explain, many mathematicians would not understand what Qiao Yu was talking about with Tao Xuanzhi.

Some people even directly summarized the content explained by the bigwigs in the mathematics community and directly made a corresponding table.

For example, vortex stretching in turbulent flow is probably equal to the anointed deformation of the complex structure in mathematics, and the corresponding modal equation explanation fragment is _tw=N_α,β(w)Vv.

For example, viscous dissipation corresponds to the anisotropic diffusion of Ricci curvature, and the modal equation fragment is vAu

Ric(g_[aβ})......

In this way, when people try to directly explain physical phenomena with mathematics step by step, it naturally makes the entire mathematical community appear to be in a gradual process of being infuriated. This is really a mathematical theory that can make the entire applied mathematics community crazy!

If Qiao Yuzhen used this method to structure the N-S equation, it means that future applied mathematicians can even directly skip physics to a certain extent, structure nature, and restore nature.·

So this is what Tao Xuan said!
Chapter completed!