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When studying these formulas, Zhu Zhengze couldn't help but think of a renowned mathematician—Pierre de Fermat!
Indeed, just from the sentence this French mathematician wrote on the verge of proposing Fermat's Last Theorem in his book "Arithmetica," we can tell how eccentric the personality of a top mathematician can be!
“I have found a truly amazing proof, but it is too long to put here.”

This one sentence has puzzled the world's mathematical community for a full 358 years!
The fact that the International Mathematical Union awarded Wiles the world's first Fields Silver Medal may well have been a tribute to that statement.

That's infuriating!
If Fermat's Last Theorem is ultimately proven wrong, then it will prove that Fermat was just bragging back then.

But when Wiles turned Fermat's Last Conjecture into Fermat's Last Theorem, it became a historical enigma, and no one can prove whether Fermat actually came up with a brilliant proof at the time.

In short, Wiles' discovery not only earned him contemporary honors but also gave the former mathematical giant a boost in prestige.

Now Qiao Yu is taking a completely different path.

"Oh dear, I've discovered a world-class problem. This problem is truly incredibly difficult; even top mathematicians can't solve it. Why don't you give it a try!"

Just a few days after he made that statement, he casually threw out a bunch of simple formulas, and everyone suddenly realized that the problem he had raised earlier had been easily solved.

Moreover, Qiao Yu's approach is so ingenious that it has once again pioneered a new method for integrating multiple types of mathematics into the generalized modal axiomatic system.

This will open up a new modal algebra architecture, which is equivalent to directly constructing a functional space that covers multi-scale problems.

To put it in Qiao Yu's words:

Zhu Zhengze has already seen that this space can simultaneously accommodate the Hodge decomposition of the differential form of classical partial differential equations, the hierarchical cohomology of algebraic varieties, and the deformable quantization parameter.

This is equivalent to providing a direct mathematical interface for modern physics.

Therefore, within this framework, the nonlinear terms of the Navier-Stokes equations have been proven to be equivalent to the Chern number calculation of a certain feature class, and a global regularity criterion can be derived from this.

Before today, Zhu Zhengze truly never imagined that mathematics could be used in this way.

In addition, to make the logic of the entire theory even more impeccable, Qiao Yu also introduced a generalized covariant derivative.

This involves a completely new differential operator:

This directly allows the geometric curvature and fluid viscosity to reach a dynamic equilibrium...

A near-perfect idea!

Zhu Zhengze dares not say that there will be no one like him in the future, but he is certainly unprecedented in his ability to manipulate mathematics in this way.

Even if historical mathematical giants like Newton, Gauss, and Riemann were reborn, they would still be humbled by their inferiority.

It can only be said that Qiao Yu's stroke of genius brought mathematics to a whole new level.

Of course, based on what he has seen so far, Zhu Zhengze does not dare to make a definitive statement as to whether Qiao Yu's formulas are valid.

This is because the mathematical structure involved in this approach is quite complex, and the rigor of mathematical proof is the lifeline of a theoretical system.

Simply put, for a theory to be valid, it must have a rigorous proof process that no one can find fault with.

The dozen or so formulas that Tian Yanzhen sent him only helped him understand Qiao Yu's problem-solving approach.

Therefore, even if Zhu Zhengze could understand that the Qiao Yu-supplemented theoretical framework of Qiao Yu showed an amazing internal consistency, without seeing the complete derivation process, probably no one had the ability to be sure that this theory was necessarily correct.

Furthermore, the modal covariant derivative introduced by Qiao Yu is clearly based on his self-created infinite Vissian manifold, which requires a lot of verification and proof...

The applicability of Frobenius' theorem; geodesic completeness, such as whether there is an explosive solution when the viscosity tensor ν(X,Y) has singularities; dimensional compatibility, such as whether dimensional analysis is applicable when the curvature tensor is directly used as the viscosity coefficient...

In particular, the constitutive relation analysis of the Navier-Stokes equations must ensure that the geometric rewriting using this method does not affect the physical properties of the equations themselves...

That's a huge amount of work!

Especially with high-dimensional fiber bundle expansion, Zhu Zhengze knew just by thinking about it that when the dimension of the fiber bundle exceeds 6, processing the tensor integral solution of the M(X) space would inevitably require massive computing power...

As Zhu Zhengze studied Qiao Yu's formulas, he kept jotting down his thoughts and ideas in his notebook. It took him nearly two hours to get through all ten or so formulas.

The main issue is that the thinking process takes a lot of time, even though this is based on the premise that Zhu Zhengze is very familiar with the generalized modal axiomatic system.

If a mathematician were to take the place of a mathematician who is not very familiar with the axiomatic system of generalized modes, it would take a long time just to look up the meaning of the various symbols.

Most importantly, if the continuity of thought cannot be guaranteed, it will be difficult to keep up with Qiao Yu's train of thought as the formula changes.

After roughly understanding Qiao Yu's line of thought, Zhu Zheng picked up the phone and dialed Tian Yanzhen as instructed in the letter.

At times like this, there's no need to consider time differences. He believes that even if it's late at night in China, Tian Yanzhen will wait for him to call back.

But to his surprise, the notification tone indicated that the other party was on another call, but Zhu Zhengze did not hang up.

Tian Yanzhen's private number has the hold function enabled.

Once the other person finishes talking, his call can be answered immediately. And even if there's a call, there will be a notification.

Sure enough, the call was connected after about fifty seconds, and the next moment Tian Yanzhen's strong voice entered his ears.

"Xiao Zhu, you finally called me back. If you hadn't called, I was about to call you myself."

To be honest, this voice made Zhu Zhengze feel somewhat emotional.

He noticed that since Academician Tian turned seventy, his personality and voice have become increasingly similar to those of his former mentor, Mr. Yuan.

Even though the two had a very unpleasant falling out back then, which even became an indelible case in the history of Chinese mathematics.

After all, it's rare for mathematicians of this caliber to have an argument that's known to the average person...

Of course, the main reason was that Mr. Yuan made many public statements.

If it weren't for Qiao Yu, these two would probably still be strangers to each other by now.

While feeling inwardly moved, Zhu Zhengze didn't stop talking and quickly explained, "I saw your email as soon as I got to work this morning. I've been studying those formulas, and I only dared to call you back after I finished reading them and gained some insights."

"What are your thoughts?" the other person immediately asked.

Zhu Zhengze subconsciously smiled wryly and said, "I have too many thoughts! The only thing I dare not think about is how Academician Qiao came up with this idea."

Such an ingenious design—I don't know how to describe it. But I can't wait to see Academician Qiao's related papers.

"If Academician Qiao can truly solve the Navier-Stokes equations problem using this method, it means that mathematics will not only take a giant leap towards grand unification, but also that the boundaries of mathematics will expand infinitely." After saying this, Zhu Zhengze paused and tentatively asked, "However, this is a very big topic, and there must be a lot of work to be done. Does Academician Qiao plan to set up a dedicated research group?"

To be honest, I haven't figured out a new research direction since my last paper was published. If Academician Qiao doesn't mind, I happen to have some time to lend you a hand..."

If the previous affirmations about Qiao Yu made Tian Yanzhen happy, then this last sentence truly made Old Tian's heart flutter with joy.

It's true that when people reach a certain high position, it's very difficult to hear the truth.

After all, people have limited energy and can only access so much information each day. The higher you stand, the farther and broader you can see, but you can't see the details clearly.

Given that he already had too many things to worry about, he could only rely on a trusted team to extract the information he needed and then take a look at it.

Once someone reaches this position, it's difficult for those around them to say anything discouraging.

After all, that's how society operates at this stage. The opinions of those in power can truly determine the future of their subordinates.

This means that those who still have aspirations for the future will choose to cater to their preferences.

There is no doubt that Qiao Yu currently holds this position in the mathematics community.

Although Qiao Yu declined invitations to review several major international mathematics awards, including the Fields Medal, the award committees for mathematics awards would specifically consult him for his opinion every year during the award season.

Back in China, let's put it this way: the entire Department of Mathematics and Physics of the Chinese Academy of Sciences basically follows Qiao Yu's lead. Even though Qiao Yu usually doesn't bother with all that nonsense, if he speaks, it definitely has an effect.

This has created a situation where many professors and researchers who are striving to climb the academic ladder may not dare to hope that Qiao Yu will speak up for them, but they still have to hope that Qiao Yu will not have any bad impression of them.

If, at a crucial moment in life, such as when applying for an award or applying for a research project to get a prestigious title, Qiao Yu were to say that this person is not up to the task, then the whole thing would definitely fall through.

In this situation, it's naturally difficult to hear the truth.

In particular, most of the professors who are good at mathematics and can thrive in universities are very clever and insightful.

If they wanted to, they could talk a better game than they could sing. But while they can fool people with words, they can't fool them with actions!
If we were to delve into such a massive project, it would essentially take several years, or even more than a decade, to complete.

If results are achieved, everyone is naturally happy. But if no results are achieved, it means that more than ten years have been wasted.

For those who have little ability and just want to rely on powerful figures for support, wasting more than ten years might not be a big deal.

But Tian Yanzhen and Yuan Lao have such sharp eyes...

People who just want to coast through life are beneath their notice. They certainly won't give such people a chance to get close to Qiao Yu.

To put it simply, none of Yuan's biological descendants had the opportunity to enter academia because Yuan himself believed they were simply not cut out for it.

Therefore, most of those who were recommended by the two elders and achieved success are still those with strong academic aspirations.

These ambitious people certainly wouldn't waste more than a decade of their prime just to follow the big shots.

Globally recognized mathematicians typically achieve significant results between the ages of thirty and fifty. After all, mathematics is unlike engineering; while experience is helpful, it can also be a constraint.

After the age of fifty, various brain functions begin to decline rapidly, making it indeed impossible to adapt to work that requires highly abstract thinking.

For example, for some ordinary people, choosing a cup of coffee and a donut in the afternoon to enjoy a leisurely afternoon tea is just a daily routine.

From their perspective, coffee cups and donuts are completely different things.

Functionally, one is a cup and the other is food; in terms of shape, the two are completely different.

But in the eyes of mathematicians, especially those who study topology, coffee cups and donuts are essentially the same thing.

Only by having this idea can one understand the very important and famous concept of "homeomorphism" in topology.

First, mathematicians need to separate the two from the physical characteristics that ordinary people use to distinguish them, such as color, material, and size.

Then look for commonalities, such as both having only one hole.

The hole in a donut is in the very center, while the hole in a coffee cup is in the handle.

Next comes the symbolic representation, using the concept of genus in algebraic topology to represent holes for classification. Undoubtedly, both have a genus of 1.

Then we can use the Euler characteristic formula to prove that they have the same topological properties.

There's a specific term for this called concept distillation, which is a method that directly abstracts a three-dimensional entity into a two-dimensional surface while transforming seemingly unrelated things into a computable mathematical model.

What's the use of this?

Its applications are truly vast. From small-scale network connectivity analysis and pore structure research in materials science, to large-scale machine learning applications of data manifolds and topological conjectures about the shape of the universe…

This topological way of thinking is required in all of them.

Enabling machines to understand and accurately express information seen and heard by humans is a highly abstract concept.

Only this kind of abstract way of thinking can turn all of this into reality.

There's a Chinese saying: "It's impossible for a person to imagine something they've never seen before."

But this statement doesn't apply to mathematicians who are in their prime.

Their brains are truly capable of abstracting far more than ordinary people can imagine. It is precisely this originality that drives the continuous progress of mathematics.

This kind of abstract thinking is obviously too difficult for men over fifty.

Therefore, most scholars, as they get older, will mentor students or lead teams, rather than personally engaging in frontline research.

Of course, most of the time when the team produces results, the first author name is still these scholars when publishing papers. But, in any case, the rich experience of encountering setbacks, which helps the younger members of the team avoid detours, is also a great contribution.

Not to mention that the project was able to secure funding entirely because of the connections of these bigwigs.

This is perfectly understandable. That's just how the world works.

People are more inclined to believe those who have a history of success than those who are unknown and have no notable achievements to prove themselves.

Therefore, actions speak louder than words! (End of Chapter)