A genius? I just love studying.

Chapter 235 Fatal Divergence
Chloe is right, sunbathing really can effectively promote sleep.

After returning to the hotel from King's College, Chen Hui had a good night's sleep. When he woke up, he felt refreshed and fully recharged. He went to his desk and began to sort through the emails he had exchanged with Dennis over the past few days.

At 8:30, after having a simple breakfast in the hotel restaurant, Chen Hui headed towards Queens College, where Dennis had arranged to meet him.

Because of the commission from the Chengdu Aircraft Industry Research Institute, Chen Hui has not been able to continue their previous research for more than two months. However, Dennis has been sending him emails to keep him updated on his latest progress. When he heard that Chen Hui was coming to the British Empire, he even made a special trip from the United States.

"Hui, it's been a long time!"

Beside the Mathematics Bridge at Queens College, Chen Hui met a spirited Dennis, whose happiness could be felt from his emails over the past few days.

"Long time no see, Dennis."

Chen Hui responded with a smile.

"I feel like we're very close to solving that ultimate mystery!"

Dennis looked at the math bridge next to him, his eyes shining with excitement.

Legend has it that this wooden bridge was built using precise calculations by Newton and did not require the use of mortises or nails.

A professor was so impressed by its structure that he dismantled it to learn about its construction details. Although the professor later rebuilt the bridge, he could not do so without nails.

Chen Hui did not refute, but he had doubts about Professor Dennis's confidence. "I still have a few questions about the plan you sent me in the email."

"Haha, that's why I came to see you today."

Dennis smiled and stepped forward. "Let's go, I've already applied for a classroom."

Soon, the two entered an empty classroom, where several notebooks, related papers, and their recently published paper, "Elastic Deformation of Vortex Fiber Bundles," were spread out on the lectern.

Dennis pointed to a diagram of vortex tube annihilation on the blackboard, saying, “Three-dimensional vortex tube annihilation is the core mechanism of turbulent energy cascades and the key to understanding singularity formation. The success of our framework largely depends on whether we can describe this process elegantly.”

“That’s right. In our vortex fiber bundle model, annihilation corresponds to a dramatic change in the fiber bundle topology—the breakage and reconnection of fibers. We need to find suitable mathematical tools to characterize this ‘operation’.”

Chen Hui nodded; this was the same proposal that Dennis had sent him in emails these past few days.

Dennis remained silent, turned around and drew on the blackboard. The two intersecting rings represented the vortex rings that were about to collide, and he labeled them as γ1 and γ1 respectively. Then he drew a complex winding structure near the intersection point, and finally drew the vortex lines that broke and reconnected to form a new configuration.

“My idea is,” Dennis marked in red chalk, “to view the vortex tube as a one-dimensional closed chain embedded in a three-dimensional fluid domain M, and the annihilation process is the abrupt change in the intersection of these closed chains in M.”

"Before annihilation, γ and γ are independent as homology classes [γ], [γ]∈H(M;), and their intersection number may be non-zero. At the moment of annihilation..."

Dennis began to explain his solution: "Closed-loop homology naturally describes topological invariants, which are observable in experiments and numerical simulations, and can clearly characterize the overall topological class changes caused by annihilation."

"Of course, how to rigorously map the time evolution and the effect of viscous dissipation νΔω in the dynamic process to this discrete topological change framework requires the development of a 'differential cohomology' theory to describe the topological transition rate, which is very tricky!"

Dennis shrugged, acknowledging that his plan presented considerable challenges.

"But I believe that if we join forces, we will definitely be able to solve this problem!"

Chen Hui picked up a blue chalk and drew a small neighborhood U near the annihilation point. He then enlarged it and drew a complex, highly distorted vortex structure within U. "The core region U of the annihilation is where the singularity is born. Physical quantities change drastically, and the traditional smoothness assumption fails."

“I believe that in this singularity neighborhood, we need to go beyond a purely topological perspective.” He drew a box on U and labeled it “quasi-convex region?”

Chen Hui's eyes grew brighter and brighter, and he vaguely felt as if he had touched something extraordinary.

His research on scramjet engines these days has not been fruitless. Although engineering fluid mechanics is very different from mathematical fluid mechanics, engineering practice has still brought him a lot of inspiration.

For mathematicians, studying simple problems occasionally can bring unexpected inspiration.

“Our previous vortex bundle model was essentially real geometry, but the strong singularity of the annihilation point reminded me of tools in complex geometry, especially the -Neumann problem of the nonhomogeneous Cauchy-Riemann equation u = f on strongly quasi-convex domains.”

“We can try to treat the fluid domain U near the annihilation point as a strongly quasi-convex domain, then the -Neumann operator □=*+* and its related estimation theory can provide a powerful set of tools.”

“Prove that within U, the vorticity field ω belongs to some Sobolev space, or more ideally, prove that ω is Holder continuous or even smooth within U, which is equivalent to achieving some kind of regularization at the singularity…”

Chen Hui spoke faster and faster, countless ideas gushing into his mind: "This can bypass the dynamics of directly dealing with topological mutations themselves, and instead prove that even at the most violent interaction points, the solution is still 'good' in a weak sense, and the singularity is 'controllable'."

Dennis's brow furrowed deeper, his fingers tapping the table rhythmically. "Your idea is mathematically very elegant, reminiscent of Mr. Qiu's style, but..."

He paused, pointing to the global annihilation graph on the whiteboard, “-Neumann theory deals with local regularity, while the essence of annihilation is a change in global topology! Closed-chain homology describes the state transitions before and after an annihilation event, which is the core of physical observation.”

"Even if your method succeeds, it only tells us that the solution in that small region U is not 'too bad,' but it does not, or rather it is difficult to, directly tell us how the topological class changes and the rate at which this change occurs. Closed-loop homology naturally characterizes such discrete events!"

“More importantly,” Dennis emphasized, “observability!”

“Experimental physicists and those who do numerical simulations see changes in vortex line configurations and redistribution of vortex fluxes—these are topological and homogeneous. How do you get them to understand a highly abstract complex structure domain and the -Neumann operator? How do you make it correspond to measurable quantities?”

Chen Hui retorted, his tone calm but firm, "Dennis, I understand the observability advantage of topological description, but the dynamic mechanism of topological transitions itself may be hidden in the analytical structure of the singularity neighborhood!"

The regularity provided by Neumann theory may be key to understanding 'how' topological changes occur, rather than just 'what the outcome is.' The effect of viscosity ν is crucial at singularities, and it is difficult to describe precisely within the topological framework.

Chen Hui pointed to the annihilation point, "Closed-chain coherence treats annihilation as a 'momentary' topological operation."

However, physically, this is a process with both spatial and temporal scales, and the Neumann framework has the potential to analytically capture this process rather than treating it as a black box transition.

As for observability, if the theory is successful, we can find its inferences—such as predictions of energy dissipation spectra, which can be verified!

Dennis shook his head repeatedly as he listened, and after Chen Hui finished speaking, he spoke up again to refute him.

The discussion lasted for several hours.

The two repeatedly deduced, gave examples, and cited classic results from their respective fields on the whiteboard. Dennis used braid groups and Flor homology to illustrate, while Chen Hui brought up the power of complex methods in the proof of the Calabi conjecture. They understood each other's mathematical logic, but they had fundamental differences in their evaluation of its importance, feasibility, and relevance to the core of physics.

Dennis argues that the essential properties of fluids are vorticity and their topology. Any model must prioritize a clear description of topological evolution. Physical interpretability and observability take precedence over mathematical completeness and elegance. Closed-loop coherence is the most promising path to this goal!

Chen Hui argues that understanding singularities requires powerful analytical tools, and complex geometry provides the most profound framework for dealing with strong singularities. The mechanism of viscosity in the neighborhood of a singularity must be rigorously and analytically characterized, and topological description requires a deeper analytical foundation. Neumann theory offers this potential!
Finally, the two stopped arguing. Dennis put down the chalk wearily and looked at the clearly defined blue and red areas on the whiteboard. "I think we both saw the core of the problem, but we also saw the fundamental difference in our approaches."

He sighed softly, tidying up the drafts on the table. "Yes, our starting points and understanding of 'key' have pointed in different directions. Closed-chain homology-Neumann is like two different coordinate scales describing the same phenomenon, but their transformation functions... seem too complicated at the moment, and may even be incompatible."

Dennis nodded, a hint of regret mixed with determination. "Well then, perhaps it's time."

Dennis paused for a long time after speaking, but still said, "Let's each go along the path we believe in and see whose framework can ultimately illuminate the abyss of Navi Stokes, or... perhaps it will ultimately prove that a combination of both is needed, but that's a matter for the future."

Chen Hui extended his hand, "Okay, explore separately, keep in touch, and I wish you success in your closed universe."

Dennis grasped his hand, "And I wish you a successful Geometry Expedition as well."

The two walked out of the classroom. The sun was still shining brightly in the sky, leaving only the paper "Elastic Deformation of Vortex Fiber Bundles" lying quietly on the lectern.

……

The United States, Washington, D.C., conference room.

The top executives were gathered together, watching the images on the big screen.

Those were images transmitted back by their air formation that had been defeated during their expedition to the oil depot. Hundreds of the most advanced fighter jets didn't even reach the Desert Fox's territory, didn't fire a single shot, and in the end, less than thirty fighter jets escaped.

This was only because they had flown out of the Desert Fox's territory; otherwise, if the enemy had continued to pursue them, this aerial combat formation would likely have been completely wiped out.

"Could this be a sixth-generation fighter jet from China?"

The white-haired old man stared at the familiar aircraft on the screen, completely baffled. He remembered that his think tank had told him that China's sixth-generation fighter jet was a strategic bomber. How could a sixth-generation air superiority fighter jet suddenly appear?

Even if this is the trump card of the Rabbit Kingdom, how can we sell our trump card to others?
Unless, he has an even stronger trump card up his sleeve!

How many tricks does the Rabbit Kingdom have up its sleeve?

The white-haired old man felt a chill run down his spine, and the high-ranking officials from the United States in the conference room all felt a cold shiver down their spines.

In order to maintain their dominant position, they have put a lot of pressure on China from other aspects. They were originally fearless, but now they look at their sixth-generation fighter jet, which is still in the PPT stage, and then look at the Qingluan, which has already dominated the oil depot area.

They suddenly became worried: what if the rabbit really bites someone if they are pushed to the limit?
It seems that the small size of the United States might not be able to withstand a bite from a rabbit.

"Sir, I would like to remind you to pay attention to someone, namely Chen Hui, who recently solved the Millennium Problem, the Yang-Mills equation."

Nick Fury, the director of S.H.I.E.L.D., stood in the old man's room and said with a serious expression, "I suggest taking this opportunity to hand Chen Hui over to Satan. A person like him cannot be allowed to return to the Rabbit Country."

Fury was a tall, imposing Black man with a black eye patch over his left eye, which gave him an extremely intimidating and menacing presence when he spoke.

"ridiculous!"

Before the white-haired old man could speak, Wan Si interrupted angrily, "What era are we living in? Still playing these tricks?"

"Can you make Satan take in one person, or make Satan take in all his enemies?"

Even when facing Wan Si, Fu Rui remained unyielding. "He's different. Killing one of him is worth ten divisions!"

"Reliable intelligence indicates that Chen Hui is working closely with the Rabbit Kingdom's fighter jet research institute. Some time ago, he disappeared from Jiangcheng for more than two months. After he reappeared, these blue phoenixes also appeared in the airspace of the Desert Fox."

"The timing is too coincidental. We can reasonably suspect that these blue phoenixes are related to him!"

"You all probably still remember gallium oxide, right?"

"That kid was the one who invented the fourth-generation semiconductor."

A sharp glint flashed in Fury's remaining eye. His intuition, honed over years of intelligence work, told him that the Chinese fighter jets were closely related to that little guy.

"This is a once-in-a-lifetime opportunity. If we don't seize it, it will be much more difficult to make a move once he returns to the Rabbit Kingdom."

The white-haired old man looked at his advisor with some hesitation.

Strategist Roger smiled and shook his head. "I know the genius you're talking about. It's a real shame that someone like him was born in China. But he's just a theorist. You think he can help China develop a new generation of fighter jets in two months?"

"Are you kidding me?"

Everyone in the conference room nodded.

Their sixth-generation fighter jet PowerPoint presentation has been sitting there for six years and it's still just a PowerPoint presentation. How can a young theoretical guy turn a PowerPoint presentation into a sixth-generation fighter jet in just two months?
What a joke.

Perhaps this kid did play an important role, but to say that he single-handedly completed the development of the sixth-generation fighter jet is a bit of a fantasy.

"As for gallium oxide."

"Isn't this publicly available technology?"

“We have all benefited from this. Such a scientist should not fall into the hands of you dirty butchers. He should play a greater role!”

Roger said firmly, "And are we prepared to bear the wrath of the Rabbit Kingdom?"

The white-haired old man nodded. "Let's not talk about this anymore. S.H.I.E.L.D. cannot act on its own. The most urgent thing is to figure out what these Azure Phoenix-20s are all about. We won't be able to make any big moves in the oil depot area until we figure it out."

"You will regret it!"

Fury glanced at Roger with his one eye, then swept it over the others in the conference room. His fierce gaze had become hollow and filled with helplessness.

He certainly knew why these people opposed his proposal.

Because everyone here is a beneficiary of gallium oxide!
After gallium oxide was industrialized, the stock prices of the consortiums behind these individuals surged by at least 30%.

They're probably still hoping that kid will come up with another new material, which will make their net worth skyrocket again.

(End of this chapter)