A genius? I just love studying.

Chapter 253 Mathematics Is Not a Person's Tombstone (Second Update)

June 7th is the day of China's annual college entrance examination.

In Philadelphia, a major quadrennial conference for the mathematics community is also being held in the lobby of the convention center.

The opening ceremony has come to an end, and the main lecture hall of the International Congress of Mathematicians is already packed with people. Even the large exhibition hall, which can accommodate 3,000 people, is now completely full, and many people can only stand in the gaps.

However, the hall remained unusually quiet as everyone craned their necks towards the main stage, waiting for the opening report to begin.

Many reporters, armed with cameras and microphones, were already positioned in various corners of the venue, eager to witness the proof of the Navier-Stokes equations!
Many of them may not understand the content of this report, but that doesn't stop them from using that legendary figure as the subject of their fantasies. They know that today's report is destined to become a breaking news story.

A near-sacred tension permeated the air, as if awaiting the eruption of a new star.

In the front row, deep red seats, Shing-Tung Yau, with his silver hair, stood solemnly, his knuckles unconsciously tapping the armrest with a rhythm as precise as the zero distribution of the Riemann zeta function. Perelman was hidden in the shadows in the corner, his eyes like those of a hawk beneath his signature curly hair. Andrew Wiles wiped his glasses, a hint of a smile of witnessing history playing on his lips.

In the shadows of the northwest corner of the auditorium, Dennis Sullivan stood alone against the wall, his gold-rimmed glasses reflecting the cold light of the stage. He toyed with a model of a braided braid made of copper wire in his hand, his knuckles turning white from the force.

The aisle behind was packed with young scholars standing, their cell phone screens glowing like a sea of ​​stars, and every camera lens was focused on the slender figure on the stage—Chen Hui.

On the huge screen behind him, there was only one concise title, yet it struck like a thunderbolt: "A Complex Geometric Proof of the Smoothness of Short-Time Solutions to the Navier-Stokes Equations".

Speaker: Chen Hui (Huaxia)
When Chen Hui stepped onto the podium, all the whispers vanished instantly.

"First of all, I would like to thank Professor Dennis for his help in the proof process. Without his help, I could not have completed the proof so quickly. 30% of the results of this proof belong to Dennis."

Chen Hui began by saying that the Chinese academic community has always only considered first authors, while in the Western academic community, second and third authors are equally valuable, usually distinguished by their level of contribution. Although the two are no longer collaborators, he did indeed use Dennis's work.

Dennis, sitting in the corner, looked at Chen Hui on the stage, opened his mouth, but ultimately said nothing.

After searching the audience without success, Chen Hui returned to his speech. He lightly pressed the controller in his hand, and the screen behind him instantly switched to a dense array of formulas.

"Professor Dennis once told me that the essence of vortices lies hidden in the crevices of topology, but I want to tell you that complex geometry can give it its soul!"

Chen Hui looked down at the crowd with confidence and high spirits, "Today, what I want to prove is how this 'bone' and 'soul' can tame the rage of the NS equation."

"How does a four-dimensional complex Kellerian manifold embed three-dimensional spacetime? How does dx∧dy∧dz∧dt in the Kellerian form both carry the measurement of physical spacetime and implicitly contain information about vortex dissipation?"

Chen Hui began explaining his core structure, “Notice the ν parameter here,” he pointed his laser pointer at the screen, “it’s not an artificially introduced correction term, but a harmonic factor naturally derived from the strong quasi-convexity of the complex manifold…”

Chen Hui was completely immersed in his own world, expressing his thoughts and ideas without reservation, forgetting the passage of time.

Gromov, sitting in the front row, suddenly sat up straight and drew a heavy line on his notebook with his pen.

Terence Tao's fingers stopped, and his pupils contracted slightly—this ingenious construction, which embeds the energy dissipation term of the Navier-Stokes equations into the regularization framework of complex geometry, is a key breakthrough in the "nonlinear regularization" problem that has plagued the academic community for thirty years.

Dennis nodded slightly, acknowledging Chen Hui's core design.

“Next is the Neumann estimation,” Chen Hui’s voice rose with excitement, “After proving the strong quasi-convexity of the boundary, we get a counterintuitive conclusion: the upper bound of the L norm of the operator □1(ˉω) is a constant C independent of the Reynolds number.”

A collective gasp filled the room.

The Reynolds number is a key parameter in fluid mechanics for describing turbulence. In traditional methods, any estimate that is not related to the Reynolds number is considered "impossible"—because the complexity of turbulence explodes exponentially as the Reynolds number approaches infinity.

At this moment, an equation appeared on the large screen: ωL2=(∫Ω∣ω∣2dx)1/2
Like a golden key, it was inserted into the most robust lock of the NS equation.

Fefferman, Schultz, and others in the front row listened with rapt attention until they saw the equation, at which point they realized the final moment was fast approaching.

"The energy dissipation of vortex annihilation is precisely controlled by the first Chen class c1."

Sure enough, Chen Hui's voice rang out the next moment.

He brought up a simulation animation of magnetic powder particle flow. Silver "stardust" outlined the contours of complex fiber bundles in the void, eventually converging into a shimmering formula: Φ≤Λ∣c1(V)∣

"This means that as long as the Chern class of the complex fiber bundle is finite, the short-time solution of the Navier-Stokes equations must be smooth!"

After Chen Hui finished speaking, he took half a step back and looked at the crowd below.

The hall was silent for several dozen seconds before applause gradually broke out, and then it exploded like muffled thunder.

Lord Atiyah was the first to stand up in the front row, followed closely by Gromov. Terence Tao's eyes were red-rimmed, and Dennis Sullivan's hands were red from clapping, his knuckles turning white.

"Professor Chen!" the host, Yves, raised his voice, "Before we proceed with the Q&A session, please allow me, on behalf of the mathematics community, to pay tribute to you and Professor Dennis!"

The applause erupted once again.

After the applause subsided slightly, Chen Hui spoke again, "Does anyone have any questions about the proof process?"

He has completed the proof of the Navier-Stokes equations for over a month, but there is still no sign of free attribute points. He doesn't know if it's because he had already proven the Yang-Mills equations, and proving a conjecture of the same level only grants one free attribute point, or if there is some other reason.

However, Chen Hui believes that based on the previous proof of the Yang-Mills equation, he needs to gain recognition from the international mathematics community to obtain the free property points. Therefore, he hopes to explain as clearly as possible, even more so than the audience.

"Professor Chen, in the four-dimensional complex Kellerian manifold you constructed, does the proof of strong quasi-convexity implicitly impose restrictions on the initial conditions? If the initial vorticity distribution is extremely irregular, such as satisfying the collapse of the H^s norm as s→∞ lim f(s), will your estimate still hold?"

Chen Hui smiled slightly and pulled up a backup slide. "Professor Dennis's question hits the nail on the head. In fact, our strongly quasi-convexity condition depends only on the complex structure of the base space X, not on the specific form of the initial data. The key is..."

He circled the ν(μ)(μ)gd4x term in the Keller form with a laser pointer. “This term automatically compensates for the singularity of the initial data through the curvature tensor of the complex manifold, just like the ‘braid group correction factor’ that you introduced in the topological method, Professor Dennis.”

Dennis suddenly realized, stared at the screen for a long time, and finally retreated to the wall, saying nothing more.

He never expected that Chen Hui would solve the problem that had troubled him for so long using a method he had used before. He simply hadn't anticipated it.

A few more questions were asked, but most of them were not related to the content of Chen Hui's report. Although there were several thousand people in the hall, no more than a handful could hear Chen Hui's report. Those who could understand it had already read Chen Hui's paper and had privately discussed it with him the day before.

However, Chen Hui answered all of these questions one by one.

The presentation lasted an hour, with only fifteen minutes allotted for questions.

As Chen Hui finished packing up his speech and stepped off the stage, a barrage of comments suddenly flashed before his eyes.

[Congratulations, host! You have completed the proof of the Navier-Stokes equations. Free attribute points +1]

Chen Hui smiled happily. Sure enough, a result of the Millennium Problem's caliber did not have the same level of penalty restrictions on acquiring free attribute points.

"Congratulations to Professor Chen for proving another Millennium Problem."

Emily, a senior science reporter from Nature magazine, came up to him. Seeing the smile on Chen Hui's face, she was delighted. "Professor Chen, could you give us a few minutes?"

"of course."

She's very lucky; Chen Hui is indeed in a good mood right now.

Reporters from other media outlets had already swarmed over, and upon hearing this, they beamed with joy, jostling to get in front of Chen Hui.

Thank you for accepting our interview!

Emily flashed a bright smile, revealing her pearly white teeth, and handed the microphone to Chen Hui. "First of all, could you explain to our readers in the simplest terms what you have proven?"

Chen Hui looked at the coffee cup next to him and said with a smile, "Imagine you have a cup of hot cocoa with milk foam floating on the surface—the Navier-Stokes equations are like the equations of motion for this cup of cocoa; they describe how the fluid flows and how it dissipates energy."

But for a century, mathematicians have been unable to solve one problem: when this cup of cocoa is violently agitated, such as by high-speed air or water flow, can its 'smoothness' be mathematically guaranteed? Could a 'singularity' suddenly appear, causing the entire model to collapse?

“And the work,” Chen Hui gestured with his hand in a spiral motion, “is to use complex geometry to ‘weave a net’ for this cup of cocoa. This net not only encapsulates the movement of the fluid, but also uses Chen-type mathematical rulers to precisely measure the rate of energy dissipation.”

In short, we have proven that as long as the fluid is not infinitely insane (i.e., the Reynolds number is finite), this net can 'catch' it and prevent it from reaching a singularity.

"Sounds like you've insured the turbulence?" the host of the tech-focused social media account "Mathematical Universe" interjected, the camera almost touching Chen Hui's face.

"To be more precise, it's like insuring the existence of a smooth solution," Chen Hui corrected, his gaze sweeping across the audience—many people hadn't left after the presentation; several young mathematicians were standing at a distance recording with their phones.

“Traditional methods are like trying to tie up a flood with ropes, the more you tie it, the more tangled it becomes; our method is like building an ingeniously structured bridge, allowing the flood to flow in a regular pattern within the bridge arches.”

The reporters seemed to understand but not quite, while Edward Witten, who was still in the front row and hadn't gone far, nodded approvingly.

At this point, a science reporter from The New York Times raised a more pointed question: "Professor Dennis's topological approach and your current complex geometry framework have always been considered 'two different paths.' Do you think this breakthrough is a victory for the 'topological school' or the 'complex geometry school'?"

"The two are never in opposition."

Chen Hui shook his head. "Topology is the skeleton, which defines the basic structure of space. Complex geometry is the soul, and differential equations describe the dynamics. Without the skeleton, the soul has nowhere to reside; without the soul, the skeleton is just a stone!"

"One last question, Professor Chen," a BBC science reporter raised his hand. "Many young scholars, upon hearing that you have proven the short-time smoothness of the Navier-Stokes equations, might think, 'The Millennium Problem has finally been solved.' What are your thoughts on this?"

Chen Hui's smile carried a hint of weariness, yet it made it all the more sincere.

He recalled the young mathematicians with red eyes in the lecture hall, and the mountains of failed drafts in his office. "The story of the Navier-Stokes equations is never about solving them, but about understanding them."

He said, "We have proven the smoothness of short-time solutions, but what about longer timescales? What about the ultimate structure of turbulence? These questions may require the next generation of mathematicians, and the generation after that, to explore."

"Just like when Hilbert posed twenty-three problems in 1900, no one expected that the seventh one (Waring's problem) would be solved a century later, while the eighteenth one (Riemann Hypothesis) remains a mystery to this day."

The charm of mathematics lies precisely in the fact that there is always another peak to climb!

Chen Hui was indeed in a good mood, but he couldn't stay there and let them ask questions forever. He apologized and walked towards the backstage area.

The reporters immediately felt a pang of regret. They had interviewed many scholars, but their answers were often vague and difficult to understand.

Chen Hui, however, is different. Every answer he gives is easy to understand, even for ordinary people who don't understand mathematics. This ability to express himself in a simple and concise way is also rare in the academic world.

They naturally prefer to interview scholars like this.

Of course, what they value even more is the popularity Chen Hui generates.

At just nineteen years old, he has already solved two Millennium Problems and become a celebrity in academic circles. Any news with Chen Hui's name on it often gets a lot of clicks and even trends on social media.

Chen Hui had no idea what the reporters were thinking. He only vaguely heard Emily, a reporter from Nature, tidying up the recording as she left, muttering, "This metaphor of the mathematical bridge is brilliant. It's sure to be tomorrow's headline."

As soon as he stepped backstage, an elderly man with white hair stopped him—it was Gromov, the titan of differential geometry.

“Young man,” the old mathematician patted him on the shoulder, “what you just said about the mathematical bridge reminds me of the proof of the Calabi conjecture in 1957. At that time, Qiu Chengwu also used geometric structures to connect analysis and topology. The progress of mathematics has always been a relay race like this.”

Looking at the starlight in the old mathematician's eyes, Chen Hui suddenly remembered what his teacher Yuan Xinyi often said: "Mathematics is not one person's tombstone, but a group of people's everlasting lamp."

He did complete the proof on his own, but without the cooperation of Dennis, he could not have completed it so quickly. Dennis's contribution is undoubtedly a part of this proof.

(End of this chapter)