Chapter 421: Smoothness Exists!

Translator: Henyee Translations  Editor: Henyee Translations

Lu Zhou originally thought he was used to this type of feeling.

He didn’t expect to feel his heart beating out of his chest.

This was different than the Princeton Institute for Advanced Study report; he wasn’t only facing the number theory world but the entire world of mathematics…

Lu Zhou stood on stage and took a deep breath as he tried to calm his heart down.

He looked at his watch.

The second hand was getting closer and closer; he put a serious look on his face and gave himself some courage.

“It’s about to begin!”

At exactly 9 a.m…

There was no need for anyone to maintain order; the noisy chaotic venue instantly became quiet.

A title appeared on the silver projector screen.

[Proof for the existence of a three-dimensional incompressible smooth Navier-Stokes equation solution.]

Lu Zhou looked at the crowd and began his report.

“Why doesn’t a car on the highway spontaneously disintegrate, why doesn’t a lake suddenly combust?

“We have been asking these questions for a long time, but the truth that we crave for is covered in disguise.

“In the 19th century, we have already invented the equations that generalize the laws of fluid emotion and made these equations succinct. However, to this day, we still don’t have a deep understanding of the mathematics and physics that are behind these equations.

“Mathematics is a rigorous discipline that involves the proposition of numbers, and there is no such thing as ‘maybe’ in mathematics.

“Back to my original question. Why doesn’t a car on the freeway disintegrate? Why doesn’t a lake spontaneously combust? Is there a mysterious singularity on an infinite time scale that causes our equation to diverge?

“It’s time to answer these questions.”

After his brief opening remarks, Lu Zhou turned the PowerPoint to the next slide.

This was the main section of the report.

Lu Zhou spent three seconds thinking of a summary. He then faced the audience and spent a minute giving a brief overview of his proof.

The crowd was silent.

Everyone stared at the pictures and calculations on the projector screen. Everyone was listening intently; they didn’t want to miss a single detail.

[μ(t)=e^(t△)·μ0+∫e^(t-t’)△B(μ(t’), μ(t’))dt’]

[…]

“When we introduce a Schwarz-free divergence vector field μ0 to the equation and set the time interval I ⊂ [0, + ∞), then we can define a generalized solution H10 of the Navier-Stokes equation as an integral equation μ (Continuous mapping of t), ie μ→H10df(R3)…”

The PowerPoint presentation was on the projector screen.

Lu Zhou had a laser pointer in his hand, and he used it to point at the screen while explaining.

This part was nothing special.

Any Navier–Stokes equation research theses would contain similar things.

However, the crucial part was his bilinear operator B’ and the L Manifold.

The next part was the key to the whole proof process!

Lu Zhou would introduce the concept of differential manifolds into partial differential equations.

This was the core idea of using topology methods to research partial differential equations!

…

Xu Chenyang was in the crowd, and he lightly tapped his notebook with the pen in his hand.

After a while, he whispered to Zhang Wei, “Do you understand?”

Zhang Wei shook his head and said, “I don’t know much more about partial differential equations any more than you. If you’re having a hard time, then so am I.”

Zhang Wei’s area of research was similar to his mentor Zhang Shouwu’s; he mainly focused on representation theory, Langlands program, and Dirichlet distribution.

He wasn’t knowledgeable at partial differential equations; he only briefly learned about the Navier–Stokes equation out of interest.

After all, not everyone was a genius like Tao Zhexuan. Not everyone could prove the weak Goldbach’s conjecture, study the Navier–Stokes equation’s abstract proof, and read all of Shinichi Mochizuki’s theses…

There were people in mathematics that knew everything.

But they were extremely rare…

Xu Chenyang looked at the calculations on stage and said, “I can’t believe it…”

Zhang Wei: “Can’t believe what?”

Xu Chenyang: “Number theory, abstract algebra, functions analysis, topology, differential geometry, partial differential equation… Is there anything he’s not good at?”

Zhang Wei said in an uncertain tone, “Maybe… algebraic geometry?”

However, he suddenly remembered Lu Zhou’s mentor was Deligne. Deligne’s mentor was Grothendieck, the founding father of algebraic geometry as well as the “pope of mathematics”.

The core theory of modern algebraic geometry was basically derived from the few books that Grothendieck wrote.

Zhang Wei was certain that Lu Zhou was well-versed in algebraic geometry as well.

He was certain Lu Zhou would eventually come up with new algebraic geometry research results…

…

The report continued.

Lu Zhou began to speak faster and faster; his ideas were becoming clearer and smoother.

The introduction of the L Manifold played a crucial role in the Navier-Stokes equation.

It was like a hammer that was breaking the maze wall.

This confusing situation became clearer and clearer.

They finally arrived at the climax of the report.

Fefferman sat in the corner of the venue with a smile on his face.

Tao Zhexuan was sitting on the other side of the venue, and he muttered to himself, “I see.”

His eyes sparkled with excitement.

Vera was sitting in the back row of the venue, and she could feel the enthusiasm in the atmosphere. Her heart rate began to increase, and she felt proud of her supervisor…

Faltings was also sitting in the back row; his rigid face finally turned into a smirk…

Deligne noticed his old friend smirking and asked, “What do you think?”

Faltings put on a poker face as he replied, “It’s okay.”

Deligne smiled and said, “You’re really saying that?”

Faltings ignored his old friend’s banter and looked at his watch. He then stood up.

Deligne asked, “It’s almost over, aren’t you going to stay till the end?”

“There’s no need.”

Faltings had understood everything already.

As for the boring questions, others could deal with it.

Faltings walked through the crowd and exited the hall.

The report ended the moment Professor Faltings left the lecture hall.

The last line of calculations was on the project screen; it was almost like Lu Zhou didn’t have to do any explanation.

Because the audience could see the answer for themselves.

“… Combining all of the above inferences, the result is obvious. There exists a smooth solution to the three-dimensional incompressible Navier Stokes equation!”

His voice was crisp and confident.

It wasn’t sonorous, but it was magically charming.

And the source of that magic was knowledge.

The second Lu Zhou finished speaking, the crowd stood up from their seats.

Then, a seemingly endless thunderous applause echoed throughout the lecture hall.

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