Chapter 820 The Key to the Poincare Conjecture

Perelman did not simply rewrite the steps that Chang Haonan had sent him at the beginning-

For a mathematician of his level, doing such a thing is somewhat unworthy.

Instead, he made some optimizations based on that proof method.

"In order to further reflect the beauty of this proof, we first introduce a concept: Τ-length..."

"Do you still remember this...?"

While Perelman was writing equations on the blackboard, a young teacher looked at the blackboard that had just been wiped clean and the notebook in front of him with several pages of dense writing, shook his sore wrist, and whispered to his girlfriend next to him.

He was not a researcher in the field of differential geometry. He just served as a ruthless note-taker, and now...

I really can't write anymore.

"Of course I have to remember it. Look, even Professor Chang is lowering his head to take notes. Are you better than him?"

Several people who heard this sentence immediately cast their eyes to the distance...

It was found that Chang Haonan, who had just been sitting and listening, had taken out a notebook from nowhere and was writing and drawing on it.

"Hiss..."

Another sound of inhaling.

Then there was the sound of turning pages.

Finally, there was the rustling sound of paper and pen rubbing against each other...

However, if someone who was close to Chang Haonan took a look, they would find that what Chang Haonan wrote on the paper was not what was on the blackboard.

Instead, he drew a ball with a pencil.

This is an extremely rare situation.

Because for the research in the field of differential geometry, high-dimensional space is often easier than low-dimensional space.

Take the Poincaré conjecture as an example. The Poincaré conjecture in five-dimensional or even four-dimensional space has actually been proved long ago.

But the barrier of three-dimensional space has never been overcome.

And as we all know.

It is impossible to draw a high-dimensional space on paper.

It can only be imagined or calculated.

In fact, even what Perelman said on the blackboard at this moment is mainly based on four-dimensional space.

However, the contents he optimized on the blackboard pointed out a new possibility to Chang Haonan...

"If this is a free isometric quotient space generated by the action of a finite group, then it seems to be differentially homeomorphic to a three-dimensional compact manifold..."

Chang Haonan could no longer hear Perelman's voice in his ears:

"It seems that we can't draw such a conclusion directly."

He frowned slightly:

"But if we add a limiting condition...make the Ricci curvature of this manifold non-negative..."

"..."

Under the stage, Chang Haonan was lowering his head, immersed in his own thoughts.

On the stage, Perelman was giving a lecture as usual.

According to the plan, after comparing the three types of singular models, he will be able to derive the same conclusion as before.

After using up one blackboard again, Perelman walked to the next one as usual.

But this time, he didn't start writing immediately.

He raised his hand to wipe the sweat from his forehead.

He had been speaking continuously on the stage for nearly two hours.

He really couldn't keep up with his energy and physical strength.

In fact, the idea on the blackboard was even thought of by him on the plane to China. He used it as the content of the lecture, which also meant to introduce and verify it at the same time.

Therefore, it is much more laborious than a simple lecture.

Fortunately, the staff next to him had already prepared it, and took this opportunity to quickly put a cup of warm water on the small table-

If it was a Chinese scholar, hot tea would usually be served directly at this stage, but considering that foreigners might not be used to this step and get scalded, the temperature was lowered under Tang Lintian's special care.

Perelman was not polite, and went to the table to pick up the teacup. While drinking water, he looked at the first two blackboards that he had written on.

Suddenly, his hand stopped.

His sight focused on the bottom of the first blackboard.

Since it was the first time to systematically sort out this method, there were some details that even Perelman himself could not notice at the first time.

There was an inequality there.

R≥(-v)[lg(-v)+lg(1+t)-3]

Originally, he just regarded it as a normal estimate generated in the derivation process, but now looking back, it seems that some very interesting conclusions can be obtained along this direction...

For example, when the curvature approaches infinity at a moment, the smallest negative cross-sectional curvature is smaller than the largest positive cross-sectional curvature.

In other words, the three-dimensional limit solution must have a non-negative curvature operator.

That's right, three dimensions.

Perelman didn't even have time to put down his teacup, and turned to look at Chang Haonan sitting in the audience.

He found that the latter was concentrating on writing something with his head down.

At this time, Chang Haonan finally proved his conjecture on paper.

He raised his head.

His eyes suddenly met Perelman's.

Although the two did not say anything to each other, they saw one thing from their eyes-

The other party and themselves thought of the same thing.

Two top scholars in the field of differential geometry, through relatively independent thinking, finally came to the same conclusion.

That basically ruled out the possibility that this conclusion was wrong.

That is to say, it is feasible to perform surgery on Ricci flow in three-dimensional space.

For the differential geometers of this millennium, there is a consensus.

To solve the Poincaré conjecture problem in three-dimensional space, the geometric method using Ricci flow is more feasible than the direct topological method.

therefore.

This could very well be a key.

A key to Poincaré's conjecture.

Of course, even if you do open the door with the key, there's still some work to be done.

For example, it is necessary to ensure that the appropriate neck area can be found for truncation surgery within a limited number of operations.

It is also necessary to solve the problem that the general initial metric causes the Ricci flow to produce singular points.

but.

These are details.

It can even be said that it is a problem that can be solved by just spending time.

If we say that Chang's Lemma is just the first step in a long journey for Poincaré's conjecture.

So today’s conclusion——

Perhaps it can be called the three-dimensional manifold theorem, or more simply, the Perelman-Constant theorem, which can be regarded as "a man who travels a hundred miles is half a mile".

Of course, neither Perelman nor Chang Haonan would agree to use the combination of their last names in this place.

Because if they continue in this direction, their surnames will most likely be directly named after "Poincaré".

…

At the same time, other listeners below were also taking advantage of this rare buffer period to review the contents of the notes they had just written down.

Of course, these people were not involved in the initial derivation of the process on the blackboard, so their mindset dictated that they would definitely think along the steps Perelman wrote on the blackboard without seeing that, at least in It will not be obvious in a short time that one of these humble inequalities will have a historic impact on the entire mathematical world.

However, most of them are professional mathematicians after all, so it is impossible to gain nothing...

"I seem to understand..."

Tian Gang was the first to frown.

Although Perelman has not yet finished writing the entire derivation process, he has already thought of the remaining steps.

Compared to the first solution, which is a headache, the one currently written on the blackboard is obviously much simpler and easier to understand.

"It is indeed...an extremely sophisticated proof...this allows you to directly calculate the local injective radius of the compact manifold..."

"It's called the non-local collapse theorem, and it's very accurate..."

He whispered to himself, attracting the attention of several people next to him.

Soon, Tian Gang's notes were circulated.

Perelman, on the stage, still stood there with his arms folded, looking at the blackboard in front of him, without speaking.

The lecture hall that had just been quiet gradually started to hear some whispers.

"As expected of Professor Tian... he can get the results faster than I did first."

A scholar looked at the notebook in front of him, then raised his head to check the half-written content on the blackboard.

"Where... I've been standing there for two hours, and I'm halfway through pushing. It's just because of my energy that I took advantage..."

Tian Gang waved his hand.

Even so, he was still relatively happy -

Although the result definitely belongs to Perelman, and perhaps to Chang Haonan who was just mentioned by the other party, the fact that he was able to deduce it on his own before reading the proof process at least shows that he has not fallen too far behind in terms of ability...

Of course, he didn't know it yet.

Perelman stopped because he could see a path leading to a higher mountain.

In comparison, these things on the blackboard are not worth mentioning at all...

"Please give me some paper, thank you."

After a few minutes of silence, Perelman's first words were not to continue introducing his ideas, but to ask the floor manager for pen and paper.

"Um?"

"Why do you want these all of a sudden..."

"Could it be that there is something wrong with the derivation process?"

"No way... I just vaguely heard that the previous teachers have already introduced this idea, and I understand it and it feels pretty smooth..."

"Is there a possibility that even we can understand it, but it is more likely that there is something wrong?"

"Don't..."

"..."

The strange behavior made the scene agitated again.

But Perelman ignored this.

He took the pen and paper, sat in front of the temporary desk, lowered his head and started his calculations...