Chapter 185 Tools Before Climbing Mount Everest

Noticing Deligne's gaze, Witten shook his head, handed him the manuscript in his hand, and said: "I can't make too many comments on this method in a short time."

"Although this method is very familiar with algebraic geometry, topology and manifolds, it is mixed with a lot of mathematical methods and concepts such as group theory and metric space, and these fields are not within my research scope."

"It is no exaggeration to say that I may not be able to understand this paper in a short time. After all, in terms of mathematics, the knowledge I have mastered is basically related to physics, and many of the mathematics in this paper's methods are not within my research scope."

Hearing this, Deligne showed a trace of surprise. As the first physicist to win the Fields Medal, Witten's mathematical ability is unquestionable.

Mathematical physics is originally a frontier and hot field in the mathematical community, because it provides a lot of new ideas and new problems for pure mathematics. In the foreseeable future, there will be many contributions from mathematical physics in the mathematical community.

And Witten has been working in the frontier fields of theoretical physics and mathematical physics for nearly 50 years. In the field of mathematical physics, he is now in the top three, not to mention the first place.

Now he actually said that he couldn't understand the method of this paper in a short time. How could this not be surprising?

With surprise and a hint of doubt, Deligne took the manuscript from Witten, and Witten took the manuscript left by Professor Mirzakhani and read it.

The dormitory illuminated by the setting sun was quiet again, but it didn't take long this time. In half an hour, Deligne and Witten put down the manuscripts in their hands one after another.

"It's really exquisite. It introduces concepts such as multiple fractals, dimensions, group theory, metric space, Riemann surfaces, and projective clusters. It's quite complicated. These things are not so easy for me to understand."

Putting down the manuscript in his hand, Professor Deligne exclaimed.

Now he understands why Witten said that he couldn't understand the method of this paper in a short time.

Not to mention Witten, even he needs a certain amount of time to understand the method of this paper.

Half an hour is just enough for him to roughly go through the manuscript in his hand, and there is almost no time to think about the core things.

The time was too short, even he couldn't really understand the method of this paper.

But a certain scientific intuition told him that with this tool, there might be hope to solve the Hodge conjecture.

Thinking of this, Deligne looked at Xu Chuan who was busy sitting on the other side, and felt a little emotional.

His disciple's mathematical ability has not surpassed him, but it is definitely not far behind him.

More importantly, he is still young and talented. He is less than 20 years old and has just entered the golden age of a scholar or researcher.

If he is given some more time, how far can he grow?

On par with his mentor, or even higher?

Suddenly, some figures appeared in Deligne's mind.

He didn't know why such an idea appeared in his mind.

But he knew that every time such a person appeared in history, he could promote the development of science and even the entire society by himself.

For example, Archimedes, Gauss, Newton.

Deligne and Witten left, leaving Xu Chuan alone in the dormitory to sort out the manuscripts in his hand.

Suddenly, he slapped his head in annoyance.

He remembered that he was so busy with his own business that he forgot to ask Wei Teng what he wanted to talk to him about.

He picked up his phone from the table and called Wei Teng. After two rings, the call was quickly connected.

"Hello, Xu Chuan, what's the matter?" Wei Teng's inquiring voice came from the other end.

"Teacher, did you call me for something? I just forgot to tell you." Xu Chuan asked.

"Actually, it's nothing. The collision experiment at CERN in the second half of the year has started. I have a group of experiments here. I want to ask if you are interested in participating. But now you have more important things on your hands, so forget it. You should deal with the paper in your hand first."

Hearing this, Xu Chuan's eyes lit up and he quickly asked: "The experiment at CERN? What is it about?"

For him in his previous life, he officially started to contact high-energy physics after 2018, so the experiments at CERN in 2016 and 2017 were blank for him.

If he had the opportunity to contact, he would naturally not give up.

Of course, he didn't want to participate in all collision experiments. After all, the LHC conducts a large number of collision experiments every year, and many experiments are not for discovering something new, but just to supplement some previous data or models.

Xu Chuan is not very interested in this kind of experiment, even if he has never experienced it. He is interested in exploring the unknown, discovering new particles, and surpassing the standard model.

"CERN's work in the second half of this year will mainly focus on the search for subatomic particles, in order to complete the standard model, and conduct a study on the basic symmetry between hydrogen atoms and antihydrogen atoms."

On the other end of the phone, Witten explained CERN's work arrangements.

"Looking for subatomic particles and basic symmetry?"

Xu Chuan touched his chin and thought for a while, then asked: "Teacher, when will the study of the basic symmetry between hydrogen atoms and antihydrogen atoms be conducted?"

He was not very interested in CERN's collision experiment to find subatomic particles. From his perspective, he knew basically all the subatomic particles discovered by CERN in recent years, and he knew who discovered them and the characteristics of these subatomic particles.

He was not very interested in cutting off other people's discoveries and opportunities in advance and doing things that were destined to happen in history.

He was quite interested in the study of the basic symmetry between atoms and antihydrogen atoms.

At least, in his memory, he had never heard of any major discoveries or breakthroughs in the study of basic symmetry in 2017.

So he might be able to take a look at this experimental study, but he didn't know if he would be able to do it in time.

After all, compared with a collision experiment, the most important thing at present is to supplement the tools at hand and try to make a breakthrough in the Hodge conjecture.

This is the most important thing at present.

As for the collision experiment of the European Atomic Energy Research Center, there will be a lot of time to do it in the future.

If there are enough discoveries, it is not impossible for CERN to repeat a particle collision experiment with exactly the same parameters and energy levels.

"The basic symmetry research between hydrogen atoms and antihydrogen atoms?"

"This experiment will be later. The current plan is to be carried out in December by the ALPHA laboratory. What, are you interested in this?"

Witten's curious voice came from the phone. Xu Chuan smiled and nodded, saying: "Well, I am very interested in basic symmetry research. I will not go to the subatomic particle discovery experiment. I hope I can finish my work before December."

Witten smiled and said: "Don't worry, you still have time."

After a pause, he continued: "If there is any new discovery on the Hodge conjecture, remember to let me know."

Xu Chuan nodded and said: "Of course."

After hanging up the phone, Xu Chuan immersed himself in the improvement of the mathematical method in his hand again.

Although the core idea has been obtained, it is not so easy to complete it.

Sometimes, some details that seem not so important may become the most fatal flaws in the method of the entire paper.

Just like when Wiles proved Fermat's Last Theorem.

At first, it was just that mathematician Katz found a flaw in the mathematical logic framework of the paper.

At first, both Wiles and Katz thought it was just a "small" defect that could be simply fixed.

But when Wiles started to fix the defect, the fragments of the logical framework that proved Fermat's conjecture began to fall off.

At this time, Wiles realized that this seemingly "small" defect was not a simple mistake, it might even exceed the scope of a repairable defect.

If it was a crack, an unrepairable defect, it would cause the entire proof of the theorem to collapse.

As time went by, more than eight months had passed, and Wiles still hadn't been able to fix the loophole. At this time, he became more and more panicked and even began to doubt himself.

But fortunately, he finally found a useful method in the ashes of the failed technology, which completely made up for the defect and proved Fermat's Theorem.

So in mathematics, in science, never underestimate those seemingly insignificant problems.

These seemingly insignificant problems are very likely to ruin all your previous achievements.

Days passed one by one.

With the core method already finalized, Xu Chuan did not lock himself in the dormitory anymore.

After more than a month of interruption, he resumed his studies. Every day, he would go to Deligne to study, and sometimes he would listen to some basic courses with other doctoral students.

But he spent more time in the Flint Library of Princeton University.

There are too many preparations to launch a charge against the Hodge conjecture.

The Hodge conjecture is simple, so simple that it can be described in one sentence, but it is also very complex, so complex that only a very small number of professional mathematicians can truly understand this problem.

"Every (certain type) harmonic differential form on a non-singular projective algebraic variety is a rational combination of the cohomology class of an algebraic closed chain."

When this sentence appears in front of you, do you find that you don't understand every professional term in this sentence.

For the Hodge conjecture, there is no simple analogy.

Among all the millennium problems, it is undoubtedly the most difficult to understand.

It clearly illustrates that the nature of modern mathematics makes most of it almost impossible for ordinary people to understand. In fact, you can't even find the threshold.

If you want to describe it in the simplest words, perhaps it can be described like this.

That is: "A palace, no matter how good or complex it is, can be built from a pile of building blocks." In the words of literati, it means: any geometric figure of any shape, no matter how complex it is (as long as you can think of it), can be made from a pile of simple geometric figures. "

For ordinary people, this should be the easiest to understand.

It's just that in actual work, we can't draw a complex multi-dimensional figure on a two-dimensional plane.

The Hodge conjecture is to split the complex topological figure into components, and we can understand the designer's ideas as long as we install them according to the rules.

This is probably the simplest understanding.

If solving the Hodge conjecture is likened to the first time humans climbed Mount Everest, it can be said that the mathematical community does not even have a base camp for Mount Everest.

Only when the tool in his hand is completely completed, the base camp for climbing the Hodge conjecture, Mount Everest, is established for the first time, preparing climbers to move towards the summit.

As for how long it will take to build this base camp, Xu Chuan does not know.

However, the progress of things seems to be much smoother than he imagined.

Previously, he promised Deligne and Witten a month, but now, less than twenty days have passed, and he has begun to finish the tool.

And the biggest breakthrough came from his ancestor, Mr. Grothendieck.

The Flint Library in Princeton is indeed a place with treasures. Here, Xu Chuan found some ideas left by the ancestor.

Some things about Motive theory, that is about the ancestor's construction of the standard conjecture.

It is about the representation of a series of Galois groups and some things about the Hodge structure in complex geometry, which completely determine the L function. In this case, the class field theory has solved the case of the commutative Galois group, allowing Xu Chuan to turn his thinking to the original Weyl group.

With this idea, he successfully completed the final supplementary work.

Staring at the manuscript on the table, Xu Chuan took a deep breath and then exhaled a long breath.

After two months of conception and arrangement, he finally completed the final work in September and built a tool that can charge against the Hodge conjecture.

After checking the method in his hand completely and confirming that there is no problem, Xu Chuan picked up the mobile phone on the table, opened the address book, found the number of his mentor Deligne and dialed it.

"Hello~, teacher, I need to take a leave for a while, and the time may not be sure."

After the call was connected, Xu Chuan said to the other end. After hearing his voice, the other end was silent for a while before asking: "Have you completed the improvement of the method of that paper?"

Maybe the sound of the mobile phone is a little distorted? The voice of the instructor on the other end of the phone trembled slightly, but the satisfaction and sense of accomplishment in his heart had made him ignore these.

Nodding, Xu Chuan responded: "Well, I just finished it today. Next, I may use it to try Hodge's conjecture, so I am going to ask you for leave. As for how long it will take, I am not sure."

The other end of the phone was silent for a while again, and after a while, Deligne's voice sounded.

"Good luck~"