Chapter 185 Tools before climbing Mount Everest
Noticing Deligne's gaze, Witten shook his head, handed him the manuscript paper in his hand, and said: "I can't make too many comments about 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 spaces, and these fields are not within the scope of my research."< br>
"It is no exaggeration to say that I may not be able to understand this paper in a short period of time. After all, in terms of mathematics, the knowledge I have is basically related to physics. Many mathematics in the methods of this paper are not in my knowledge." Within the scope of research ”
Hearing this, Deligne showed a look of surprise. As the first physicist to win the Fields Medal, Witten's mathematical ability is beyond doubt.
Mathematical physics has always been a frontier and hot field in the mathematics community, because it provides a large number 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.
Witten has been working in the cutting-edge fields of theoretical physics and mathematical physics for nearly fifty years. In the field of mathematical physics, he is now not only the first person, but also the top three.
Now he actually said that he could not understand the method of this paper in a short time. How can this not be surprising?
With surprise and a trace of doubt, Deligne took the manuscript paper from Witten, and Witten took the manuscript left by Professor Mirzakhani and read it.
In the dormitory illuminated by the setting sun, it became quiet again, but this time it didn't take long. Within half an hour, Deligne and Witten put down the manuscripts in their hands.
"It is indeed exquisite. It introduces concepts such as multifractals, dimensions, group theory, metric space, Riemann surfaces, projective varieties, etc. It is quite complicated. Even I want to understand these things. It is not that easy."
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Putting down the manuscript in his hand, Professor Deligne admired.
Now he finally understood why Witten said that he could not understand the method of this paper in a short time.
Not to mention Witten, even him, it will take a certain amount of time to understand the method of this paper.
Half an hour was only enough time for him to go through the manuscript paper in his hand roughly, and there was almost no time to think about the core things.
The time was too short, even he could not really understand the method of this paper.
But some kind of scientific research intuition told him that with this tool, there might be hope to solve the Hodge conjecture.
Thinking about it, Deligne glanced at Xu Chuan, who was sitting on the other side and was busy doing something, and felt a little emotional.
His disciple's current mathematical ability has not only surpassed him, but it is definitely not far behind him.
What's more important is that he is still young and talented. He is less than twenty years old and has entered the golden age of a scholar or scientific researcher.
If you give him some more time, how far can he grow?
As good as the mentor, or even higher?
Suddenly, some figures appeared in Deligne's mind.
He didn't know why such thoughts came to his mind.
But he knew that every time such a person appeared in history, he could single-handedly promote the development of science and even the entire society.
Such as Archimedes, Gauss, Newton.
Deligne and Witten left, leaving Xu Chuan alone in the dormitory to sort out the manuscript materials in his hands.
Suddenly, he slapped his head in frustration.
He remembered that he had just been preoccupied with his own affairs and forgot to ask Witten about what he had called him about before.
He picked up his cell phone from the table and dialed Witten. After two rings, the call was quickly connected.
"Hey, Xu Chuan, what's the matter?" Witten's questioning voice came over from the other end.
"Teacher, did you call me for something before? You just forgot to mention it." Xu Chuan asked.
"Actually, it's nothing. The collision experiments at CERN in the second half of the year have already started. I have a group of experiments here and I want to ask if you are interested in participating. But now you have more important things on hand, so forget it. You should deal with the paper in hand first.”
Hearing this, Xu Chuan's eyes lit up and he quickly asked: "The experiment at CERN? What is it about?"
In his previous life, it was 18 years since he officially started to contact high-energy physics, so the CERN experiments in 2016 and 2017 were blank to him.
If there is a chance to contact him, he will naturally not give up.
Of course, he doesn't want to participate in all collision experiments. After all, the LHC conducts a large number of collision experiments every year, and many of the experiments are not to discover something new, but just to make up for some previous data or models. .
Xu Chuan is not very interested in this type of experiment, even if he has never experienced it. What he is interested in is the exploration of the unknown, the discovery of new particles, and the transcendence of 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 fundamental symmetries?"
Xu Chuan touched his chin and thought for a moment, then asked: "Teacher, when was the basic symmetry between hydrogen atoms and antihydrogen atoms studied?"
He is not very interested in Cern's collision experiment to find sub-atomic particles. From his perspective, he basically knows about the various sub-atomic particles discovered by CERN over the years. He knows who discovered them, and he also knows very well about these sub-atomic particles. Properties of original particles.
He is not very interested in cutting off other people's discoveries and opportunities in advance and doing things that are destined to happen in history.
Instead, 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 basic symmetry research in the past seven years.
So he might be able to take a look at this experimental research, but he didn't know if it would be too late in terms of time.
After all, compared with a collision experiment, the most important thing at the moment is to supplement the tools at hand and try to make a breakthrough in the Hodge conjecture.
This is the most important thing right now.
As for the European Atomic Energy Research Center's collision experiment, there is still 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.
"A study of fundamental symmetries between hydrogen and antihydrogen atoms?"
"This experiment will be delayed for a while. The current plan is to conduct it in December in 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'm quite interested in basic symmetry research. I won't go to the experiment of discovering subatomic particles. I hope I can You can finish what you are doing before December.”
Witten smiled and said, "Don't worry, you still have plenty of time."
After a pause, he continued: "If you find anything new about the Hodge conjecture, remember to let me know." Xu Chuan nodded and said, "Of course."
After hanging up the phone, Xu Chuan re-immersed himself in perfecting the mathematical methods in his hands.
Although the core idea is already there, it is not that easy to complete it.
Sometimes, some trivial, seemingly unimportant things may become the most fatal flaw in the entire paper method.
Just like when Wiles proved Fermat's last theorem.
At first, it was just the mathematician Katz who found a flaw in the mathematical logic framework of the paper.
At first, both Wiles and Katz thought this was just a 'minor' flaw that could simply be fixed.
But as Wiles set about fixing the flaws, the pieces of the logical framework for proving Fermat's conjecture began to fall apart.
At this time, Wiles realized that this seemingly 'small' flaw was not a simple mistake, it might even be beyond the scope of a repairable flaw.
If it were a crack, an irreparable flaw, it would collapse the entire proof of the Last Theorem.
As time went by, more than eight months passed, and Wiles still failed to fix the vulnerability. 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, completely made up for the shortcomings, and thus proved Fermat's last theorem.
So in mathematics and science, never underestimate those seemingly inconspicuous problems.
These seemingly inconspicuous problems are very likely to ruin all your previous achievements.
The days passed like this day by day.
Now that the core method has been finalized, Xu Chuan no longer locks himself in the dormitory.
After a break of more than a month, he resumed his studies. Every day he would go to Deligne to study, and sometimes he would also attend some basic courses with other doctoral students.
But most of my time was spent in the Flint Library at Prince University.
To launch a charge against Hodge's conjecture, there are too many preparations required.
The Hodge conjecture is very 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 of) 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 do not understand every professional term in this sentence?
For the Hodge conjecture, I can't even find some simple analogies.
Of all the Millennium Puzzles, it is certainly the most difficult to understand of all the Millennium Puzzles.
It illustrates most clearly that the nature of modern mathematics makes much of it almost impossible for ordinary people to understand it. In fact, you can't even find the threshold.
If you want to describe it in the simplest terms, maybe it can be described like this.
That is: "No matter how good or complex a palace is, it can be built from a pile of wood." In the words of literati: any geometric shape, no matter how complicated it is (as long as you can think of it), it can be made up of a bunch of simple geometric figures. ”
This should be the easiest for ordinary people to understand.
It's just that in actual work, we cannot draw a complex multi-dimensional figure on two-dimensional paper.
The Hodge conjecture is to break down complex topological graphics into component parts. As long as we install them according to the rules, we can understand the designer's ideas.
This is probably the simplest understanding.
If solving the Hodge conjecture is compared to humans climbing Mount Everest for the first time, it can be said that the mathematical community does not even have a base camp for Mount Everest yet.
Only when the tool in his hands was completely completed, Hodge guessed that the base camp for climbing Mount Everest was established for the first time, preparing climbers to advance to the summit.
As for how long it will take to establish this base camp, Xu Chuan doesn't know.
However, things seemed to be progressing much more smoothly than he imagined.
The time he promised to Deligne and Witten was one month, but now, less than twenty days have passed, and he has already begun to put the finishing touches on this tool.
And his biggest breakthrough point actually came from his ancestor, Mr. Grothendieck.
The Flintstone Library in Princeton is indeed a place with treasures. Here, Xu Chuan found some of the ideas left behind by his ancestor.
Something about the Motive theory is about the construction of the standard conjecture by the founder.
is something about the representation of a series of Galois groups and the Hodge structure in complex geometry, which completely determine the L function, and in this case, class field theory has solved the case of commutating Galois groups, let Xu Chuan turned his thoughts to the original weyl group.
With the help of this idea, he successfully completed the final supplementary work.
Staring at the manuscript paper on the table, Xu Chuan took a deep breath, and then exhaled a long breath.
After two months of conceiving and organizing, he finally completed the final work in September and built a tool that can challenge the Hodge conjecture.
After thoroughly checking the method in his hand and confirming that there was 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.
"Hey~, teacher, I need to take another period of leave. The time may not be easy to determine."
After the call was connected, Xu Chuan said to the other end. After hearing his voice, the other end was silent for a rare moment before asking: "Have you completed the improvement of the method of that paper?"
Maybe the phone sound is a little distorted? The instructor's voice on the other end of the phone was slightly trembling, but the full sense of satisfaction and accomplishment in his heart made him ignore it.
Nodding, Xu Chuan responded: "Well, I just finished it today. Next, I may use it to try the Hodge conjecture, so I am going to ask you for a leave. As for how long it will take, I am not sure. ”
The other end of the phone was silent again for a while, and after a while, Deligne's voice rang.
"Wish you good luck~"
(End of chapter)