Chapter 184 Pretending in Front of the Two Mentors

Chapter 184 Pretending to be good in front of two mentors

On the side, Witten raised his head and said: "I think the Weyl group mapping of algebraic clusters is more essential here. It directly twists the application of the maximal torus, which gives me a very amazing feeling."

After a pause, Witten added: "This is a brand new idea, and perhaps it can be further expanded on it."

Hearing this, Professor Deligne thought for a while, his eyes suddenly lit up, and he quickly said: "Topology and algebraic manifolds of algebraic clusters!"

Witten smiled and said: "Yes, I should be more sensitive to this than you. You know, I am good at quantum field theory, string theory and related topology and geometry."

"If we start from this aspect and continue to extend it, it may become a new tool for solving the differential form type problems generated by non-singular projective complex algebraic clusters"

Before Witten finished speaking, Deligne added: "For example, the Hodge conjecture."

On the side, Xu Chuan looked at the two mentors with a smile, and sighed in his heart.

As expected of two of the top mathematicians, they noticed the two most critical and core points in this paper on the association method of domain-theoretic algebraic clusters after just one reading.

One is used to solve the problem of irreducible decomposition of differential algebraic clusters, and the other is extended to solve the Hodge conjecture.

Even though the two mentors in front of them are already in their sixties and seventies, their sensitivity to mathematics is still terrifyingly high.

Any details, even if they are just some things that occupy a small space, cannot escape their eyes.

Hearing Deligne's words, Witten put down the manuscript in his hand and said, "Indeed, it may have such potential, but it is still unknown whether this road can be taken."

After a pause, he looked at Xu Chuan and continued, "I wonder if you have thought about this?"

Xu Chuan grinned and said, "Of course, in fact, I am almost done with this work."

Hearing this, Deligne and Witten were stunned at the same time.

Almost done, what does it mean?

"Have you solved the Hodge conjecture?" Witten couldn't help but ask tentatively. If that's what he meant, it would be too scary.

Xu Chuan shook his head and said, "No, I just extended this line of thought and used it to make a mathematical tool."

"I have been dealing with this these days. I haven't finished it all yet. I just made a core and haven't had time to organize it. Otherwise, I would have brought it here today."

Hearing that the Hodge conjecture had not been solved, Witten and Deligne breathed a sigh of relief at the same time.

If their student solved the Hodge conjecture in more than a month, it would be too scary.

This is the Hodge conjecture, one of the seven millennium problems.

The millennium is a thousand years on the calendar. As the name suggests, it means a thousand years. The first millennium is 1000 years, and the second millennium is 2000 years.

Of course, the seven millennium problems are not problems that require humans to spend a thousand years to solve.

Instead, the seven mathematical conjectures announced on May 24, 2000 are called the seven millennium problems because of the special year.

Although it does not mean that it will take a thousand years to solve, when the Clay Mathematics Institute and top professors such as Wiles and Connes formulated these seven mathematical problems, they were prepared for the mathematical community to solve them in a whole century.

A century, a hundred years, to solve seven mathematical problems, it can be seen that these seven mathematical conjectures are difficult.

And the facts also prove the difficulty of these seven problems. Up to now, more than ten years have passed, and only the Poincaré conjecture has been solved.

This was completed after countless mathematicians worked hard after the 1930s.

From the Whitehead manifold and Dean's lemma in the 1930s, to the high-dimensional Poincaré conjecture in the 1960s, to the Ricci curvature flow in the 1970s and 1980s.

Countless people have made great contributions to the Poincaré conjecture, and finally Perelman covered the roof of this century-old problem.

In addition to the Poincare conjecture, if there is any progress on the other Millennium Problems, it may be the BSD conjecture.

In 2014, Fields Medal winner and Princeton University professor Manuel Bhargava said that "the number of solved problems among the seven Millennium Problems may be one more than I expected."

Professor Bhargava recently reported a number of results related to the Behe ​​and Swinnerton-Dyer conjectures.

In one of the results, he said that he and his colleagues "proved that more than 66% of elliptic curves satisfy the Behe ​​and Swinnerton-Dyer conjectures."

This means that the progress of conquering the Behe ​​and Swinnerton-Dyer conjectures, that is, the BSD conjecture, has exceeded half.

Of course, no one knows how long it will take to conquer the remaining half.

Maybe three years, maybe five years, or maybe thirty to fifty years.

Even if you can already look up at the top of the mountain, before you reach the top, no one can know how tortuous the road ahead is, and whether there is an abyss that cannot be crossed.

Apart from this, there has not been much progress on the other several millennium problems.

And for super problems like the Riemann hypothesis, which was proposed in the 19th century and spanned three centuries, there has been almost no progress.

Finding the answer to the Millennium Prize Problem is akin to trying to climb Everest for the first time.

There are many steps along the way, which symbolize progress.

But the real question is: "Can you reach base camp? Even if you can, you know you are still far from the summit."

For problems such as the Bech and Swinnerton-Dyer conjectures and the Riemann hypothesis, the math community is clearly still in Nepal, one of the countries from which Everest is climbed.

Even if you can successfully reach Everest Base Camp, mathematicians may still need additional "equipment" to reach the summit.

Just like Peter Schulz's "theory of p-adic class perfect spaces", mathematicians can use this tool to make a series of major breakthroughs in the Langlands Program.

The same is true for solving the seven Millennium Problems. Perhaps each problem requires mathematicians to build one or more new tools to take it down from the temple of mathematics.

"You mean, you came up with a mathematical method based on the Weyl group mapping of algebraic clusters and the twisting of the maximal torus?"

After calming down his violently beating heart, Witten asked impatiently.

Although he believed in the mathematical ability of the student in front of him, no matter how he looked at it, it was incredible that he solved the problem of irreducible decomposition of differential algebraic clusters in more than a month and came up with a mathematical method that might be used in the Hodge conjecture.

Perhaps the problem of irreducible decomposition of differential algebraic clusters was helped by another Fields Medal winner, but the mathematical method used to solve the Hodge conjecture or the type of differential form generated by non-singular projective complex algebraic clusters was his own achievement.

Have the young people nowadays become so perverted?

Schultz made the "quasi-complete geometry theory method" during his doctoral studies, and his student also made new mathematics during his doctoral studies.

More importantly, the latter is younger than the former.

Hearing this, Deligne on the side also cast a concerned look. Xu Chuan nodded and said, "Some ideas and cores have been written, but they have not been organized and perfected."

After that, Witten quickly asked, "How long will it take you to organize it?"

Compared with Deligne, he is more concerned about whether the Hodge conjecture can be solved.

Because the Hodge conjecture is related to a series of physical problems such as general relativity, M theory, and three-dimensional physics.

The Hodge conjecture is one of the basic carriers of the geometric topology of the structure of general relativity and M theory, and its importance to physics is unquestionable.

As a physicist, he has solved the positive energy theorem in general relativity, and is also the main core figure of M theory and string theory. No one pays more attention to the research and attention of these two aspects than him.

Xu Chuan thought for a moment and said, "Maybe it will take about a month?"

After a pause, he added, "Now I have only made a core, which has not been verified yet. It is not an easy thing to continue to improve it."

Witten took two deep breaths, suppressed his violently beating heart and his scattered thoughts, and said, "Can I see your manuscript?"

It is actually very presumptuous to ask others to see unfinished and unpublished manuscripts, even when facing his own students,

But at this moment, Witten could not care so much, he just wanted to see hope as soon as possible.

He proposed and improved the M theory, but he also struggled on this road for most of his life. Now that he saw a glimmer of hope, he was naturally impatient.

Xu Chuan nodded and said, "Of course."

Witten made the request, and Deligne followed along.

The three of them came to Xu Chuan's dormitory and pushed open the door. The bad environment even made the two old men who walked into the dormitory have nowhere to step.

In the dormitory, there were waste papers all over the floor, some crumpled into a ball, some scattered in the corner, and in the corner, there was a bag of household garbage that had not been cleaned up.

Seeing this scene, Xu Chuan smiled awkwardly and said, "I have been organizing my thoughts these days and haven't had time to tidy up the dormitory."

But neither Deligne nor Witten had any disdain.

This is where science was born. No matter how dirty and messy the surface is, it can't cover up the knowledge contained in it.

Walking into the dormitory, Deligne leaned over and picked up a crumpled manuscript from the ground.

Opening it, the black handwriting inside occupied about half of the manuscript paper.

Looking from top to bottom, you can clearly see the writer's thoughts fluctuating, from the smooth writing at the beginning without any corrections, to the intermittent and revised smears at the end, and the last formula that was not finished and was completely crossed out by the author with a horizontal line, you can see the author's tossing on this road.

Deligne didn't care that it was a waste manuscript. After smoothing the manuscript paper with his palm, he read it with relish.

As for Witten, he was not very interested in the messy manuscripts on the ground, whether they were useful or not. In other words, he was more interested in the complete method, so when he entered the door, his eyes fell on the thick stack of printed papers on the desk.

It recorded the method to solve the differential form type problem generated by non-singular projective complex algebraic varieties.

The two top bosses were sitting in the dormitory, and Xu Chuan could not remain indifferent. While Deligne and Witten were browsing the manuscripts, he began to tidy up the dormitory.

The manuscripts on the ground, whether useful or useless, even crumpled, were temporarily picked up by him and put aside.

These things, even if they are completely discarded, have extremely high collection value. At least for him.

After all, these things witnessed the complete process of the birth of a new mathematics.

If this mathematical method can be used to solve the Hodge conjecture, their value will be incomparable. After all, knowledge is the most precious wealth at any time.

"Xu Chuan, where is the manuscript left by Professor Mirzakhani?"

After reading the waste manuscript in his hand, Professor Deligne put it on the desk and asked Xu Chuan.

Although he was also interested in Xu Chuan's research results, Witten had now occupied the manuscript and was reading it. Instead of going over to read it together, it is better to look at Professor Mirzakhani's manuscript first.

The only female Fields Medal winner in history, what she left before her death is fascinating for any mathematician.

"Wait a moment."

Xu Chuan responded, sorted the manuscript in his hand in order, found the original manuscript from the bookcase, and handed it to the mathematics tutor.

Looking at the manuscript that was completely preserved in the storage bag, Deligne's eyes showed a hint of approval.

Respect for other people's achievements is a necessary scientific research spirit.

In dormitory No. 306, Deligne and Witten were each immersed in the manuscript in their hands.

Time passed bit by bit, until the sun set over the mountain, and the golden afterglow shone through the glass window, and the two big guys woke up one after another.

"It's worthy of Professor Mirzakhani, the ideas he left behind are amazing."

Looking at the golden afterglow falling on him through the glass window, Professor Deligne pushed his glasses on the bridge of his nose with one hand and pinched the deep root of his nose.

From this manuscript, he saw the initial starting point of the problem of 'irreducible decomposition of differential algebraic clusters', and also saw the insights of this female Fields Medal winner in Riemann geometry, differential geometry, Weyl groups, and algebraic groups.

Deligne believes that this is not all of Professor Mirzakhani, and may not even be one percent.

But it is a pity that such an outstanding mathematician died this year.

With a slight sigh in his heart, Deligne looked up at Witten, wanting to see his evaluation of the mathematical method in his hand.