The top student must be diligent

Written here, Xiao Yi turned his head and smiled slightly: "Through this construction, the Hodge structure can be combined with the framework of vertex algebra, so it is the Hodge-vertex algebra configuration."

"But the following question arises, how should we use this configuration?"

"If it cannot be used, even if it is combined, it will only be like a castle in the air, without any practical significance."

"So, at this time, we have to use the moduli space and introduce the Hodge structure class at the same time."

"Consider the moduli space M of X, the points on it correspond to some geometric objects, such as equivalence classes of vector bundles, algebraic clusters, etc., and at this time, we use the Hodge-vertex algebra configuration just now to study the Hodge structure on the moduli space!"

[H^k_(global)(M, C)=_(p+q=k)H(p, q)_(global)(M).】

When Xiao Yi wrote here, there was already a wave in the audience.

Seeing these processes given by Xiao Yi, those mathematicians were completely unable to calm down.

Is this the Hodge-Vertex Algebra Analysis Method?

Such a wonderful derivation, and the effect of this method...

It almost completely connects several tools in Hodge's theory?

And the moduli space given now...

At this moment, they can only realize that algebraic geometry is going to change.

In the position of Princeton and other scholars, Deligne leaned forward a lot, as if he wanted to see the derivation process on the blackboard more carefully, and he almost stood up and walked to the blackboard.

"This method... this method... if I could use it to prove the Weil conjecture back then..." Deligne said: "The teacher should be satisfied, right?"

"You mean, this method can also be used to prove the Weil conjecture?"

Next to Deligne, Bombieri asked in surprise.

"Of course, and..." Deligne murmured: "It can allow me to get rid of other additional structures and realize the pure algebraic geometry proof of the Weil conjecture."

Bombieri understood what Deligne meant.

As one of the most important conjectures in algebraic geometry, the Weil conjecture was tried by many top mathematicians at that time.

In that time when the stars of mankind were shining, André Weil, Alexander Grothendieck, Jean-Pierre Serre, Michael Atiyah, and of course Pierre Deligne in front of him, all made efforts to prove the Weil conjecture.

In the end, Deligne became the mathematician who won the crown and won the Fields Medal.

However, Grothendieck, Deligne's teacher, was not very satisfied with his proof.

Because Grothendieck himself has always advocated using very abstract and general methods to deal with problems, making the proof more pure, highly universal and elegant.

However, Deligne's proof method used more specific and technical means, including the use of l-adic homology and monovalence theory, etc. In Grothendieck's view, this method is more like "patching" or "clever skills" rather than showing the inherent power and beauty of the theory.

In this regard, we can only say that Grothendieck has his own unique mathematical philosophy, which is unimaginable to others.

Faced with the teacher's dissatisfaction, Deligne felt a little aggrieved after all. Over the years, he has tried to use Grothendieck's ideas to prove it.

Unfortunately, he never succeeded until Grothendieck's death.

Even until now, no one in the mathematics community has ever been able to achieve this.

However, now...

Bombieri looked at the method given by Xiao Yi on the blackboard, and his eyes became more and more shocked.

If even this can be achieved, then for algebraic geometry, it will really change.

The same emotion also happened to many scholars present.

Faltings' expression was very serious, and he looked at Xiao Yi's derivation very seriously, while Schulz next to him was shocked and incredible. Hodge theory, as one of the most important mathematical tools in algebraic geometry, has made an extremely important contribution to the study of algebraic geometry.

And now... Xiao Yi's method not only makes Hodge theory more concise, but also can achieve a more important role by combining it with vertex algebra.

In particular, vertex algebra itself can be used to study the Langlands program, such as combining affine Lie algebra with W algebra, combining vertex operator algebra with automorphic form, or connecting S-duality with Langlands duality, etc.

Schultz couldn't help but sigh: "This guy... is he creating a miracle?"

...

When mathematicians were amazed by this theory, those mathematical physicists were also unable to calm down.

Vertex algebra originally originated from physics, and finally scholars discovered that it can also play an extremely important role in pure mathematics, which is why physics can also promote the development of mathematics.

But for these mathematical physicists, they are more concerned about how much help vertex algebra combined with Hodge theory can bring to their theoretical physics research!

So at this time, their excitement is no less than that of those mathematicians.

Probably, Xiao Yi was the only one on the stage who spoke in the same tone as always, as if he didn't know how much influence his words would have on the two academic circles.

At this time, he was finishing the final proof.

"...Finally, using the Hodge-vertex algebra analysis method, we can easily prove the theorem I proposed earlier."

"There is such a non-zero Chern number c on G, so it is equivalent to the lowest energy excited state of the Yang-Mills field A having a strictly positive mass gap."

"So far." Xiao Yi opened his hands and said, "We have completed the proof of the existence of Yang-Mills and the mass gap problem."

After speaking, he paused.

The scholars in the audience seemed to have not reacted and were extremely quiet.

However, Xiao Yi just smiled and continued, "In the end, according to the Hodge-vertex algebra analysis method, we can also easily generalize the Hodge structure to quantum field theory."

He demonstrated it twice on the blackboard, and soon, he easily completed this step.

However, it was not over yet. He deduced a few more steps, and these steps made the physicists in the audience unable to sit still.

This is the expression of a new particle!

"Yes, through this new method... well, I will call it quantum Hodge theory, we can easily derive a brand new particle."

"It can be seen that it is homologous to the X field, so I will temporarily name it the X particle. As for whether it can be discovered in the future, it still requires the efforts of the experimental physics community."

After saying the last sentence in a very relaxed tone, Xiao Yi put down the blackboard pen in his hand, turned his head, and walked to the front of the stage.

"Then, my report ends here."

"Thank you for your patience."

Xiao Yi bowed gentlemanly.

The atmosphere in the audience was as quiet as suffocating for a moment, and then burst into warm applause.

It seemed to overturn the ceiling of this conference hall.

Mathematicians and physicists stood up excitedly and sent all their enthusiasm to Xiao Yi.

They can all see that this report will become a carnival of mathematics and physics!

Chapter 174 The True Master

"He succeeded again."

Deligne stood up, clapping his hands, and sighed as he looked at Xiao Yi, who was greeting the audience on the stage.

At this time, he had recovered from his previous trance.

Although this new method can help him achieve a purer proof of the Weil conjecture, after all, his teacher has passed away, and he himself is already a retired old man.

The original obsessions have gradually blurred with the passage of time.

So, now let's immerse ourselves in this carnival moment of mathematics and physics.

Bombieri also smiled and nodded, looking at Xiao Yi on the stage, and said: "Back then, Grothendieck was probably about the same as him, right?"

"My teacher..."

Deligne smiled: "When he was eighteen years old, he was still a young man living in Montpellier who was unwilling to go to math classes."

Bombieri laughed suddenly.

Having said that, Grothendieck did not like math classes at the beginning, but that was because he reconstructed a very general form of Lebesgue integral by self-study without any guidance. At that time, France was in the process of post-war reconstruction, and education was naturally not up to date. Therefore, for Grothendieck, such math classes were completely unnecessary.

So much so that he naively thought that he was the only mathematician in the world at that time, and this situation did not change until he went to Paris.

However, Bombieri did not refute it. After all, in his opinion, Xiao Yi was not like Grothendieck, but the two people might be the same kind of people.

In this information age, learning is very easy. If Grothendieck was in this era and did not work behind closed doors like he did back then, who knows how many great achievements these peerless geniuses could have made?

At this time, beside them, Edward Witten put down his draft paper and pen, stood up, and applauded Xiao Yi.

Deligne smiled and said jokingly: "What? Have you finally finished the research?"

Witten said: "No, there are too many things to study. What can be studied in this short time? I am just trying to derive the quantum Hodge theory he mentioned according to the method he said."

"Quantum Hodge theory..." Deligne nodded slightly: "I don't know much about physics. How much help can this thing bring to your physics?"

Witten smiled and said: "Then you really don't know much. This kind of thing can no longer be said to be of great help to our physics, but to say how much help it can bring to all fields related to physics."

"This quantum Hodge theory is a further supplement to the topological quantum field theory, or in other words, it allows the original topological quantum field theory to evolve directly from 1.0 to 2.0. Consider the significance of quantum field theory to today's physics..."

Witten looked down at the pile of draft paper he had just used, and finally said: "All I can say is that the role is immeasurable."

Deligne raised his eyebrows: "Is that so?"

Witten nodded slightly, then said with emotion: "However, I am more concerned about how much role it can play in string theory."

"This theory is naturally meant to be used to study string theory."

Deligne rolled his eyes and said unhappily: "You will make string theory proud."