The top student must be diligent.

Chapter 173 A Carnival of Mathematics and Physics
"It feels like...he has become different from before."

In the audience, Schultz looked at Xiao Yi on the stage and couldn't help but say something.

Faltings next to him smiled and asked, "What's different?"

"Well...it feels different everywhere," said Schultz.

Faltings smiled, turned his head and looked at Xiao Yi on the stage again, and said: "To sum it up, he has a true master's demeanor."

Schultz was stunned: "Master style?"

Faltings nodded and said, "In the past, he would not chat at the beginning of his reports. He would usually get straight to the point. But now, he is in the mood to chat with so many of us. This shows that he has truly remained calm when facing us, as if we were all here to listen to him impart the truth - although this is indeed the case."

"Is that so?" Schultz nodded thoughtfully and said, "After all, he has even proved the quality gap problem, so he is definitely a master."

"That's not necessarily true." Faltings shook his head: "Do you think you are a master?"

"Me?" Schultz was stunned, then waved his hands repeatedly to deny: "How can I be considered a master..."

"You still have some self-awareness." Faltings said with a smile.

Schultz's expression suddenly became embarrassed.

“To be called a master, one must not only achieve the level of a master in terms of results. In the history of mathematics, there are only a few people who can do this. In recent years, there are probably only Grothendieck and Serre.”

Faltings said: "What's more important is to change your mentality. At least you have to recognize your own abilities in your heart. Only then can you be truly considered a master."

"You still need to work hard." Faltings patted Schultz on the shoulder. "Of course, I still have high hopes for you. After all, I boasted to the media that you are one of the three mathematicians I most recognize. Please don't embarrass me."

Schultz's expression became serious: "I will try not to disappoint you."

Faltings smiled and nodded.

But Schultz responded at this time: "So do you think you are a master?"

"I?"

Faced with the same question, Faltings just smiled and waved his hands: "I don't count."

However, no matter how Schultz looked at it, he felt that Faltings' answer was just out of modesty.

This answer is completely different from the one he just gave.

All of a sudden he understood what Faltings had just said about the "Master's mind".

It was probably like what a character said in a game he had played: A true master always has the heart of an apprentice.

"Okay, just listen to the report." Faltings reminded him, "Today we can see the true face of the Hodge-Vertex Algebra Analysis Method with our own eyes. It's really exciting."

Schultz came to his senses and then turned his attention back to the stage.

At this time, Xiao Yi had already begun to describe his proof process.

……

"Undoubtedly, proving the quality gap problem is a very long and arduous process, which requires exploring various different angles."

"I tried four directions. The first one is lattice QCD, which is familiar to everyone. It is a numerical simulation method that can easily help us verify the existence of the mass gap by discretizing time and space. However, for well-known reasons, numerical simulation cannot replace rigorous mathematical logic and cannot be transformed into a real mathematical proof."

"Then there is the Schwinger-Dyson equation. As long as we can find a non-zero solution to the gluon self-energy function, this will indirectly prove the existence of the mass gap."

Xiao Yi began to demonstrate some of his results on the Schwinger-Dyson equation method on the blackboard.

In the end, just after he made crucial progress, he had to give up because the solutions were too complicated to proceed to the next step.

"Then there is the renormalization group method, which analyzes the behavior of Yang-Mills theory at different energy scales. I found a non-perturbative fixed point in the renormalization group flow, suggesting the possible existence of a mass gap. Unfortunately, the complexity of solving this problem is still beyond imagination."

"The fourth approach, using the AdS/CFT duality to understand the non-perturbative nature of Yang-Mills theory through the duality between conformal field theory and anti-de Sitter space, provides a new perspective, but the final complexity is far beyond the acceptable range."

Watching the demonstration of these methods given by Xiao Yi, many people present were dumbfounded.

Almost every method was far beyond their imagination and far beyond the research progress of the academic community on this problem.
The physicists present at the scene were sweating profusely. The mathematics used in these methods were almost beyond their imagination. Even so, they couldn't solve it?

Everyone has a further understanding of the difficulty of the quality gap problem.

So, how did Xiao Yi solve it?
"Finally I aimed at the perspective of topological quantum field theory."

"Yang-Mills theory has rich topological structures, and trying to make a breakthrough from TQFT is a well-understood angle."

"And it turns out that the angle I chose was correct."

[For the Yang-Mills field A on S^4, its curvature form F satisfies: F=dA+A∧A.]

【Chern number c is defined as: c=1/(8π^2)∫_S4Tr(F∧F)】

Xiao Yi turned around and started writing on the blackboard, saying, "After getting started, I began to observe the performance of Yang-Mills theory on a four-dimensional sphere. As we all know, this four-dimensional spherical space has very special topological properties."

"The four-dimensional sphere S^4 is a compact, unbounded four-dimensional manifold with topological properties of simple connectivity and the nullification property of high-order homotopy groups, which makes our analysis a little simpler."

"So we will naturally be able to think of using anti-self-dual fields, and Hodge dual operators."

Xiao Yi's deduction began again.

After he constructed what he called the anti-self-dual field on the blackboard, many physicists present immediately remembered the X field that Xiao Yi had derived, which was derived from this anti-self-dual field!
After realizing this, their eyes suddenly lit up. They finally discovered the original origin of the X-field. After carefully observing the deduction process given by Xiao Yi, they also understood the mechanism of the X-field more clearly.

For a moment, they were all looking forward to Xiao Yi's final results. How much help could it provide for the research of theoretical physics?

After all, in the summary of this report, Xiao Yi clearly stated that he would explain the physical significance of the conclusions.

In this way, mathematicians look forward to the analytical theory of Hodge-Vertex algebra, physicists look forward to the physical significance of the final conclusion, and everyone has a bright future...

"…ultimately, we can derive a theorem: Let G be a compact, simple Lie group, and let A be a Yang-Mills field defined on the four-dimensional sphere S^4. If there exists a non-zero Chern number c, then the lowest energy excited state of the Yang-Mills field A has a strictly positive mass gap."

"Obviously this theorem is equivalent to the mass gap problem, so we only need to prove it to prove the existence of the mass gap."

The audience in the audience held their breath and carefully observed the theorem given by Xiao Yi.

"I see. He actually deduced topological quantum field theory to this extent..."

Sitting in the first row, Edward Witten, one of the main audience of the lecture, had a draft paper on his knees, and he was following Xiao Yi's narration and making deductions on the draft paper.

Finally, he raised his head and looked at Xiao Yi with even more shock in his eyes.

Being able to derive this equivalent relationship is to combine almost all the methods that can be used in the whole process with quantum field theory to a new extreme. The technical considerations involved far exceeded his imagination.

It includes the Chern-Simons theory he once studied, as well as a host of complex mathematical methods such as four-dimensional topological invariants and fiber bundle theory.

It is already quite difficult to master so many methods, let alone to integrate them all and use them to solve such a difficult problem as deriving the mass gap.

As a top master of mathematics and physics, Witten now has a deeper understanding of Xiao Yi's mathematical ability.

However, having used the method to this extent, can we only derive such an equivalent theorem in the end?
How to prove it next?
It should be the Hodge-Vertex Algebra analytical method that has been widely circulated, right?

Witten's heart also began to look forward to this method.

At this moment, after deriving this theorem, Xiao Yi on the stage turned around and smiled at all the audience: "We have derived the equivalent relationship. The next question is, how do we prove this theorem?"

Then he turned to the next page of the PPT.

The content on this page is the Hodge criterion conjecture that gave Xiao Yi inspiration.

"The Hodge criterion conjecture is one of a series of conjectures about algebraic cycles on algebraic varieties. It is related to the Hodge conjecture, but is relatively more specific and technical."

"Now you can look at the statement of this conjecture and think about the theorem I just gave. Can you find some connections?"

Xiao Yi stopped talking here, picked up his water cup from the side and took a sip.

More than 90% of the people off the field were confused.

No way, you really want us to observe?

Are you overestimating us a bit? What can this thing observe?

For most people, they can't even understand the statement of this conjecture.

【For a non-singular projective algebraic variety X defined over the complex number field, consider an algebraic cycle Z in the (p, p)-homology class of X, and define an operator L(Z) induced by Z: H^m(X, Q)→H^(m+2p)(X, Q), where Hm(X, Q) is the m-th homology group on X. The conjecture asserts that for appropriate p, this operator L(Z) is positive definite. 】

"Do you understand?"

Below the stage, in the area where Ye Cheng and the others were, they all looked at the thing Xiao Yi gave, all with confused expressions.

"What the hell do you understand?"

Chen Muhua yawned deeply.

At this time, they were basically in a drowsy state.

It was as if I was back to that hot afternoon in my second year of high school, listening to the teacher explaining elliptic curve problems on the stage, and I was yawning continuously, wishing I could just fall asleep.

Of course, for these math ace students, they knew everything the teacher taught them, but now when faced with what Xiao Yi said, they really couldn't understand a single thing. "Don't think about it, Brother Xiao didn't ask us to observe, he asked the experts sitting in front to observe." Lupin waved his hand and said calmly.

For them, accepting reality is the most important thing.

However, having said that, in fact, for those big shots sitting in the front row, they look left and right, but they can't see anything at all?

Also, why did Xiao Yi suddenly raise this question now?
Could it be to prove the Hodge criterion conjecture?

What a joke!
Proving the mass gap problem is not enough, you also want to prove the Hodge criterion conjecture?
It should be noted that there are many conjectures in mathematics that are no less difficult than the seven Millennium Problems, and the Hodge Criterion Conjecture is one of them. The most important thing about the Millennium Problems is not just their difficulty, but the value they can bring to the academic community after they are solved.

Of course, Xiao Yi did not wait any longer. After taking a sip of water, he continued, "After observation, we can easily connect to some tools in Hodge's theory."

"First, there's Hodge decomposition, and then there's vertex algebra."

"Hodge decomposition is one of the core concepts of Hodge theory, which decomposes the deram homology on complex algebraic varieties into (p, q)-type parts. On the other hand, vertex algebra, as an important tool in quantum field theory and algebraic geometry, can be used to describe the algebraic structure in conformal field theory."

“What can we gain from combining the two?”

Xiao Yi did not give a direct answer, but started writing on the blackboard.

"Consider a complex algebraic variety X whose Deram homology group HkdR(X, C) can be represented by a Hodge decomposition."

[HkdR(X, C)=_(p+q=k)H^(p, q)(X)]

"Vertex algebra is an algebraic structure used to describe the operator algebra in two-dimensional conformal field theory. Let V be a vertex algebra whose operators satisfy certain commutative relations and locality conditions. In particular, the vertex algebra has a state space V = _(n∈Z)Vn, where Vn is a subspace with energy level n."

"We now consider a vertex algebra V acting on the homology class of Hodge structures. Specifically, let the operators of V act on Hp,q(X) and define a mapping."

【φ:VH^(p, q)(X)→H^(p′, q′)(X)】

"where p′ and q′ are determined by the operator properties of the vertex algebra."

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

"But then the next question arises: how do we use this configuration?"

"If it can't be used, even if it is combined, it will only be like a castle in the air and have no practical significance."

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

"Consider the moduli space M of X, where the points correspond to some geometric objects, such as equivalence classes of vector bundles, algebraic clusters, etc. At this time, we can use the Hodge-vertex algebraic 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 to this point, there was already a ripple in the audience.

When the mathematicians saw the processes given by Xiao Yi, they were completely unable to calm down.

Is this the Hodge-Vertex Algebra method?
Such a wonderful deduction, and the effect of this method...

Almost completely connecting several tools in Hodge's theory?

And now given the moduli space...

At this moment, all they realized in their hearts was that algebraic geometry was about to change.

Sitting in the seats of Princeton and other scholars, Deligne leaned forward a lot at this time, 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 would be satisfied, right?"

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

Next to Deligne, Bombieri asked in surprise.

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

Bombieri understood what Deligne meant.

As one of the most important conjectures in algebraic geometry, the Weil conjecture attracted a large number of top mathematicians who were trying to solve it at the time.

At that time when human stars 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 final crown and was awarded the Fields Medal.

However, Deligne's teacher, Grothendieck, 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 proofs more pure, highly universal and elegant.

However, Deligne's proof method used more specific and technical means, including l-adic homology and monotonym theory. In Grothendieck's view, this method is more like "patching" or "clever techniques" rather than demonstrating 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 difficult for others to imagine.

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

Unfortunately, Grothendieck never succeeded until his death.

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

But now...

Bangbieli looked at the method Xiao Yi gave on the blackboard, his eyes becoming more and more shocked.

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

The same emotion also occurred to many scholars present.

Faltings' expression was very serious, and he looked at Xiao Yi's derivation very seriously, while Schultz next to him was shocked and unbelievable. 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 forms, or connecting S-duality with Langlands duality, etc.

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

……

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

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

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

Therefore, at this moment, the excitement in their hearts was no less than that of the mathematicians.

Perhaps, Xiao Yi on the stage was the only one in the whole audience who still spoke in the same tone as always, as if he had no idea how great an impact what he was saying would have on the two academic circles.

At this moment, he was finishing the work for the final proof.

"…Finally, using the analytical method of Hodge-Vertex algebra, we can easily prove the theorem I proposed above."

"There exists such a non-zero Chern number c on G, which 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 saying this, he paused.

The scholars in the audience seemed to have not reacted at all 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 briefly twice on the blackboard, and soon completed the step easily.

However, he did not finish there. He deduced a few more steps, and these steps made the physicists in the audience unable to sit still.

This is an expression for a new particle!

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

"It can be seen that it has the same origin as the X-field, so I will temporarily name it the X-particle. As for whether it can be discovered in the future, it will require 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 around and walked to the front of the stage.

"Well, this is the end of my report."

"Thank you for your patience."

Xiao Yi bowed like a gentleman.

The atmosphere in the venue was suffocatingly quiet for a moment, and then broke into warm applause.

It seemed as if it was going to blow away the ceiling of this conference hall.

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

They can all see that this report will be a carnival of mathematics and physics!
(End of this chapter)