The top student must be diligent

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

Faltins smiled and nodded.

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

"I?"

Facing the same question, Faltings just smiled and waved his hand: "I don't count."

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

It's completely different from the answer he gave just now.

For a moment, he suddenly understood what Faltings had just said about "the nature of the master's mind."

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

"Okay, listen to the report with peace of mind." Faltings reminded at this time: "Today you can see with your own eyes the true face of the Hodge-Vertex Algebra Analytical Method, which is really exciting."

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

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

…

"Undoubtedly, proving the quality gap problem is a very long and difficult process, and it requires exploring various angles."

"I have tried from four directions. The first is lattice QCD, which everyone should be familiar with. It is a numerical simulation method that can easily help us verify the existence of the mass gap by discretizing space and time. However, as we all know The reason is that numerical simulation cannot replace rigorous mathematical logic and cannot be converted into real mathematical proof.”

"Then there is the Schwinger-Dyson equation. As long as a non-zero solution to the self-energy function of gluons can be found, 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 very crucial progress, he had to give up because the solutions were too complicated to proceed to the next step.

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

"The fourth method, using the AdS/CFT duality, understands the non-perturbative properties of Yang-Mills theory through the dual relationship between conformal field theory and anti-de Sitter space. Although it provides a new perspective, The resulting complexity was also far beyond what was acceptable.”

Seeing the demonstration of these methods given by Xiao Yi, many people present were stunned.

Nearly every one of these methods was far beyond their imagination and far beyond the progress of academic research on the problem.

And those physicists at the scene were sweating profusely. Good guy, the mathematics used in these methods are almost beyond their imagination. Even so, it can't be solved?

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

So, how did Xiao Yi solve it?

“In the end I aimed at the perspective of topological quantum field theory.”

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

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

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

[Chen number c is defined as: c=1/(8π^2)∫_S4Tr(F∧F)]

Xiao Yi turned his head and started writing on the blackboard, and said at the same time: "After I got it, I began to observe the performance of Yang-Mills theory on the four-dimensional sphere. As we all know, this four-dimensional spherical space is very special in topological properties. ”

"The four-dimensional sphere S^4 is a compact, boundaryless four-dimensional manifold. It has the topological property of simple connectivity and the zeroing property of the higher-order homotopy group, which allows our analysis to be variable. It has to be 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.

And as he constructed what he called the anti-self-dual field on the blackboard, many physicists present immediately remembered that the X field deduced by Xiao Yi was derived from this anti-self-dual field!

Realizing this, their eyes suddenly lit up, and they finally discovered the original origin of the X field. Carefully observing the derivation process given by Xiao Yi also made them understand the mechanism of the X field more clearly.

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

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

In this way, mathematicians are looking forward to the Hodge-vertex algebra analytic theory, physicists are looking forward to the physical significance of the final conclusion, and everyone has a bright future...

"...Finally, we can derive a theorem: Let G be a compact, simple Lie group, and A be a Yang-Mills field defined on the four-dimensional sphere S^4. If there is a non-zero Chen 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, which also proves the existence of the mass gap."

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

"So that's it, he actually deduced topological quantum field theory to this point..."

In the first row of seats, Edward Witten, one of the main audience members of this lecture, had a scratch paper on his lap, and he was following Xiao Yi's narration and making deductions on the scratch paper.

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

Being able to derive this equivalent relationship has almost combined all the methods that can be used in the entire process with quantum field theory to a new extreme, and the consideration of technology is far beyond his imagination.

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

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

As a top mathematical physics master, Witten has a deeper understanding of Xiao Yi's mathematical abilities this time.

However, since the method has been used 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 spread, right?

Witten's heart also ignited expectations for this method.

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

Later, he also turned to the next page of the PPT.

The content on this page is exactly the Hodge standard conjecture that inspired Xiao Yi.

"The Hodge standard conjecture is one of a series of conjectures about algebraic cycles on algebraic varieties. It has a certain connection with the Hodge conjecture, but is relatively more specific and technical."

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

Xiao Yi said this, then stopped, picked up his water glass from the side and took a sip.

More than 90% of the people in the audience were confused.

No, you really want us to observe?

Are you looking down on us a little too much? What can this thing observe?

For most people, they cannot even understand the statement of this conjecture.

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

"Do you understand?"

Under the stage, in the area where Ye Cheng and others were, they all looked confused when they looked at what Xiao Yi gave.

"Can you understand a ghost?"

Chen Muhua yawned deeply.

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

It seemed like I was back on that hot afternoon in my second year of high school, when I was listening to the teacher talking about the topic of elliptic curves on the stage, but I was already yawning so much that I wanted to fall asleep directly.

Of course, for them, the top students in mathematics, they knew everything the teacher said back then, but now they really can't understand anything Xiao Yi said.

"Don't think about it. Brother Xiao didn't let us observe, but the big guys sitting in front." Lu Ping waved his hand and said calmly.

For them, accepting reality is the most important thing.

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

Also, what does Xiao Yi want to do when he suddenly raises this question?

Could it be that he wants to prove Hodge's standard conjecture?

What a joke!

Proving the mass gap problem is not enough, you also want to prove the Hodge criterion conjecture by the way?

What you need to know is that there are many conjectures in mathematics that are no less difficult than the Seven Millennium Problems, and the Hodge Standard Conjecture is one of them. The most important thing about the Millennium Problem is not only that it is difficult; The value they can bring to the academic community after solving the problem.

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

"First, Hodge decomposition, then vertex algebra."

"Hodge decomposition is one of the core concepts of Hodge theory. It 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 get if the two are combined?"

Xiao Yi did not give a direct answer, but began to write on the blackboard.

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

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

"Vertex algebra is an algebraic structure used to describe operator algebras 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."

"Now we consider a vertex algebra V acting on the homology class of the Hodge structure. 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 characteristics of the vertex algebra."