A genius? I just love studying.

Chapter 125 I'll be the speaker?

"Some time ago, we collaborated with Google to train a neural network using this new theory to predict novel fractional topological phases. The resulting DeepMind model has successfully predicted candidate fractional Chen insulator materials in two-dimensional heterojunctions."

After writing out the proof, Professor Edelian looked excited and was quite proud of his latest research. "Moreover, the research on this theory can also help explore the mechanism of high-temperature superconductivity, such as studying the potential connection between the pseudogap state and fractional Chern number of copper-based superconductors, and constructing a theoretical model for the coexistence of d-wave superconductivity and Z-topological order."

"If we can eventually unify the theories of high-temperature superconductivity and topological quantum liquids, then perhaps we really will be able to find a room-temperature superconducting material!"

"Isn't this an exciting study?!"

Professor Edrian stood on the stage with a broad smile, gesturing wildly as he envisioned the future.

It's clear that this professor has some Indian blood.

Chen Hui frowned, feeling that the professor was being overly optimistic. Not to mention finding room-temperature superconducting materials, the theory in this paper itself was incomplete and had many problems in practical applications.

Of course, the value of this paper is beyond doubt.

Condensed matter physics still has many problems to be solved. For example, the traditional K-theory topological classification fails in strongly correlated systems, there is a gap between experimental observations and theoretical descriptions of fractional Chern insulators, and the existing Chern-Simons theory cannot uniformly handle non-Abelian statistics and topological order.

The core contradiction in these problems is that the mathematical description of integer versus fractional topological numbers needs to break through the limitations of commutative algebra.

In this paper, Professor Adelian introduces layer cohomology to replace traditional vector bundle theory, constructs a modified Atiyah-Singer index theorem on noncompact manifolds, extends the non-Abelian Chern-Weil theory with non-integer coefficients, constructs a layer-theoretic realization of fractional topological charges, establishes a noncommutative algebraic representation of the generalized Bloch theorem, and discovers a direct correspondence between the fractionalization of Berry phases and the layer cohomology group H(X, /).

For the first time, a rigorous mathematical formulation of fractional Chern numbers was achieved; a new cohomology theory for handling fractionalization phenomena was developed; a hidden connection between topological order and number theory (modular form) was discovered; a deep link between noncommutative geometry and algebraic topology was established; and a new type of computable topological invariant was established…

These achievements are groundbreaking!

Therefore, the fact that this paper was eventually published in SIAM Review is also an affirmation of its work.

Although SIAM Review has an impact factor of only 10.8, most papers in the field of mathematics are like this. Even the top mathematics annual journal among the four major journals only has an impact factor of 5.7. In the field of mathematics, the impact factor is not the only criterion for judging a journal.

However, it is the undisputed absolute authority in the field of applied mathematics, with only 28 articles published annually, which shows how difficult it is to get published in it. It is incomparable to those low-quality journals that publish hundreds of articles a year.

If we had to rank them, then SIAM Review would be considered the top journal below the four major journals.

However, this paper undoubtedly has many areas for improvement, and Chen Hui also has many questions.

"Okay, now everyone can ask any questions. We have fifteen minutes to discuss them."

Edrian threw down the chalk, sat back down in the lecture hall, and said with an excited smile that he was also looking forward to communicating with the young people from China.

"Professor Edrian, hello."

Just as Chen Hui was about to raise his hand, a loud voice suddenly rang out from the side. A young man standing in the aisle raised his hand high, and his youthful voice echoed in the quiet lecture hall.

"children of curiosity, just say!"

With a bright, Western-style smile, Adrien looked at Ma Weiyang. These young Chinese people were just too reserved. If they could be more open, perhaps they could go further on the path of mathematics.

Taking the microphone from the staff, Ma Weiyang began, "In your paper, you mentioned predicting observable fractional quantization phenomena and gave a formula to predict the fractional values ​​of σ_xy= e/(3h). However, in real materials, disorder and interactions may lead to the renormalization of topological numbers, such as the integer quantum Hall effect becoming fractional scaling behavior. Existing experimental methods cannot directly distinguish between intrinsic fractionalization and effective phenomena. How did you solve this problem?"

"!"

Adelian decided to retract his earlier assessment of the little guy. Well, these little Chinese kids really don't hold back at all!

Undoubtedly, this is a fair, definite, and incisive question...

"Good question!"

Edelman gave a heartfelt compliment. The question asked by the Chinese student was exactly the problem their team was planning to tackle next, which was enough to show that the little guy not only understood what he was saying, but also had his own thoughts, which was already very impressive!
"As this child said, the differential geometric implementation of fractional Chern classes is a problem that urgently needs to be solved. When defining Ch^α=(i/2π)∫Tr(Ω^α), it is necessary to verify whether the fractional curvature form Ω^α satisfies the Bianchi identity."

Traditional Chern classes are based on principal bundle connections, but the fractional case may require breaking through the framework of principal bundle theory, and there is no mathematical consensus yet.

"We have also explored some solutions, such as directly generalizing the traditional Chern class to rational coefficients and defining the fractional Chern number as..."

Adelian picked up the chalk again, erased all the proofs on the blackboard, and wrote down the formula again: Chpm/n=nmChpZ.

"Unfortunately, it cannot explain why only specific denominators were observed in the experiment, rather than all rational numbers, such as n=3 in magic-angle graphene."

"Of course, we can also construct the principal bundle of a non-commutative canonical group (such as SU(n)/Zk) and define the modified curvature Ωα=dω+ω∧ω+αΘ..."

Edelman continued writing on the blackboard, “This approach cannot prove that the modified curvature satisfies the Bianchi identity, thus violating the self-consistency of differential geometry.”

"Clearly, as you can see, our team doesn't have a good solution to this problem yet."

Adrien shrugged, indicating some regret, and then added with a touch of American humor, "If this problem could be solved, this paper would have been submitted to one of the Big Four accounting firms, not SIAM Review."

"If you are interested in this question, you are welcome to join my team."

Edlian is considered a relatively conservative professor at Stanford University. Chinese people have long been among his colleagues and team members. It is said that those Chinese people have performed quite well and are hardworking and conscientious.

He used to find it hard to believe, but now that he has seen it with his own eyes, he has also developed the idea of ​​recruiting a few Chinese students into the team.

Many graduate students at Yanbei University looked on with envy at Ma Weiyang, never expecting that simply asking a question would earn him the favor of such a renowned scholar.

Stanford University is considered top-tier in the US, and Professor Adrien is a leading figure in condensed matter physics research. It's foreseeable that joining such a team would lead to a bright future.

However, Ma Weiyang was somewhat disappointed with this answer.

He majored in physics and hoped to develop high-performance materials. His ultimate goal was to prepare room-temperature superconducting materials. His interest in mathematics was limited to solving physics problems.

He asked the question to get an answer, not to receive an invitation.

"grateful……"

The sounds of the outside world faded from Chen Hui's mind, and inspiration was bursting forth in his world, like a burst of fireworks.

Directly generalizing the traditional Chern class to rational coefficients cannot explain why only specific denominators were observed in the experiment. So why not introduce the framework of the Langlands program?
The Fourier coefficients of the modular form are often rational numbers. For example, the coefficients an∈Q of the modular form f(z) with weight 2, and the denominator is constrained by the modulus N. n=3 corresponds to N=27, which is naturally consistent with the denominator selection mechanism in the experiment!

Meanwhile, the irreducibility of Galois representations in the Langlands program corresponds to the stability of topological phases, which can provide a number-theoretic basis for the classification of fractional topological orders.

The combination of modular periodic integrals and Chern-Simons theory can rigorously derive the fractional quantization condition σxy=e2/(nh).

"That's right! That's right!"

Therefore, this problem can be solved by mapping fractional Chern numbers to specific coefficients in modular form, and by using Langlands correspondence to establish a strict connection between topological invariants and automorphic representations!

But how do you do that? Chen Hui's brain raced.

The papers related to the Langlands program that I've been reading these past few days have been flooding my mind, colliding with the condensed matter physics knowledge I was reading a few days ago, exploding into a series of dazzling fireworks.

First of all,

选取与物理系统对称性匹配的模形式，例如对于具有C3旋转对称性的魔角石墨烯，选取权k=2、级数N=3的模形式 f(z)∈S2(Γ0(3))，其傅里叶展开为：
f(z)=∑n=(1,∞)anq^n, q=e^(2πiz)
Then construct the fractional Chern number...

The interaction between Da Niu and the students startled the high school students who were still daydreaming. These future academicians looked at Ma Weiyang with longing in their eyes.

Will they be able to do the same when they go to college?

Li Zehan, who was studying, had already raised his head and looked at Ma Weiyang, who was asking the question, feeling a surge of excitement.

This is what a man should do!

To be a student at a lecture, to converse with top experts, and to even gain their appreciation—what kind of novel protagonist's script is this?
I want to be like that person when I grow up!
He subconsciously turned his head to look at Chen Hui beside him, only to find Chen Hui engrossed in writing and drawing on a draft paper, seemingly doing nothing.

That's normal. It seems that Chen Hui is only interested in what he is studying, and nothing else can affect him. No matter where he is, he can immerse himself in his own world.

In the past, Li Zehan was quite envious of Chen Hui's state, but now, he felt that Chen Hui must have felt a little regretful that he couldn't see this scene.

He had originally set Chen Hui as his lifelong goal, but after seeing Ma Weiyang, Li Zehan suddenly realized that the world was vast and far beyond the CMO and math competitions.

Perhaps Chen Hui will grow into such a person in the future, but right now he is still too immature.

"Perhaps I can broaden my perspective further!"

Li Zehan turned his gaze away from Chen Hui, looked at Ma Weiyang beside him, and then at Professor Edrian on the stage.

Many other CMO contestants also looked over at Chen Hui, and were somewhat disappointed to see that he hadn't made any comment.

Indeed, compared to a towering tree that has already grown, even a genius is ultimately inferior and pales in comparison.

Chen Hui was unaware of Li Zehan's and the other students' thoughts; he simply wrote rapidly on the draft paper, completely immersed in his own world.

"Here we introduce an important theorem derived from the Langlands conjecture: for an automorphic form f, its coefficient ap generates an algebraic integer ring over a specific prime field, such that the denominator of Chpm/n is constrained to be a factor of the modulus N. Therefore, the denominator n of a fractional Chern number must be divisible by the modulus N, explaining the phenomenon that only n=2, 3, 5, etc. were observed in the experiment!"

Then, the modular form f(z) is embedded into high-dimensional Chern-Simons theory to complete the differential geometric implementation of the fractional Chern number, defining the fractional curvature form:

Ωm/n=f(z)Ωstd
Using the modulus transformation property f(cz+daz+b)=(cz+d)kf(z), we prove that the modified curvature satisfies the generalized Bianchi identity!
After writing the proof in one go, Chen Hui looked up and found that the lecture hall was already empty, with only Yuan Xinyi, Li Zehan, and a few others still waiting for him.

Your math level has improved from 2 (76%) to 77%.

Your math level has improved from 2 (77%) to 78%.

……

Your math level has improved from 2 (81%) to 82%.

At the same time, a series of comments flashed before my eyes, and my math proficiency suddenly increased by 6%!

This reminded him of the moment he had an epiphany a few months ago under Teacher An's guidance. Indeed, whether it's studying or anything else, in addition to diligent practice, a few lucky moments of inspiration are also indispensable.

The effect of this epiphany was almost equivalent to him reading a thesis for half a month.

However, he also knew that the so-called sudden enlightenment was more accurately described as the result of years of hard work and accumulation. Without these days of relentless accumulation, there would have been no sudden enlightenment at this moment.

"Is this a differential geometry implementation of Chern's fractions?"

After staring at Chen Hui's draft paper for a moment, Yuan Xinyi asked a question with surprise in his eyes.

He had roughly figured out what Chen Hui was doing more than ten minutes earlier, so he didn't disturb Chen Hui after the lecture ended.

"Yes, teacher, could you please check if there's anything I've overlooked?"

Chen Hui was somewhat uneasy.

In his view, the problem that even the Stanford team of top experts couldn't solve couldn't be that simple, so there must be some flaw in his method.

Yuan Xinyi nodded. He hadn't watched the entire proof process because he was worried about disturbing Chen Hui.

"Introducing the Langlands framework provides a rigorous mathematical definition for the number theory foundation of Chern classes of fractions, overcoming the limitations of the differential geometry framework!"

The entire deduction process was not complicated in Yuan Xinyi's eyes. He only spent a few minutes reading through the entire deduction process.

"How many of the papers I gave you have you read?"

"Due to time constraints, I only managed to read five articles."

"!!!"

Yuan Xinyi's heart was already filled with turbulent waves.

In just two days, he read through five papers. From this deduction process, we can see that he not only read them, but also thoroughly understood them.

What kind of learning efficiency is this?
He suddenly realized that he couldn't quite see where the little guy's potential lay.

After closing his eyes and thinking for a moment, he solemnly said to Chen Hui, "From what I see now, I think this plan is feasible!"

Although his tone was calm, it was still full of emotion.

Although he was preoccupied with his own research and didn't delve into Professor Edelian's paper, Professor Edelian was an expert on the Langlands program. This little guy had come up with a solution that Professor Edelian hadn't even considered, and not only that, he provided a rigorous proof in just over half an hour!

"Professor Edrian would be very happy to know that the problem that has been troubling them for more than half a year has been solved!"

"Professor Adrien happens to be at Yanbei University. This afternoon, I've invited him and several professors from Tsinghua and Yanbei in related fields to hold a small seminar. Prepare well; you'll be the speaker this time!"

Yuan Xinyi was also decisive; after speaking, he took out his phone and started calling for backup.

Chen Hui was excited to receive his teacher's affirmation, but he was also a little nervous about the upcoming seminar.

"The speaker?"

(End of this chapter)