A genius? I just love studying.

Chapter 263 Unexpected Students

"Deng Leyan?"

Opening the six resumes in his email, Chen Hui was somewhat taken aback by the first one, as it turned out to be from an acquaintance.

When Chen Hui first participated in the provincial competition, Deng Leyan's name had already spread throughout Chengdu. He won a bronze medal in the CMO in his third year of junior high school. He was indeed exceptionally talented and a natural-born talent for studying mathematics.

Unexpectedly, Deng Leyan did not attend university in China after graduation, but applied directly to Princeton University, and may even become my student. Life is truly amazing.

Without hesitation, Chen Hui kept Deng Leyan's resume. This was not out of favoritism; he knew Deng Leyan well. With his abilities, getting into Princeton would be no problem at all. Of course, if he wanted to be his student, he would need to pass his interview.

Leaving your resume is only the first step.

The remaining resumes were also very impressive, with most having experience in the IMO, or even being gold medalists. Those who could pass Princeton's initial screening were certainly not ordinary people. In fact, these people were probably Chen Hui's classmates. Of course, Chen Hui didn't have much of an impression of these names.

Of the remaining five resumes, Chen Hui kept two more.

Fefferman gave him three admission slots, but Chen Hui's requirements were still very high. If he didn't meet his requirements, he didn't necessarily have to use all three slots.

Elena from the seminar is also quite good; if she passes the assessment, Chen Hui is willing to give her a chance.

After replying to Fefferman's email and scheduling an interview, Chen Hui got up and left the room, heading towards the senior club. Now a member, he had been eating at the club for the past few days, and compared to Jiangcheng University, the food at the club was undeniably of much higher quality.

Chen Hui decided that when he returned to Jiangcheng University, he would definitely give the school some suggestions to try to improve the quality of the food in the school cafeteria.

Although Li Zehan is also at Princeton, he is equally busy. Li Zehan plans to complete his credits in two years, obtain a master's degree, and then pursue a doctorate.

Of course, he would wait for Chen Hui at the senior club before mealtime.

From frugality to luxury is easy, from luxury to frugality.

With his physique, he truly couldn't resist the temptation of food.

After receiving Chen Hui's recommendation, Li Zehan underwent another interview, and this time he successfully joined the senior club.

"Hey, what time is it?"

"If you don't eat enthusiastically, there's something wrong with your brain!"

Li Zehan came to greet them from afar, chattering on and on, as if he wanted to pull Chen Hui into the club as quickly as possible.

After studying the Riemann Hypothesis all morning and running five kilometers, Chen Hui's stomach was already growling, even though he had eaten breakfast. His brain was still in a state of residual heat from high-speed operation and he urgently needed high-calorie food.

It was lunchtime, and the club was already quite full. Chen Hui carried his tray and found an empty seat.

"Excuse me, could you squeeze in a bit?"

Li Zehan came over carrying a large plate of casserole and a cup of coffee, and squeezed his way to sit down next to Chen Hui.

Chen Hui doesn't like drinking coffee. He feels that it not only doesn't help him stay alert, but it also makes him feel bloated and unable to digest his food for a short time, which affects his study efficiency.

He can eat bread once or twice occasionally, but eating it as a meal would be a real ordeal. He prefers things like grilled steak; he loves meat.

Sitting opposite the two was a disheveled young man with a notebook in front of him filled with symbols of despair. His eyes were vacant, and his right hand was unconsciously stirring the long-cold coffee with a coffee spoon.

Someone has issued a challenge to you!

Li Zehan took a sip of coffee, took a big bite of chamois, and mumbled, "Someone named Elena suggested that your appointment system idea was brilliant, but it overlooked human nature—people might lie, might be late, might change their minds at the last minute, like suddenly having an inspiration and wanting to calculate a little longer."

Chen Hui didn't take it seriously and didn't pay much attention to Li Zehan's description; the questions on the blackboard were just things to do after class.

He is currently looking at the young man's notebook across from him.

Judging from the contents of his notebook, he was trying to prove that even with small perturbations in the system parameters, the multifractal spectrum of the chaotic attractor it generates—a key measure describing the distribution of its different “roughness” regions—is stable in the sense of a certain “renormalization group.”

He has constructed a complex framework, but is stuck on a key lemma: when estimating the upper and lower bounds of a certain Hlder exponent, he needs to deal with an extremely ill-conditioned oscillation term that stems from the non-uniform separability of dynamics at different scales in high-dimensional phase space.

Traditional box counting and measure theory techniques appear clumsy and powerless here. It is clear that he is struggling to prove the stability of multifractal spectra in specific non-autonomous systems. He has been sleep-deprived for several weeks, and his eyes are tired but burning with a stubborn flame.

He must have felt utterly helpless, like someone trying to grasp a smooth thread in a thick fog.

"I'll write it for you."

After eating two cups of coffee, Li Zehan finally stopped feeling like a starving ghost. He picked up a pen and started scribbling on a napkin, unaware that his elbow was constantly pushing the coffee cup.

With a clatter, the coffee cup tipped over, and a few drops of dark brown, scalding hot coffee spilled onto the young man's notebook across from him.

"Sorry!"

Li Zehan suddenly realized what was happening and quickly grabbed a tissue to wipe himself.

But Chen Hui didn't move, staring intently at the dripping and rapidly spreading coffee stain. The dark liquid spread wildly across the rough napkin fibers, its edges forming extremely irregular jagged patterns of varying shades. This natural, random process of penetration created the pattern.

In an instant, a violent triple collision occurred in her mind with the abstract morbid oscillation term on her notebook and the turbulent fractal structure next to it!
The irregular boundaries of coffee stains are a natural, random fractal front.

Could the destructive effect of the oscillation term also be understood as a kind of "phase interference" in a high-dimensional space, which hinders the smooth distribution of the measure?

Does the changing pattern of coffee stain diffusion on different paper textures metaphorically represent the "stability" she needs?
Why is it necessary to forcibly 'control' that oscillation term?

Chen Hui asked.

Ella, who had been glaring angrily at Li Zehan, suddenly frowned, looked down at her notebook, and her dark-circled eyes began to light up little by little.

Why not regard it as the 'power source' of the system itself, the generator of fractal structures?
Just like the random permeation of this drop of coffee defines its boundaries!

Perhaps I need a completely new framework based on random phase or dynamic renormalization, not to suppress oscillations, but to tame them and let them naturally define the spectral stability boundary under perturbation!
A crazy yet clear thought cleaved through the fog in her mind.

Completely forgetting the occasion, Ella snatched the notebook stained with coffee and syrup fractals from Li Zehan's hand, grabbed her pen that had fallen to the ground, and began to write frantically next to this sacred stain.

Li Zehan immediately understood what had happened and looked at the young man opposite him with some envy.

Looking at Chen Hui, then at the young man opposite him, he muttered under his breath, "No wonder everyone says Princeton is full of monsters. You really are born to be here."

[Your math level has improved from 65% to 66%] An unexpected improvement.

Chen Hui didn't disturb Ella, who was writing furiously. After finishing his grilled steak, he left the food club.

Before returning to his apartment, he took a detour to the large blackboard on the second floor of the Finn Building. Sure enough, there was a lot more content there. Below this content, there was a coffee bean pattern surrounded by random perturbations and the initials of Elena's name.

After a casual glance, Chen Hui already had an idea. He stepped forward, picked up the blue chalk he had used last time, and wrote on the blackboard.
Variance reduction is too shallow!

MDP is overly complex and is only suitable for airport scheduling; it's like using a sledgehammer to crack a nut.

Credit point penalties plus a flexible buffer are sufficient. The core principle is transparent rules and minimal cost incentivizing honesty and punctuality; budget constraints were not considered! Instead, the 'coffee fund' contributed by latecomers should be used for machine upgrades…

After finishing all this, Chen Hui took his thesis to the seminar room on the first floor.

The afternoon was the weekly seminar time, and he was too lazy to go back and come back again, so he simply waited in the seminar room for the seminar to start. He was late last time, so he would make up for it by arriving early this time.

Chen Hui didn't make much progress after a whole afternoon of research, but he wasn't in a hurry. As long as his proficiency continued to improve, he would be able to solve this problem sooner or later.

"Good morning, Professor Chen."

Elena was the first student to arrive at the classroom and was surprised to see Chen Hui sitting at the oval table reading a paper.

"morning."

Chen Hui responded with a smile.

“I have been thinking about the question you raised a week ago.”

Elena walked straight to the blackboard, picked up a piece of white chalk, and said, "Today, I will try to give a preliminary answer, the core of which lies in introducing a multi-scale oscillation and an effective degradation distortion index coupled with it..."

Elena's chalk danced smoothly on the blackboard, no longer deconstructing Chen Hui's framework a week ago, but reconstructing it with creativity.

Definition: For a point x and a scale r > 0, let B_r(x) be a sphere. We consider the local average of ω on B_r(x)...

As time passed and the students arrived one after another, Elena remained immersed in her own world, continuing to write on the blackboard.

Chen Hui did not interrupt, and the other students sat quietly in their seats, focusing on Elena's proof process.

Consider the behavior of the coefficients ω(x + sy) under the scaling transformation y = (z - x)/s.

EDDI aims to measure the potential maximum rate of resonant amplification of nonlinear terms caused by oscillation modes under the weighted space norm as the scale s varies.

The specific construction involves performing local Fourier analysis or wavelet analysis on the oscillation modes, extracting the dominant frequency/scale components, and calculating their influence factors on the weighted energy estimation...

Elena demonstrated the core idea of ​​the construction, and although she did not provide all the tedious details, the mathematical rigor and innovativeness of the concept left the audience breathless.

EDDI(ω, x, r) essentially characterizes the destructive potential of coefficient oscillations on the classical iterative scaling process at a scale r of point x. Elena has embedded these two new tools, MOM and EDDI, into Chen Hui's original framework like precise gears.

In explaining how to calculate EDDI, Elena demonstrated remarkable geometric intuition by drawing a two-dimensional diagram with the horizontal axis representing the scale (log s) and the vertical axis representing the oscillation frequency (log ξ).

"Imagine that the oscillating energy is distributed across this scale-frequency phase diagram near each point x. The core of EDDI is to identify which regions in this phase diagram will have the most severe resonance amplification effect on the weighted energy estimate?"

"By introducing MOM and EDDI, we provide a set of quantitative and actionable robustness criteria for handling PDEs with highly oscillating degradation coefficients."

It not only patched the potential loopholes in the original framework under extreme conditions, but also opened a door—allowing us to understand more systematically and precisely how coefficient oscillations affect the regularity of the solution.

This might lead us to a more general theory of regularity classification for oscillatory degenerate PDEs.

She turned to Chen Hui and the other classmates, "My report is finished. Please feel free to offer your corrections."

This seminar had essentially become a stage for her personal performance.

"Bravo! This is not just an answer, it's a work of art! You have perfectly combined profound insights, innovative tool construction, and clear expression."

Your MOM and EDDI, especially the intuitive presentation of that scale-frequency phase diagram, are absolutely amazing!

Chen Hui spared no praise, which reminded him of his first student. But unlike Chloe, Elena stood on the podium, calmly enjoying everyone's gaze without the slightest fear.

A round of enthusiastic applause erupted in the seminar room.

Many people in the classroom went back and thought about the problem, but none of them were able to solve it as thoroughly as Elena. It was precisely because they had thought about it deeply that they understood how difficult it was to solve the problem completely.

Elena deserves their applause.

"This idea about the regularity classification theory of 'oscillatory degenerate PDEs'... it deserves a separate and important paper, and perhaps you should write the next chapter of Feinlou's book."

After the applause subsided, Chen Hui smiled and said, "This achievement is enough to publish a first-tier SCI paper."

"So, Professor Chen, am I qualified to be your student?"

Elena stared intently at Chen Hui, as if the thesis was completely unimportant and becoming Chen Hui's student was the most important thing.

"of course."

Chen Hui replied affirmatively, "Come to my office tomorrow."

Fefferman had already arranged an office for Chen Hui when he allocated the enrollment quota, and now it could finally be put to use.

"what?"

"Can I become Professor Chen's student by answering this question?"

The other students in the seminar were all heartbroken and regretful that they hadn't put their full effort into thinking about this problem in the previous days. Otherwise, they would be the ones standing on the podium answering questions and becoming Professor Chen Hui's students.

With Chen Hui's current level of expertise and reputation, coupled with several seminar courses, these students have long been completely convinced, and becoming Chen Hui's student is undoubtedly their dream.

Chen Hui smiled and announced the end of the class. He walked out of the seminar room and looked at the ancient outline of Fein Building in the twilight and a few squirrels scurrying by. A gratified smile appeared on his face.

(End of this chapter)