A genius? I just love studying.

Chapter 260 Seminar

At 8:05 a.m. on Monday, the line for the only espresso machine in Princeton University’s common room was so long it almost blocked the way to the restrooms.

Chen Hui stood at the end of the line, watching the time on his watch tick by. There was less than an hour left before the 9 o'clock discussion class.

He has been at Princeton for over a week and has gradually integrated into the community. The outside world's distractions have long since faded away. He continues his research and even gets to communicate with some of the world's brightest minds.

This kind of life is really good; Princeton is even more suitable for him than he imagined.

It's summer vacation now, but not many students have left campus. Princeton has a laissez-faire policy towards students, but the students are extremely strict with themselves and stay at school to study hard even during the summer vacation.

Fefferman did not assign Chen Hui any courses for the time being, but asked him to lead two undergraduate seminars first, and then decide what kind of course he wanted to teach when the semester started. According to Fefferman, it would not matter if Chen Hui did not teach any courses. With Chen Hui's current achievements, he could enter the Institute of Advanced Studies to focus on research without taking any teaching position.

Chen Hui naturally declined. He also wanted to communicate with these brilliant minds at Princeton, which he thought would be a lot of fun.

His teacher, Yuan Xinyi, once told him that thinking about some simple mathematical problems can bring a lot of inspiration, and Chen Hui has always agreed with this.

He took on a teaching position at Jiangcheng University, so it was only natural that he would not have close contact with the students at Princeton.

Today is the first group discussion of the seminar he is leading, and he is looking forward to it.

Lost in thought, Chen Hui finally got his turn. Unfortunately, the coffee machine needed cleaning and the hot water wasn't hot enough, so he received a lukewarm, bland cup of comfort food.

"This is a living example of Pareto inefficiency!"

Li Zehan, who was in front of him in line, cursed. He had found Chen Hui the day after Chen Hui arrived and acted as a local, showing Chen Hui around Princeton for a day.

Although his coffee was slightly better than Chen Hui's, it still annoyed him. Heaven knows he had come all the way from China to Princeton, only to have to wait in line.

America's land area is only a few hundred thousand square kilometers smaller than China's, and its population is only a quarter of that of China, yet its queuing is even worse than China's. This is something Li Zehan finds unbearable. "This terrible infrastructure deserves to be drowned in a pig cage."

Chen Hui stared at the long queue, his mind no longer preoccupied with topological manifolds, but instead filled with cold queuing terminology such as waiting time, service rate, and queue length. "Perhaps, we can do something about it."

An idea began to take shape.

However, he needed to hurry to the seminar room on the first floor of Fein Building in the Princeton Mathematics Department. Li Zehan also had his own tasks to complete. He parted ways with Chen Hui in the corridor leading to Fein Building.

Blah blah blah...

As I walked briskly down the corridor, it suddenly started raining on campus. Many students who were strolling around the campus quickly ran to the corridor to take shelter from the rain.

When Chen Hui entered the seminar room, a dozen or so students were already sitting around the oval mahogany table. Their gazes toward Chen Hui carried a hint of scrutiny and an almost imperceptible desire to challenge him. Those who could come to Princeton were naturally not ordinary geniuses.

Chen Hui has indeed become famous worldwide, and they have all read his papers. Of course, even they cannot fully understand his papers, so if Chen Hui wants to impress them, he obviously has to show his true abilities.

"Sorry, I'm late."

Chen Hui smiled gently, because he was indeed a few minutes late waiting for this terrible cup of coffee.

"Today we will discuss the topic of 'regularity theorem for nonlinear elliptic equations with degenerate or singular coefficients'."

Without giving the students time to react, Chen Hui continued speaking, as if being late was a trivial matter.

As he spoke, Chen Hui had already elegantly reviewed several classic results on the blackboard, such as the generalization of the De Giorgi-Nash-Moser theory under specific degenerate conditions, and proposed a new framework for global Holder estimation of a certain class of strongly degenerate quasi-linear equations, which he had recently been thinking about.

Putting down the chalk, Chen Hui looked at the students in the seminar room. He was about to hand over the stage to these brilliant minds.

"Professor Chen, thank you for your excellent overview. I am very interested in your strategy for dealing with oscillatory terms near the degenerate point in your framework. Specifically, in the key estimate of your Lemma 3.1, you used a cleverly weighted Morrey space embedding to absorb higher-order perturbations of the nonlinear terms."

At this moment, a Russian girl sitting in the back row stood up. She spoke slowly, but her questions were as precise as a scalpel: "However, I noticed that when you were deriving inequality 3.7, you seemed to assume that the degenerate weight function ω(x) satisfies a certain almost monotonicity in a neighborhood of the singular set S."

She paused, her gaze sweeping over the formulas on the blackboard, then looked directly at Chen Hui, her pale blue eyes shooting at him like icy blades. "My question is, for the kind of highly oscillating degradation coefficient ω(x) that you are considering, which may even have a fractal structure on S, is this almost monotonicity assumption universally true?"
If there exists a counterexample that causes ω(x) to oscillate violently on any point set close to S, thus breaking this monotonicity, would your core estimate of Lemma 3.1 collapse?

Furthermore, will this shake the very foundation of your Holder continuity proof?

The seminar room fell silent instantly, with only the sound of rain outside the window. The other students nodded or looked thoughtful.

Chen Hui laughed happily. There was no sign of panic or offense on his face. Instead, he showed an even brighter, almost excited smile. "Excuse me, what's your name?"

"Elena."

The girl stared directly at Chen Hui, her lips full and pale, as if kissed by the cold wind of the snowfield. Even a simple sentence carried this stubbornness.

"Excellent question, Elena. Truly excellent!"

Chen Hui's tone was sincere and appreciative. Elena's question was not nitpicking, but rather pointed directly to a potential and extremely hidden vulnerability in Chen Hui's method—a "devil's detail" that might be overlooked in a smooth or mildly degenerate situation, but must be questioned.

This question is of a very high level, demonstrating her profound understanding of PDE regularity theory and her keen critical thinking.

Chen Hui turned to face the blackboard, picked up a blue chalk, and drew a circle next to the inequality (3.7) that Elena pointed out.

“You’ve got it right. This almost monotonic observation is very insightful. Yes, in my initial derivation, this assumption is implicit, and it does depend on the fact that ω(x) cannot behave ‘too pathologically’ near the degenerate set.”

He did not defend himself, but rather frankly acknowledged the existence of the hypothesis. Then, his tone suddenly became more insightful and inspiring: "But let's think about why we need this almost monotonousness."
"What is its essential function?" Chen Hui asked himself, answering the question as the chalk moved quickly across the blackboard, no longer deducing, but outlining the thought process. "It is essentially to control the behavior of the nonlinear perturbation term under the weighted Morrey norm during scale changes!"
Specifically, this is to ensure that when we perform the standard iterative scaling process to prove Holder continuity, that pesky nonlinear term will not exhibit uncontrollable, destructive resonant amplification at different scales due to violent oscillations in the weights.

Chen Hui paused, glancing at the others in the seminar room to make sure they were following his train of thought, before continuing, "So, if ω(x) really does oscillate like crazy Brownian motion near the singular set, as you say, what should we do? Abandon this framework?"

Chen Hui made an exaggerated gesture, mimicking Brownian motion, and then quickly gave his answer, "No!"

"This suggests that we may need to introduce a more sophisticated tool to characterize the effect of such oscillations on scale transformations—perhaps some novel concept based on oscillation integrals or multiscale analysis!"
Could we consider defining an effective degradation exponent that depends on the oscillation frequency, or, more radically, treat the oscillation mode itself as part of the free boundary of the equation?

Instead of focusing on fixing the loopholes, Chen Hui used the counterexamples proposed by Elena as a springboard to instantly elevate the discussion to a deeper and more cutting-edge level!
In just a few words, he outlined several possible research directions, each of which directly addresses the current challenges at the forefront of PDE. He cleverly transformed the "defects" into inspirations for "new opportunities."

The scrutiny and challenges in the eyes of the students in the seminar room had long been replaced by contemplation. They were completely immersed in the path that Chen Hui had outlined for them, and they saw amazing prospects. The scenery along the way was so fascinating.

They couldn't wait to think about these questions and come up with answers.

Elena's furrowed brows relaxed, and her eyes sparkled with excitement as she fell into deep thought.

Chen Hui didn't disturb them. He quietly returned to the podium, took out a paper, and began to study it carefully. Before the seminar even started, everyone fell into a collective silence.

The clock ticked away, and two hours passed in the blink of an eye. Looking at the students lost in thought, Chen Hui smiled and added, "Elena's counterexample is like a mirror, reflecting the boundaries of our current approach."

We are grateful to her for reminding us that in the deep sea where PDEs have degenerated, there are always more bizarre sea monsters waiting for us to understand and tame.

This is precisely what makes this field so fascinating—every seemingly insurmountable problem may hold the key to unlocking a new door.

Chen Hui put down the chalk, placed his hands on the podium, and stared intently. "So, instead of getting hung up on the possibility that my Lemma 3.1 might fail in extreme cases, let's take this line of thought a step further and see if we can build a more robust framework to quantify the 'illness' of coefficient oscillations."

Is it possible to find a universal rule that tells us when classical methods still work and when we must resort to 'heavy weapons' like oscillatory integrals or multiscale methods?
This might be a starting point for our next discussion!

The initial tension and skepticism in the seminar room had long since dissipated. Elena led the applause, her eyes filled with sincere admiration.

The other students also applauded, their faces filled with excitement and admiration.

Conquering these geniuses is simple: just be stronger than them.

Chen Hui did not use his Fields Medal title to intimidate them, but rather used his profound insights into the essence of mathematics, his candid attitude towards the problems, and his creative thinking in turning challenges into opportunities to completely win over this group of proud Princeton elites.

The rain outside the window seemed to have lessened a bit, and the ancient walls of Fein Building seemed to resonate silently as well.

Chen Hui put away his paper, smiled slightly, and stepped out of the seminar room, heading towards the second floor of Fein Building.

"Professor Chen."

A cool voice called out to Chen Hui from behind.

Turning around, he saw Elena, who had just raised a question in the seminar room. Her long, light golden hair was as soft as wheat waves under the sun, casually tied into a loose ponytail, with a few strands falling beside her ears. Her blue-gray eyes, like the thin mist of a Siberian winter, shone with a determined light as she looked at Chen Hui. "Professor Chen, may I apply to be your graduate student?"

"of course can!"

Chen Hui agreed without hesitation. This Russian girl had already proven herself in class, and in terms of mathematical talent, she was far superior to Chloe.

Of course, that's not enough. Since it's Princeton, we have to use Princeton's standards. Chen Hui said with a smile, "I hope you can bring me some surprises at the next seminar."

Elena nodded. "Understood, Professor."

After saying that, she turned and left.

"Interesting guy!"

Chen Hui smiled slightly and stepped to the largest public blackboard in the second-floor corridor.

The discussion didn't end until 11 a.m., and by then the blackboard was already occupied. Li Zehan was excitedly waving chalk, and the bold title written on the blackboard read, "The Nash Equilibrium Trap of the Coffee Queue? Prove It's Not Optimal! Challenge All the Minds in Feinhouse!"

Below is the model meticulously constructed by Li Zehan.
Participants: N rational students/professors who crave coffee.

Strategy: Select arrival time t (discretization).

Payoff function: Negative utility = f(waiting time) + g(coffee quality degradation factor) + h(lateness penalty factor). Where, waiting time depends on the arrival time distribution and the coffee machine's service rate μ.

Nash equilibrium (NE): He drew a bell-shaped distribution at the peak, labeled as the observed NE, and calculated the average waiting time W_ne and the average coffee quality deterioration Q_ne for each person at this time.

Challenge objective: Design a mechanism, rule, or coordination method such that, under a certain equilibrium (not necessarily a non-negative equilibrium, such as a strong Nash equilibrium or a related equilibrium), everyone's coffee utility (which is the negative of the negative payoff function) is strictly higher than the utility under the current non-negative equilibrium, i.e., achieve a Pareto improvement.

Li Zehan drew a small ghost image and the initials LZH in the signature area. Passing students began to whisper among themselves, some frowning in thought, and others showing knowing smiles.

Li Zehan was very proactive, but Chen Hui looked at the rough model and his lips curled down slightly. He picked up a red chalk and wrote smoothly in the blank space next to the model.

The Q(t) model is crude! The anxiety factor β(L(t)) dependent on the queue length should be considered, β'>0, as anxiety affects taste perception.

Should the punishment for being late be applied uniformly, h(t)? A distinction should be made between teaching responsibility and learning responsibility, with different weights assigned to them.

(End of this chapter)