The classroom fell silent at the teaching assistant’s words.
The doctoral students sitting around the U-shaped table all glanced at one another.
"*The* Li Dong?"
"East God?"
"...I’ve never seen him in person."
The voices were hushed, but the U-shaped table wasn’t that big. Everyone could hear whoever spoke.
A PhD student in his thirties nudged the younger student next to him with his elbow.
"Should we... make some room for him?"
The younger student quietly shifted his laptop, which was running a numerical program, further inward.
The seats around the U-shaped table were already reserved for Academician E Weinan’s ten or so doctoral students, and space was tight.
Squeezing another person in would be a bit of a crush.
When the few undergraduates in the back rows heard that Li Dong was coming, the looks in their eyes began to change.
A small, skinny student on the end was biting his lip so hard it was about to bleed.
He was a senior from the School of Information Science, with the surname Zheng.
Coincidentally, like Qin Fei, he had chosen his senior thesis topic six months ago.
"Multithreaded Parallel Optimization of the Nontrivial Zeros of the Riemann ζ-Function Based on the OS Algorithm."
He had been full of ambition when he chose the topic. His advisor had even clapped him on the shoulder and said it was a solid direction that would guarantee his graduation.
But then...
Li Dong’s paper on the dimensionality reduction algorithm dropped.
Its nonlinear FFT grid expansion with Chebyshev interpolation had violently smashed the computational efficiency for nontrivial zeros, boosting it to the order of 10^23.
Once the algorithm was made open-source, everyone working on that entire research path, including him and his senior thesis, was completely and utterly screwed.
The more the small student thought about it, the angrier he became, grinding his teeth.
Then he turned his head.
He noticed that the complexions of the two people next to him weren’t any better than his.
He recognized the one on his left wearing glasses.
A senior from the School of Mathematics, surname Shen. They had both audited Teacher Han’s Real-Valued Function course before.
He lowered his voice and nudged the other guy.
"Brother Shen... why do you look even worse than I do?"
The senior, Shen, gave a bitter smile.
"My prospective graduate advisor..."
"...told me to chew on that paper."
For a moment, the small student didn’t get it.
"Whose?"
Brother Shen didn’t speak. He just lifted his chin toward the door, indicating the "Daniel Wu" in question.
The small student understood instantly.
Li Dong’s paper.
"I just started reading it last night,"
Brother Shen said in a low voice.
"The introduction alone took me all night, just to look up the references and barely make sense of the notation."
"And after that..."
He shook his head and said no more.
The small student was speechless.
He had thought he had it bad.
Compared to Brother Shen, maybe his situation wasn’t so tragic after all.
At least he could try to pivot his senior thesis to a downstream application and coast a bit. Brother Shen had to actually digest that entire paper.
Just then...
CREAK.
The back door was gently pushed open a crack.
A head poked through.
"Teacher E—"
"Hope it’s okay if I audit your class."
Seeing Li Dong’s casual entrance, E Weinan couldn’t help but smile.
He beckoned him in.
"Come on in."
Only then did Li Dong step into the classroom.
E Weinan pointed at Li Dong and said,
"This is Li Dong. You all know him, right? I won’t bother with an introduction."
After a pause, he added,
"He’s my student."
He said it with such natural ease.
As if Li Dong were truly one of his own PhD students.
Which wasn’t wrong, technically.
He was, in fact, one of the fifteen advisors Yan University had prepared for Li Dong.
As for the other fourteen...
’Who cares about them?’
"Find a seat for yourself."
The moment Academician E Weinan finished speaking,
the PhD student at the U-shaped table who had just moved his laptop shot to his feet.
"Hey..."
"You can sit here."
Li Dong glanced at the already crowded seats and quickly waved his hands in refusal.
"No, no, senior, you sit."
"I’m fine sitting in the back."
"After all, I haven’t even graduated from my undergrad program yet."
"My PhD doesn’t start until the second half of the year."
"How could I take a seat from all the seniors here?"
The PhD student: ’???’
’He’s only starting his PhD in the second half of the year?’
The expressions on the faces of the nearby PhD students were indescribably complex.
A female doctoral student sitting at the far end of the table silently cursed in her mind, ’Holy shit!’
She was thirty this year and had just passed her qualifying exams.
Li Dong’s words had just rendered her completely speechless...
Li Dong quietly thanked the senior who had offered him the seat.
Then he ambled over to the last row.
He found an empty spot next to the three undergraduates who looked equally miserable and sat down.
The small, skinny student by the window shot him a resentful glare.
Li Dong smiled and gave him a nod by way of greeting.
The small student: ’??? What are you smiling for? Can’t you see the resentment in my eyes?’
E Weinan paid no more mind to the drama in the back row.
He cleared his throat and began the lecture.
"That thing we were discussing last month..."
"The inverse problem with singular source terms, the discontinuous Galerkin variant, the relationship between the interior penalty parameter and stability—you all remember that, right?"
The dozen or so doctoral students below naturally nodded to show they did.
E Weinan continued,
"This research direction has been very hot in the community these last couple of years."
"Especially after being deployed with high-performance computing, its handling of inverse problems with singular source terms is much more elegant than traditional finite element methods."
"This is considered one of the more cutting-edge applications in applied mathematics right now."
"And this is what’s most interesting about applied mathematics."
"You think it’s just a tool, but then you look back and realize it’s actually pointing out for you whether the physical problem itself is ’clean’ or not."
At this, he glanced at a PhD student in the front row.
"Li Wei."
The PhD student looked up.
"That problem you mentioned to me last month, did you solve it?"
Li Wei nodded.
"Yes, it’s solved."
"Last week I tried a different approach. I weakened the dual form of the bilinear functional and stuffed an extra term into the penalty term..."
Li Wei had only gotten that far...
...when E Weinan cut him off.
"Good enough."
"Since you’ve solved it, we won’t waste class time on it."
Li Wei, getting the hint, swallowed the rest of his words.
This was one of E Weinan’s long-standing rules for his students.
If you solved it, just write it down on a small note and hand it to him after class.
This one hour in class, every single minute and second of it, was for breaking new ground.
E Weinan looked down at his A4 paper again.
"Today, I want to talk to you about something that’s caught my attention these past two days..."
"I can’t say for sure whether it’s a topic or just a problem yet."
"In any case, it’s quite cutting-edge, so I wanted to share it with you."
As he spoke, he picked up a piece of chalk and drew a simple diagram on the blackboard.
The horizontal axis: energy.
The vertical axis: the eigenvalue density of a certain operator.
Three peaks.
The first one was sharp.
The second one was barely discernible.
The third was just a blurry mess.
Sitting in the back row, Li Dong perked up when he saw the diagram.
’This diagram...’
’It looks a lot like the one Professor Levan pulled up for me on his tablet yesterday at Jingchun Garden.’
Of course, the one E Weinan drew was simpler, and the physical scenario he described wasn’t the ligand field tensor inversion Levan was working on.
But at their core, they were the same class of problem.
A class of inverse problems for ill-conditioned operators.
E Weinan tapped the third peak with his chalk.
"There are a few hurdles in here I want you to pay attention to..."
"First, its kernel is incomplete. You can only invert for partial information from certain segments of experimental data."
"Second, during the inversion, different basis functions will ’fight’ with each other. If the phase is off, you get this kind of blurry mess on the graph."
"And third..."
"This is what I’m personally most interested in."
"In this class of problems, there might exist an implicit constraint."
"I haven’t yet found a way to quantify this constraint."
"But my intuition tells me it should be able to pin down the third peak."
E Weinan put down the chalk as he spoke.
"As for which direction to look for this constraint..."
"I’ll give you a general framework. You can go back and feel out the details yourselves."
Using the simple diagram, he explained several possible entry points.
For each one, he only pointed in a general direction without delving into the specifics.
This was Teacher E’s style of mentoring PhD students.
He gives you the framework; you figure out the details.
Since you’re doing a PhD, you shouldn’t think of yourself as a student anymore. You have to start seeing yourself as an independent researcher.
You can’t just expect others to pave the entire road for you.
Li Dong sat in the back, listening intently.
A direction was exactly what he needed.
He was already planning in his head to ask Academician E Weinan after class which dozens of books would be best to tackle first to dig deeper into these entry points.
However, the undergraduates sitting next to him were starting to fall behind. Some had already started recording the lecture on their phones.
Truthfully, Li Dong didn’t understand everything either.
Of the several entry points for the "implicit constraint" that E Weinan mentioned, there were two he had never even heard of.
But he didn’t panic.
He simply opened a new slot in his Memory Palace and stored the two incomprehensible points there, exactly as he’d heard them.
He would chew on them slowly after class.
Just then, E Weinan stopped talking.
He surveyed the students in the classroom.
This time, he didn’t look at the circle of PhD students at the U-shaped table.
Instead, his gaze fell directly on the back row.
"You few..."
"What did you take away from this?"
The undergraduates in the back row all lowered their heads in unison.
Their "takeaway" was currently stored in their phone recordings.
They’d have to go back to their dorms, pause the video, and painstakingly cross-reference it with other materials just to unpack the lecture piece by piece.
And he wanted them to talk about their takeaways now?
They truly had nothing to say.
E Weinan waited for a moment. When no one in the back spoke up,
he turned to Li Dong.
Then, he smiled.
"Li Dong."
"What about you?"

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