Chapter 57
The academic conference began.
One professor was the first to speak.
“To start with the most direct impact, there are now thousands of conditional theorems proved on the assumption that the Riemann Hypothesis is true. We can now say that all of them have been proved true and have become unconditional theorems.”
The professor who specialized in analytic number theory nodded.
“That’s right. In analytic number theory alone, results on the zero distribution of L-functions come pouring in one after another. I feel relieved that when I write papers from now on, I won’t have to add the phrase ‘assuming the Riemann Hypothesis is true.’”
“I’m most excited about the impact on cryptography.”
said the professor specializing in cryptography.
“Since an error bound is now guaranteed in predicting primes, prime-generating algorithms have gained a solid theoretical foundation. It could also affect the Miller-Rabin primality test. Researchers will likely dig into this direction. With a theoretical foundation, cryptography will become even more robust.”
I shot my hand up!
A six-year-old daring to raise her hand and voice her opinion while the adults were discussing something important.
“Smashing Confucian thought wabababat-style—baby Ayeon, as expected!”
“I found that part interesting. There are some byproducts from the process of constructing operator H, and I think a few of them could be applied to generating functions such as L-functions or the Selberg zeta function.”
The professors all turned to look at me.
“Which part?”
“The way I set up the boundary conditions to ensure the operator’s self-adjointness. I designed it with the Riemann zeta function in mind, but I think the same structure could be used when extending it to Dirichlet L-functions or the L-functions of modular forms. Of course, the detailed conditions would differ in each case.”
The professor who heard me quickly jotted it down in his notes and said.
“That direction could also connect to the Langlands program.”
“I think so too.”
Professor Park Mu-yeol glanced at me briefly before opening his mouth.
“The part I paid the most attention to in Ayeon’s paper was the construction of operator H. It was so different from the approaches mathematicians had tried before that it was hard to understand at first. Could you explain that part a little more?”
I nodded and walked up to the stage toward the blackboard.
I picked up a piece of chalk as I stepped onto the platform.
“Yes. Most existing approaches tried to tackle it this way. The part I drew inspiration from while looking through Professor Shin Ijun’s research notes was—”
Clack-clack-.
Formulas began to fill the blackboard.
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The meeting went on for more than two hours.
A variety of opinions poured out on the subject of my paper.
The impact of the proof of the Riemann Hypothesis on number theory.
The possibility of extending it to L-functions.
Its connection to the Langlands program.
Its impact on cryptography and computational number theory.
And future research directions.
Chatter—.
Even though it was a difficult paper, the mathematicians talked without missing a beat.
Looking around, I was struck anew by the fact that every mathematician here had carefully read my 148-page paper.
“Woo-hehing-.”
It felt like becoming a bestselling author, I guess?
Mom also chimed in here and there.
She didn’t know the details of my paper’s proofs, but she had a rough idea of the direction.
In the parts related to complex analysis or topology, Mom could offer plenty of opinions too.
After hearing Mom’s opinion, one professor asked with interested eyes.
“Ms. Seol-ae Han. Do you plan to resume your research in the future? You said you studied under Professor Shin Ijun, right? Maybe because of that, but you seem to have quite sharp insight.”
Mom thought for a moment and said.
“I’m thinking about it. I haven’t decided yet.”
“If you continue researching in this field, you’re welcome. There’s a direction I’d like to work on together.”
Mom quietly smiled and lowered her head.
I thought to myself as I listened to the conversation.
‘I hope Mom goes back to academia too...’
Losing a talent like Mom was a huge loss for the mathematics world.
But that kind of decision has to be Mom’s own choice.
More than anything, we needed someone who could take care of me and Arin.
For now, I’d just have to be satisfied with reading journals with Mom at home.
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By the time the conference was drawing to a close.
One professor said, unfolding the printed copy of the paper with a shrrk-.
“Something has kept running through my head ever since we started talking today.”
Everyone looked at that professor.
“The operator H proposed by Yoo Ayeon, I mean. Its construction, I mean. It looks like it could have a significant impact on future number theory research. Especially the part in the paper proving that an operator satisfying certain conditions is self-adjoint, [Theorem 4.18]. I think that’s a pretty powerful theorem.”
Hearing that, one professor immediately nodded.
“That part stood out to me the most too. Honestly, the proof of the Riemann Hypothesis itself is amazing, but I thought this [Theorem 4.18] that came out of it could be used much more generally in the future.”
A professor specializing in operator theory continued.
“Exactly. In operator theory, whenever we study operators, lacking self-adjointness causes all sorts of headaches because the desired results just don’t come out. But the fact that an operator satisfying Ayeon’s conditions is guaranteed to be self-adjoint is, in itself, an incredibly powerful tool. I think it may even be used in theoretical physics.”
“Oh, come on. Don’t hype me up so much—”
Overwhelmed by the unexpected praise, I squirmed with embarrassment.
Then Professor Park Mu-yeol quietly said,
“Heh, I think so too. Perhaps Ayeon has invented something even more valuable than proving the Riemann Hypothesis. A tool that can be used broadly in theoretical mathematics and theoretical physics in the future.”
Even one of the researchers who had handled the review at the Clay Mathematics Institute added a word.
“I also felt while reviewing the paper that this could be extended to a result far more general than I expected. Exactly how general will take a little more research, though.”
The auditorium fell briefly silent.
I felt my face go hot in an instant.
Pshhh-.
I was so embarrassed that I covered my face with both hands.
“Hmph! No matter how much you praise me, you won’t get anything out of me!”
Bung-bung-!
I kicked my feet and shook my head in denial.
Hahaha-.
The professors laughed.
But even after the laughter faded, no one changed the subject.
Everyone had rather serious expressions.
No way. This wasn’t a joke...?
One professor thought for a moment and said.
“Come to think of it, this theorem doesn’t have a name. In the paper, it’s just listed as Theorem 4.18.”
“Right. Shouldn’t we give it a name?”
The professors looked at one another.
‘No... wait. This flow is...’
Since I had already guessed what would happen next,
I raised my hand and tried to object a little.
“Ah, um, that... that’s a little...”
“Ayeon Yoo, why is that?”
“Why, why...? Because... it’s embarrassing.”
“Ayeon, in mathematics, naming things after the person who first proposed them is perfectly standard.”
Professor Park Mu-yeol replied.
“Hng... b-but-but-but...”
I covered my face again with a swish.
Only then did I realize for the first time that I, too, had a timid side no less than Arin’s.
Pat-pat.
Mom patted my head.
The professors started discussing it.
Leaving me completely out of it.
“What should we call operators that satisfy this condition?”
“It seems appropriate to call them Ayeon Operators, since they are operators proposed by Yoo Ayeon.”
“Then what about the theorem that an operator satisfying that condition is self-adjoint?”
A professor thought for a moment and said.
“Let’s call the theorem that every Ayeon Operator is self-adjoint Ayeon’s Self-Adjointness Theorem.”
The mathematicians nodded.
“Good.”
“That seems right.”
“I agree too.”
With the consensus of dozens of mathematicians, the name was decided right there and then.
And they didn’t even ask the person concerned!
I sat motionless, covering my face with both hands.
Pshhh-.
Heat steamed up from my face.
I hoped my bright-red face wouldn’t show beyond my palms.
But why,
behind my tiny little palms,
twitch-.
twitch-twitch-.
was this clueless corner of my mouth trying to keep lifting up...!
I desperately tried to hold back my laughter, but I couldn’t.
A giggle kept leaking from the corners of my mouth.
“Hihihihuhu-hihihuhu...”
✒️✒️✒️
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Theorem 4.18 (Ayeon Self-Adjointness Theorem, Ayeon’s Self-Adjointness Theorem)
Let H be a densely defined symmetric operator on a separable Hilbert space.
If H satisfies
(1) the Arithmetic Boundary Regularity condition,
(2) the Spectral Compactness condition,
then,
H has a unique self-adjoint extension H‾.
Moreover, the spectrum of H‾ is purely discrete.
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And this theorem would, far in the future, even be included in a graduate textbook.