My Study Chat Group is Full of Real Big Shots

Shanghai International Conference Center, Shanghai.

The breakout sessions at the International Congress of Mathematicians these past few days were nothing short of a clash of titans.

Countless academic titans, figures rarely seen in public, took the stage one after another to present a series of spectacular, cutting-edge research results.

In the morning, a professor from Shuimu University stood on stage, speaking eloquently.

"...Therefore, by introducing special test configurations and performing a refined lower bound estimation of the Mabuchi functional, we can arrive at a definitive conclusion."

"The existence of a Kähler-Einstein metric on a Fano variety is strictly algebraically equivalent to its K-stability."

This presentation, titled "Kähler-Einstein Metrics and K-stability on Fano Varieties," delved deep into the intersection of Algebraic Geometry and differential geometry, drawing gasps of astonishment from the audience.

In another session, a Chinese-American scholar from MIT presented on "Partial Progress on the Global Regularity of the Three-Dimensional Incompressible Navier-Stokes Equations."

He wrote a series of complex a priori scaling inequalities on the blackboard, then turned to his peers in the audience and said,

"As long as we can strictly control the vorticity dissipation rate of the nonlinear advection term within a specific initial energy threshold, then the finite-time singularity blow-up of a three-dimensional fluid can be mathematically suppressed..."

Using a priori estimates for nonlinear partial differential equations, he had single-handedly pushed one of fluid dynamics’ Millennium Prize Problems a small step forward.

Any one of these presentations would have been enough to headline an ordinary international conference.

They were not only extremely cutting-edge but also held immense academic value.

But no matter how significant or stunning these achievements were, the attention of the truly influential titans in attendance was not entirely focused on them.

Every one of them was waiting.

Waiting for the invited talk on the afternoon of the second day, to be delivered by an undergraduate from Yan University!

...

Soon, it was the afternoon of the conference’s second day.

With twenty minutes to go before the invited talk, the main auditorium was already packed to the rafters.

Not only were there no empty seats, but the aisles on both sides and the open space at the back were also filled with people standing.

In the front row sat Peter Sarnak, Zhang Yitang, Tao Zhexuan, and a host of other titans.

Of course, the most eye-catching figure was a foreign old man with thinning hair and a wrinkled face.

Enrico Bombieri.

This living fossil of analytic number theory, now in his eighties and having won the Fields Medal back in 1974, was also waiting for Li Dong to take the stage.

"Mr. Li Dong, you can get ready to go on stage now."

A staff member in the waiting area reminded Li Dong softly.

"Alright, thank you."

Li Dong was unexpectedly calm.

Facing an audience of scholars who could shake the world of mathematics with a single stomp of their feet, his heart wasn’t racing as he’d imagined, nor were his legs turning to jelly.

’Nervous?’

’Why should I be nervous?’

’The titans in the audience were indeed living legends, their names immortalized in textbooks and top-tier journals.’

’But on my phone, in that chat group called the Cyan Dragon Study Group, there’s a whole bunch of bona fide True Gods popping up every day!’

Having been baptized by deities of that caliber, Li Dong felt the pressure melt away.

He calmly walked onto the stage.

When Li Dong’s ridiculously young face appeared on stage, the entire auditorium instantly fell silent.

Countless gazes—filled with envy, jealousy, and admiration—fixed on him in unison.

Standing in the aisle, Qin Fei watched Li Dong on stage with an indescribable feeling in his heart.

’We’re both university students. I’m still stressing over my graduation project, yet he, a freshman younger than me, is already standing on a world-class stage like this, presenting to a group of Fields Medalists?’

The immense gap between them made Qin Fei let out a long, silent sigh.

’Are we even attending the same university?’

On the podium, Li Dong gently tapped the microphone.

He tested the sound, confirming the equipment was working.

"Good afternoon, esteemed colleagues and fellow students."

"I am Li Dong from Yan University."

"I assume most of you have already read the paper I published in *Mathematics Computing*."

"Today, I don’t intend to simply rehash the already public code and the lengthy Chebyshev interpolation formulas on a PowerPoint."

Li Dong smiled.

"I’d like to talk to you all about the foundational thinking behind this algorithm."

"For a long time, when dealing with the Riemann-Siegel formula, whether it was Hardy-Littlewood’s Euler-Maclaurin summation or the later Odlyzko-Schönhage algorithm, we seem to have been stuck in a certain mindset."

"We’ve grown accustomed to confronting those extremely high-frequency oscillating remainder terms head-on, right on the critical line where the real part is 1/2."

"We have tried using larger matrices and more powerful supercomputers to brute-force the errors caused by floating-point truncation."

"But in essence, this is still just piling on computational power, not a victory for mathematics."

The titans in the audience nodded. What Li Dong said was indeed the truth.

Li Dong casually drew a coordinate system of a complex plane on the whiteboard behind him, then drew a curve.

"My Logic is, since a head-on confrontation is too costly, why not utilize the perspective of complex analysis via contour integration?"

"What if we pull our perspective away from the single real line and apply a slight shift to the integration path of the Riemann-Siegel formula’s remainder term, moving it toward the deeper topological manifold of the imaginary part?"

"Through this local topological deformation, we can introduce the method of steepest descent."

"On this new integration path, those chaotic, disordered, high-frequency oscillating terms are transformed, through topological isomorphism, into controllable exponential decay!"

"This is the key to the dimensional takedown that brings the time complexity down from O(T^(1/2))!"

As soon as these words were spoken, everyone in the audience who could understand gasped audibly.

And those who couldn’t understand simply watched those who could gasp.

Sitting in the fourth row, Zhou Shenzhi was staring intently at the diagram of the shifted integration path that Li Dong had drawn on the whiteboard.

As Professor Jiang Yubai’s star pupil, he had been wrestling with this algorithm for some time now.

He could never figure out how Li Dong had managed the error convergence of the higher-order remainder terms.

’So that’s how... using topological deformation of the contour integral, combined with the saddle-point method to cancel out the high-frequency oscillations?’

He tried to follow Li Dong’s Logic and piece together the missing part.

’If the integration path is shifted... then the error term from the prime powers can be scaled down to within an extremely small bound...’

’But, if you follow that derivation, how do you ensure the principal term isn’t contaminated when dealing with the residues at the poles?’

Zhou Shenzhi rubbed his temples forcefully, completely unable to deduce that final, crucial step—the one that brought the whole thing to life.

Li Dong’s presentation wasn’t long. In just over thirty minutes, he had finished explaining the core, foundational Logic.

"That concludes my presentation. Thank you."

Li Dong gave a slight bow to the audience.

However, the expected thunderous applause did not come.

Everyone was still immersed in the algorithmic Logic Li Dong had just laid out.

This Logic struck everyone present as both novel and ancient.

It was novel because, in an era where modern computing power has skyrocketed for decades, no one had ever thought to use such a classical, geometric-topological approach to tackle a computational number theory problem so heavily reliant on numerical analysis.

But it was also ancient, because this way of thinking, with its profound classical mathematical beauty, was so reminiscent of the pure mathematical masters from over a century ago!

It was like the timeless shock of 1932, when Carl Ludwig Siegel, in the library of the University of Göttingen, unearthed the Riemann-Siegel formula from Riemann’s dusty manuscripts.

In fact, he had dug even deeper than Siegel!

Sitting in the front row, Professor Arthur Penrose’s body was trembling slightly from excessive excitement.

His eyes shone with a fanatical light.

He felt as if he had just met a mathematical deity in the flesh and had finally glimpsed the very tip of the iceberg of how that deity thought.

His student, Sarah, sitting beside him, stared at her professor in utter astonishment.

This was the first time she had ever seen her professor lose his composure like this.

Meanwhile, Li Dong, standing on the stage and seeing the complete lack of reaction from the audience, was speechless.

’Seriously...’

’When other people finish their presentations, you all give them a token round of applause whether you understood it or not! How come when it’s my turn, there’s nothing?’

’What’s the deal? What about the Q&A session? Is no one going to say anything?’

To break the awkward silence, Li Dong could only give a couple of soft coughs.

"Ahem..."

"Um, esteemed colleagues, does anyone have any questions regarding the presentation?"

As soon as he finished speaking, there was finally a stir from the audience.

An eager voice called out from the fourth row.

"Li Dong, I have a question."

Zhou Shenzhi stood up.

"Regarding the data set of zeros on the order of 10^23 that you posted on arXiv a few days ago..."

Zhou Shenzhi stared intently at Li Dong.

"The algorithm used for that must be different from the one you just shared, the one published in *Mathematics Computing*, right?"

Zhou Shenzhi had asked the question that was on the mind of every scholar present.

"Could you share the Logic of that Advanced Edition algorithm with us here?"