348: Chapter 348 You solved the P vs NP problem?

Less than half an hour later, Terence Tao arrived at the villa.

As soon as the door opened, he saw the dense derivations on the whiteboard in the study and the mountain of draft papers piled on the desk.

And there was Zhao Yang, sitting in the middle.

Terence Tao asked Zhao Yang, his breathing somewhat rapid, "You solved it? The P vs NP problem???"

Terence Tao's expression was excited, his eyes filled with fervor as he looked at Zhao Yang.

"Here, it's all right here. Why don't you take a look first!" Zhao Yang pointed to the pile of draft papers beside him, smiled, and said to Terence Tao.

"Alright!"

Without any hesitation, Terence Tao picked up the stack of draft papers from the top of the desk and began to flip through them.

He flipped through them very slowly, occasionally stopping to look at a certain line of derivation repeatedly, or picking up a pen to make a brief calculation mark in the small blank space on the side.

Zhao Yang's study was very quiet, with only the sound of turning pages and the occasional rustle of a pen tip across the paper.

Terence Tao stopped when he flipped to the third chapter regarding the construction of the normal cone of polynomial ideals.

He stared at one of the pages for a long time, then pointed his finger at a section of the derivation, his lips moving as if talking to himself, but no sound came out.

He flipped through a few more pages, wrote a line of brief calculation on the blank space of the draft paper, and then continued flipping.

He flipped all the way to the last line, to that Q.E.D., then placed the draft papers on the table, let out a long sigh, and looked at Zhao Yang beside him.

"Based on what I have seen so far, I am inclined to believe that you have really done it. But how exactly did you get past this step? Also, I have many points of confusion regarding this paper and hope you can answer them for me."

Terence Tao looked at Zhao Yang with a glint of yearning in his eyes.

For a pure mathematician like Terence Tao, the thirst for truth clearly transcended many things.

He had known Zhao Yang for so long and had collaborated many times; every single time, Zhao Yang was able to take one more step when everyone else thought they had reached the end.

But this time was different.

This was the P vs NP problem!

It was one of the seven Millennium Prize Problems, the Holy Grail of computer science, and a chasm that countless mathematicians had failed to cross despite their best efforts.

Zhao Yang was too magical, and too incredible.

"Of course." Zhao Yang smiled and said to Terence Tao.

"Before this, I had gone around in circles more than once. I couldn't get past the Elimination theorem for fields of positive characteristic, so I stopped using elimination. I directly used a generalized version of Hilbert's Nullstellensatz, combined with flat morphisms, to pull the problem back to a field of characteristic zero to handle it."

"On this problem, Pierre Deligne's variant of the Trace map helped a great deal. The hardest part was when estimating the dimension at the boundary of the normal cone; I discovered a new constraint condition. Even Grothendieck hadn't mentioned this condition back in his day, and the article from that MIT team didn't mention it either, yet it exists."

Terence Tao stared blankly at the draft papers, then picked them up and flipped through them from the beginning again.

He turned to a page in the middle and pointed to a section of the derivation: "Here, you are pulling it back directly using the Trace map, without needing any additional smoothing?"

"No need. I've verified it; the boundary conditions are exactly covered by that new constraint condition. The degenerate cases you were worried about do exist, but they only appear in a parameter interval I hadn't noticed before. After adjusting the parameters, I bypassed them directly."

Zhao Yang smiled and answered.

"I'll help you organize the paper."

"Alright!" Zhao Yang nodded.

Terence Tao pulled up a chair and sat down at the desk; his expression became exceptionally serious and devout!

Terence Tao had never solved a mathematical problem of this level before!

With Terence Tao's help, the two of them spent a full three days organizing Zhao Yang's original manuscript into a standard paper format.

It was over two hundred pages in total, divided into six chapters, with each chapter corresponding to a major logical stage.

Zhao Yang wrote the main framework, while Terence Tao helped to organize the connection parts for Harmonic Analysis and refine the phrasing of several key lemmas.

At the same time, he restated the smoothing treatment that Zhao Yang had used to bypass the Elimination theorem for Finite Fields during his seclusion, strictly using standard academic language.

After organizing the paper, Zhao Yang opened his computer, uploaded the paper to arXiv, and simultaneously submitted it to the Annals of Mathematics.

Less than an hour after the paper appeared on arXiv, in an old apartment building in the suburbs of Saint Petersburg, Russia, Professor Perelman was facing his computer screen, scrolling through today's arXiv updates.

He was wearing a plaid shirt that had been washed until it was somewhat faded, and on the table was a cup of black tea that had already gone cold.

Since proving the Poincaré Conjecture, he had almost disappeared from the public eye.

He had declined the fields medal, declined the one-million-dollar prize from the Clay Mathematics Institute, and declined all media interview requests.

He lived alone in this old apartment, his life so simple it was almost austere.

But he had not stopped doing mathematics.

He had maintained the habit of opening arXiv every morning for over a decade, starting from before the Poincaré Conjecture was published.

In today's update list, he saw Zhao Yang's name.

Clicking on the title, he saw P vs NP.

He did not hesitate, downloaded the PDF directly, and started reading from the abstract.

After reading the abstract, he stood up, walked to the kitchen, picked up the teapot to pour himself a fresh cup of hot tea, then returned to the computer and began reading the introduction.

When he read the third chapter regarding the construction of the normal cone of polynomial ideals, he took off his glasses, grabbed a piece of blank draft paper, and began to perform calculations while comparing them to the paper.

There were many people like Professor Perelman.

Not long after Zhao Yang's paper came out, it quickly topped the arXiv paper download list.

There was no helping it; Zhao Yang was a celebrity, and there were many people watching him.

Ever since he proved the Goldbachs Conjecture and the Kakeya Conjecture, mathematicians all over the world, whenever they saw the name Yang Zhao appear on arXiv, would reflexively click in to take a look.

Some wanted to follow the cutting-edge breakthroughs as soon as possible, some wanted to see what big news this young man had cooked up again, and a small minority were there with a critical mindset, wanting to find a fatal flaw in this paper.

Professor Grothendieck also received this paper from his student.

Pierre brought the printed paper to the hospital room in a folder.

Grothendieck was leaning against the hospital bed, with an IV line still inserted in his arm, but his eyes changed the moment he saw the title of the paper.

He asked Pierre to crank the head of the hospital bed up a bit, put on his reading glasses, and began to read page by page.

After reading for a while, Grothendieck's eyes were full of admiration.

He flipped to the middle of the paper regarding the construction of flat morphisms in fields of positive characteristic, his finger sliding slowly across the page, and he softly said a sentence.