After taking a leave of absence from Deligne, Xu Chuan got up and walked out of the dormitory.

He still has a lot of work to do before he officially enters the uncharted territory of Hodge's conjecture. Both in life and in mathematics.

Solving the Hodge conjecture is like the first time that humans sailed the vast sea, and no one knows if there are other landmasses in the unknown ocean, and no one knows if they will be able to reach another coastline smoothly.

The only thing he had was a boat that had just been built.

And whether this small boat, after entering the unknown ocean, will be overturned by the wind and waves, whether it will sink to the bottom of the sea, whether it will hit the reef and get stuck and unable to move, Xu Chuan does not know.

But despite this, he still has to try.

Because even if it's just ten meters away, it's a great breakthrough.

.......

After purchasing a batch of daily necessities in the store, Xu Chuan borrowed a batch of manuscripts and materials related to Hodge's conjecture from the flint library.

Some of them were things he had seen before, and some of them were things he hadn't read yet.

These are all precious knowledge left by predecessors, and some of them are simply not searchable on the Internet. Because they are just some ideas and original theories of a certain mathematician, and they have not yet been formed.

These things, whether you have seen them or not, will be useful for him to charge Hodge's conjecture.

But when it came to borrowing these things, he had a lot of trouble.

In charge of the flintstone library is a raunchy-looking old man with messy hair like a bird's nest, a top expert in the field of paper preservation, but also unusually stubborn.

And this stubborn old man has always been reluctant to lend so many documents to the outside world, believing that he is likely to damage or lose these precious manuscript papers.

In order to obtain these materials, Xu Chuan spent a day grinding in the Flintstone Library, and the final effort was only to get the other party to agree to put them together and read them in the library.

But for Xu Chuan, proving Hodge's conjecture in the library was not a very reliable path.

It's quiet, but it's full of people every day.

In the end, he could only find David Xiu, the dean of the Princeton School of Mathematics, and made a series of assurances, learned some methods of preserving paper materials, and even signed a letter of guarantee before reluctantly agreeing to the other party.

With a lot of information, Xu Chuan returned to the dormitory.

In fact, he didn't need to be reminded by the bad old man from Germain, he would have taken good care of these things.

Now, however, in addition to being well preserved, the greater value of these materials is to play their part in the Hodge conjecture.

Presumably, the mathematicians who created this knowledge in the first place must have thought the same way.

No one wants to see the knowledge they have created put on the shelf, and if a piece of knowledge cannot be circulated and used, it has no value for knowledge.

.......

After dealing with the preparations before entering Hodge's conjecture, Xu Chuan locked himself in the dormitory again.

Time passed like this, and in the blink of an eye, the golden autumn of October arrived, and the sugar maples, sycamores and other trees outside the Rockefeller Residential College began to glow with a hint of gold. Occasionally, a few fallen leaves fall slowly in the wind.

In dormitory No. 306, a figure stood in front of the window, looking out at the plane tree full of plane fruits.

The early morning sunrise is bright in the dark blue clouds, and the golden and dark green leaves outside the window are intertwined, and the heavy plane nuts are inlaid in it.

Looking at the scenery outside the window, Xu Chuan had a smile on his face.

Autumn is the season of harvest.

Although the research on Hodge's conjecture did not go as smoothly as he expected, he was always confident in the final result.

And two months later, in the unknown ocean of Hodge's conjecture, he finally found a piece of coastline in front of him.

That's the New World!

Looking at the scenery outside the window, Xu Chuan turned around and returned to the table with a smile.

Although Hodge's conjecture had not yet been perfectly resolved, he had already seen the horizon where the coasts intersected, and the new continent that towered over the horizon.

All that's left is to row your boat over with all your might.

.....

Picking up the ballpoint pen on the table, Xu Chuan picked up the pen in the place where he had not finished writing before, and continued:

“...... Let v be an algebraic cluster in the projective space, and v̊ is the set of regular points of v. The cohomology group on vˊ relative to the fubini-study measure l?2-de rham is isomorphic to the cross cohomology group on v....."

"If y is a cluster of closed sub-algebra defined as x with a corest dimension of j on k, we have the standard mapping: tr : h2(n?j)(y?k k, q')(n? j)→ q`...... Here (n? j) is ?? q`(n? j)。

This mapping is the same as the limit mapping: h2(n?j)(x?k k, q')(n? j)→ h2(n?j)(y, q')(n? j)"

“........”

"According to Poincar′e duality theorem: HOM(H2(N?J)(X?k K, Q')(N? j)， q`)～= h2j (x?k k， q`)(j)......“

.......

Time passed little by little under his pen, and Xu Chuan devoted himself to the final breakthrough.

Eventually, the pen in his hand suddenly turned.

“..... Based on the mapping tr, the limit map, and poincar′e, the duality theorem is all compatible with the action of gal(k/k), so the role of gal(k/k) on the upper cohomology class defined by y is also trivial. then aj (x) is the q vector .......space generated by the cohomology class of a closed algebra cluster on k in h2j (x?k k, q')(j) with a coincidence of x defined as j on k."

"When i≤n/2, the quadratic x→(?1)il?r?2i(x.x) on ai (x)n ker(l?n?2i+1) is positively definite."

"From this, it can be concluded that on a nonsingular complex projective algebraic cluster, any Hodge class is a rational linear combination of algebraic closed-chain classes."

"That is, the Hodge conjecture is true!"

The ballpoint pen in his hand clicked the last dot on the white manuscript paper, Xu Chuan breathed a long sigh of relief, threw the ballpoint pen in his hand aside, lay back, leaned on the back of the chair and stared at the ceiling in a daze.

When the last character landed on the manuscript paper, his heart was not excited, not happy, not satisfied and accomplished.

Rather, there was some incredibly confused feeling.

It took more than four months, starting from the manuscript left to him by Professor Mirzahani, to the solution of the problem of 'unreducible decomposition of differential algebraic clusters', to the refinement of algebraic clusters and group mapping tools, and finally to the solution of the Hodge conjecture.

He has been through so much on this path.

After staring at the ceiling for a long time, Xu Chuan finally came back to his senses, and his eyes fell on the manuscript paper on the desk in front of him.

After going through all the manuscript papers and making sure that this was really the result of his own work, he finally showed a bright smile on his face, as bright as the sunlight shining through the window.

If there were no accidents, he succeeded.

Successfully solved the problem of the century of the Hodge conjecture.

This is the most important breakthrough in the problems related to the Hodge conjecture since the mathematician Lefschetts proved the Hodge conjecture of class (1,1) in 1924.

Although he does not yet know if it will stand the test of time by other mathematicians.

But in any case, he took another big step forward in mathematics.

.......

After finishing the paper proving Hodge's conjecture, Xu Chuan took some time to go through the items on the manuscript paper again and refine some other details.

Once he had processed this, he began to organize it into his notebook.

Then it's ready to be made public.

The proer is not qualified to give an evaluation of the correctness of any mathematical conjecture.

Only when it is fully public, peer-reviewed and time-tested, can we determine whether it has indeed succeeded.

It took a whole week, and Xu Chuan finally entered all the nearly 100 pages of manuscript paper in his hand into the computer.

More than one-third of these hundreds of pages of proof are explanations and demonstrations of the algebraic cluster and group mapping tools that solve the Hodge conjecture, and one-third of the pages are theoretical frameworks for the Hodge conjecture and the algebraic cluster and group mapping tools.

The rest is the process of proving Hodge's conjecture.

For this paper, the tools and frameworks are the core foundation of it.

If he wished, he could have split the tools and theoretical framework separately and publish them as stand-alone papers.

It's like Peter Schultz's 'P-Class-Perfect Space Theory'.

These things, if finally accepted by the mathematical community, would be enough for him to win a Fields Medal.

It's not that the Fields Medal is cheap, it's the importance of mathematical tools to mathematics.

A great mathematical tool that solves more than just a problem.

Just like an axe, it can be used not only to cut down trees, but also to be used as a tool for carpentry, to process items, and to use as a weapon for fighting.

In the same way, the tools he constructed for mapping algebraic clusters and groups were not limited to the Hodge conjecture.

It can be used to try many algebraic clusters and differential forms, as well as polynomial equations, and even difficult problems in the direction of algebraic topology.

For example, the 'Bloch conjecture', which belongs to the same family of conjectures as the Hodge conjecture, the problem that Hodge's theory of algebraic surfaces should determine whether the chow group of zero loops is finite-dimensional, and the cohomology problem of the cohomology group on some motives of finite coefficients is mapped to the cohomology problem on the etale, etc.

These conjectures and problems support each other, and mathematicians are constantly making progress on one or the other, trying to prove that they have led to great advances in number theory, algebra, and algebraic geometry.

If the algebraic cluster and group mapping tool can solve the Hodge conjecture, then it can be fully adapted to the same type of conjecture, but it can at least play a partly role.

Because the Hodge conjecture is a conjecture that studies the correlation between algebraic topology and the geometry expressed by polynomial equations.

It does not study state-of-the-art mathematics, but establishes a fundamental connection between the three disciplines of algebraic geometry, analysis, and topology.

To solve this problem, it takes a prover with a deep understanding of all three areas of mathematics.

For the vast majority of mathematicians, it is not easy to have an in-depth study in one of the three major fields of algebraic geometry, analysis, and topology, let alone be proficient in all three.

For Xu Chuan, analysis and topology were the mathematical fields he had mastered in his previous life, but algebraic geometry was not in the scope of study.

But in his life, he followed Deligne to study mathematics in depth, and with such a tutor, he made incredible progress in algebraic geometry.

......

After finishing all the proof papers of Hodge's conjecture and entering them into the computer, Xu Chuan converted them into PDF format and sent them to the two supervisors, Deligne and Witten, by email.

After thinking about it, he uploaded it to the arxiv preprint website.

Although today's Arxiv preprint site has become a place for computers, there are still a large number of mathematicians and physicists on it.

Throwing your unpublished papers on them can not only occupy the pit in advance to prevent plagiarism, but also expand the influence of the paper in advance.

For a proof paper for a problem like the Hodge conjecture, it will undoubtedly take a long time to complete the verification thoroughly.

For example, the three-dimensional case of the 'Poincaré conjecture' was proved by mathematician Grigory Perelman around 2003, but it was not until 2006 that the mathematical community finally confirmed that Perelman's proof solved the Poincaré conjecture.

Of course, this also has something to do with Perelman's rejection of almost any award presented to him, and he is deeply hidden.

After all, it is almost impossible for a provers of conjecture to quickly understand this method if they do not promote their own proof methods and processes.

Especially in the field of mathematics.

For a proof paper, it is difficult for other mathematicians to understand the paper thoroughly without the original author to explain it and answer the confusion of other peers.

In addition, the process of acceptance by the mathematical community is generally longer for the major conjectures of the millennial mathematical problem.

After all, the stakes are immense whether it's right or not.

Just like the Riemann conjecture, since it was proposed by the mathematician Bonhard Riemann in 1859, there have been more than thousands of mathematical propositions in the literature of mathematics, which presuppose the establishment of the Riemann conjecture (or its generalized form).

If the Riemann conjecture is rejected, not to mention the collapse of the edifice of mathematics, it will at least involve the huge field of the Riemann conjecture, from number theory, to functions, to analysis, to geometry...... It's fair to say that almost all of mathematics is going to change dramatically.

Once the Riemann conjecture is proved, the thousands of mathematical propositions or conjectures built around it will be promoted to theorems. The history of mathematics of mankind will usher in an incomparably vigorous development.

In fact, the speed of reviewing the proof of a problem or conjecture depends to a large extent on the popularity of the problem or conjecture and the extent to which the research work on the problem or conjecture has progressed.

In addition, there are methods, theories, and tools used to prove this problem or conjecture.

For example, when he proved the weak weyl_berry conjecture, he only made some innovations in the field of the symmetric structure theory of Banach space and spectral asymptotic on the connected region with fractal boundaries, and used the fractal drum to make an opening for the associated counting functions.

So the process of proving the weak weyl_berry conjecture was quickly accepted by Professor Gowers.

In proving the weyl_berry conjecture, he made a breakthrough in his previous method, using the Dirichlet field to limit the fractal dimension of the Ω and the spectrum of the fractal measure, supplemented by the expansion of the domain and the transformation of the function into subgroups and the establishment of connections between the intermediate domain and the collection.

The acceptance of this approach by the mathematical community has been much slower.

Even if his paper was finally reviewed by six top elders, four of whom were Fields Medal winners, and he was on hand to answer questions throughout the process, it still took a long time to be confirmed.

To this day, there are still not many people in the entire mathematical community who can fully understand the process of proving weyl_berry conjectures.

Even if he later extended this method to the astronomical community, increasing its importance.

As for the process of proving Hodge's conjecture in his hands now, that's not to mention.

God knows how long it will take for the mathematical community to fully accept this paper.

A year? Three years? Five years? Or longer?

During this long period of time, Xu Chuan did not want to see his thesis shelved.

He hopes that more mathematicians and even physicists will participate in it, expand and apply it to more and wider fields.

........