Great Country Academician

After taking a vacation with Deligne, Xu Chuan got up and walked out of the dormitory.

He still has a lot of work to do before officially entering the uncharted territory of Hodge's conjecture. Whether it is in life or in mathematics.

Solving the Hodge conjecture is like sailing the vast ocean for the first time. No one knows whether there are other lands in the unknown ocean, and no one knows whether they can reach another coastline smoothly.

The only thing he owns is a boat that has just been built.

And Xu Chuan doesn't know whether this small boat will be overturned by wind and waves after entering the unknown ocean, whether it will sink to the bottom of the sea, or whether it will hit a rock and get stuck and unable to move.

But even so, he still wanted to try.

Because even sailing only ten meters away is a great breakthrough.

After purchasing a batch of daily necessities in the store, Xu Chuan borrowed a batch of manuscripts and materials about Hodge's conjecture from the Flint Library.

Some of them have been read before, and some of them have not been read yet.

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

These things, whether he has seen them or not, are very useful for him to launch a charge against Hodge's conjecture.

But when borrowing these things, he encountered a lot of trouble.

The man who manages the Flintstone Library is a shaggy-looking old man. This old man with disheveled hair like a bird's nest is a top expert in the field of paper materials preservation, but he is also extremely stubborn.

And this stubborn old man was still reluctant to lend so many documents to the outside world, thinking that he might damage or lose these precious manuscripts.

In order to obtain this batch of materials, Xu Chuan spent a day in the Flint Library, and his final effort was nothing more than getting the other party to agree to put them together and browse them in the library.

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

Although it is very quiet here, there are people coming and going every day.

In the end, he had no choice but to find David Xiu, the dean of the Princeton School of Mathematics, to make a series of guarantees, learn some ways to preserve paper materials, and even sign a letter of guarantee before he reluctantly got the other party to agree. .

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

In fact, he didn't need that bad old man from Germany to remind him, he would take good care of these things.

But now, in addition to being well preserved, the greater value of these materials is to play their role in Hodge's conjecture.

Presumably the mathematicians who created this knowledge must have thought so too.

For a scholar, no one wants to see the knowledge he has created being shelved. If a piece of knowledge cannot be passed on and used, it is of no value to knowledge.

After finishing 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 came, and the sugar maple, sycamore and other trees outside the Rockefeller Residential College began to turn golden. Occasionally, a few fallen leaves slowly fall with the wind.

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

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

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

Autumn is the harvest season.

Although the research on Hodge's conjecture was not as smooth as he expected, he was always full of confidence in the final result.

And two months later, in the unknown ocean that Hodge guessed, he finally found a coastline that appeared in front of him.

That is the New World!

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

Although Hodge's conjecture has not been perfectly resolved, he has already seen the horizon where the coasts intersect, and the new continent towering in the sky.

All that was left was to row his boat across.

Picking up the ballpoint pen on the table, Xu Chuan picked up the pen where he hadn't finished writing before and continued:

". Let V be an algebraic variety in a complex projective space, and V' be a set of regular points of V. The L2-de Rham cohomology group on V' relative to the Fubini-Study metric is isomorphic to the crossed cohomology group of V of."

"If Y is a closed subalgebraic variety of X defined on k with codimension j, we have a standard mapping: Tr : H2(nj)(Yk k, Q`)(n j)→ Q` where (n j) is n j times Tate twist Q`(n j).

This mapping and restriction mapping: H2(nj)(Xk k, Q`)(n j) → H2(nj)(Y, Q`)(n j)"

"."

"According to Poincare's duality theorem: Hom(H2(nj)(Xk k, Q`)(n j), Q`)= H2j (Xk k, Q`)(j)"

Time passed under his pen bit by bit, and Xu Chuan concentrated himself on the final breakthrough.

Finally, the pen in his hand suddenly turned.

".Based on the mapping Tr, restriction mapping and Poincare, the duality theorem is compatible with the action of Gal(k/k), so the action of Gal(k/k) on the cohomology class defined by Y is also trivial. Then Aj (X ) is the Q vector space in H2j (Xk k, Q`)(j) generated by the cohomology class of closed subalgebraic varieties defined on k with codimension j of X"

"When i≤n/2, the quadratic form x→(1)iLr2i(x.x) on Ai(X)∩ker(Ln2i+1) is positive definite."

"Therefore, it can be obtained that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes."

"In other words, the Hodge conjecture is established!"

Xu Chuan let out a long sigh of relief as he tapped the last dot on the white manuscript paper with the ballpoint pen in his hand, threw the ballpoint pen aside, lay back, leaned on the back of the chair and stared at the ceiling in a daze.

When the last character fell on the manuscript paper, what came out of his heart was not excitement, joy, satisfaction or accomplishment.

But with some unbelievable confusion.

It took more than four months, starting from the manuscript left to him by Professor Mirzakhani, to the solution of the problem of "irreducible decomposition of differential algebraic varieties", to the improvement of algebraic varieties and group mapping tools, and finally The solution of the Hodge conjecture.

On this road, he has experienced too much.

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 confirming that this is really the result of his own work, a bright smile finally appeared on his face, as bright as the sunlight coming in through the window.

If nothing unexpected happened, he succeeded.

Successfully solved the century-old problem of Hodge's conjecture.

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

Although he doesn't know yet whether it will stand the test of other mathematicians and time.

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

After completing the thesis proving Hodge's conjecture, Xu Chuan spent some time going over the things on the manuscript paper and perfecting some other details.

After processing these, he began to organize them into notebooks.

Then prepare to make it public.

For the proof of any mathematical conjecture, the prover is not qualified to evaluate whether it is correct or not.

Only full disclosure, peer review and the test of time can we be sure that it has actually been a success.

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

More than one-third of the hundred-page proof is aimed at the explanation and demonstration of the algebraic variety and group mapping tools that solve Hodge's conjecture, and another third is aimed at Hodge's conjecture A theoretical framework built with tools for algebraic varieties and group mappings.

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

For this paper, tools and frameworks are its core foundation.

If he wants, he can separate the tools and theoretical framework separately and publish them as independent papers.

Just like Peter Schultz's "p-adic perfect space theory".

These things, if finally accepted by the mathematics community, are enough for him to win a Fields Medal.

This is not the cheapness of the Fields Medal, but the importance of mathematical tools for mathematics.

A great mathematical tool that can solve more than just one problem.

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

In the same way, the algebraic varieties and group mapping tools he constructed are not limited to the Hodge conjecture.

It can be used to try many algebraic varieties and differential forms, polynomial equations, and even algebraic topology problems.

For example, the "Bloch conjecture", which belongs to the same conjecture family as the Hodge conjecture, "The Hodge theory of algebraic surfaces should determine whether the zero-cycle Chow group is finite-dimensional", and some motivations for finite coefficients Homology group isomorphism maps to etale cohomology problem Guess and so on.

These conjectures and questions supported each other, and mathematicians continually made progress on one or the other, trying to show that they led to great advances in number theory, algebra, and algebraic geometry.

Algebraic varieties and group mapping tools can solve the Hodge conjecture, so it can not be said to be fully adaptable to the same type of conjectures, but at least it can also play a part.

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

What it studies is not the most advanced mathematical knowledge, but to establish a basic connection between the three disciplines of algebraic geometry, analysis and topology.

To solve this problem, the prover needs to have a deep understanding of mathematics in these three fields.

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

For Xu Chuan, analysis and topology were the areas of mathematics he was proficient in in his previous life, only algebraic geometry was not within the scope of his research.

But in this life, he has followed Deligne to study mathematics in depth. With such a mentor, his progress in algebraic geometry is beyond imagination.

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

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

Although today's arxiv preprint website has gradually become a place where computers occupy holes, there are still a large number of mathematicians and physicists on it.

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

For proof papers on such issues as Hodge's conjecture, it will undoubtedly take a long time to complete the verification thoroughly.

For example, the three-dimensional case of the previous "Poincaré Conjecture" was proved by the mathematician Grigory Perelman around 2003, but until 2006, 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 his reclusiveness.

After all, if a conjecture prover does not promote his own proof method and process, it is almost impossible for others to quickly understand this method.

Especially in the field of mathematics.

For a proof paper, if there is no original author to explain and answer the confusion of other colleagues, it is very difficult for other mathematicians to thoroughly understand this paper.

In addition, for the major conjectures of the Millennium Mathematical Problem, the process of acceptance by the mathematics community is generally relatively long.

After all, whether it is correct or not is extremely important.

Just like the Riemann conjecture, since it was proposed by the mathematician Bernhard Riemann in 1859, there have been more than thousands of mathematical propositions in the literature of the mathematical community, and the establishment of the Riemann conjecture (or its extended form) as a premise.

If the Riemann Hypothesis is proven or rejected, not to mention the collapse of the building of mathematics, at least it involves the huge field of Riemann Hypothesis, from number theory, to functions, to analysis, to geometry, it can be said that almost the entire mathematics will undergo major changes .

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

In fact, the review speed of a proof of a problem or conjecture depends largely on the popularity of the problem or conjecture and the progress of the research work on the problem or conjecture in the mathematics community.

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 before, he only made some innovations in the fields of Banach space symmetry structure theory and spectral asymptotics on connected regions with fractal boundaries. made an opening.

So the proof process of the weak Weyl_Berry conjecture was quickly accepted by Professor Gowers.

When proving the process of the Weyl_Berry conjecture, he made a breakthrough in the previous method. He restricted the fractal dimension of Ω and the spectrum of the fractal measure through the Dirichlet field, supplemented by the expansion of the field and the transformation of the function into sub Groups are connected with intermediate domains and collections.

The mathematical community has been much slower to embrace this approach.

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

Today, there are still not many people in the entire mathematical community who can fully understand the proof process of Weyl_Berry's conjecture.

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

As for the proof process of Hodge's conjecture in his hands now, let alone.

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 being shelved.

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