Chapter 186 proves Hodge’s conjecture!
After asking for leave from Deligne, Xu Chuan got up and walked out of the dormitory.
He still has a lot of work to do before officially entering the unknown territory of Hodge's conjecture. Whether it's in life or mathematics.
Solving the Hodge conjecture is like the first time humans sailed on the vast ocean. No one knows whether there are other lands in the unknown ocean, and no one knows whether they can successfully reach another coastline.
The only thing he owned was a newly built boat.
And Xu Chuan didn't know whether this small boat would be overturned by wind and waves after entering the unknown ocean, whether it would sink to the bottom of the sea, or whether it would hit the rocks and become stuck and unable to move.
But despite this, he still had to try.
Because even if it only sails ten meters, it is a great breakthrough.
After purchasing a batch of daily necessities in the store, Xu Chuan borrowed a batch of manuscripts and information about Hodge's conjecture from the Flint Library.
Some of them he had read before, and some he had not read yet.
These are precious knowledge left by 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 have not yet been formed.
These things, whether he has seen them or not, are very useful for him to charge towards Hodge's conjecture.
But when borrowing these things, he ran into a lot of trouble.
The Flintstone Library is managed by an old man who looks slovenly. This old man with messy hair like a bird's nest is a top expert in the preservation of paper materials, but he is also extremely stubborn.
And this stubborn old man has always been unwilling to lend out so many documents, thinking that he is likely to damage or lose these precious manuscripts.
In order to obtain this batch of materials, Xu Chuan spent a whole day in the Flintstone Library, and his final effort was just 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 is not a very reliable path.
Although it is very quiet here, people come and go every day.
He had no choice but to find David Hugh, the dean of the Princeton School of Mathematics, and made a series of guarantees, learned some methods of preserving paper materials, and even signed a guarantee letter before he reluctantly got the other party to agree. .
With a lot of information, Xu Chuan returned to the dormitory.
In fact, there is no need for the bad old man from Germany to remind him, he will also take good care of these things.
But now, in addition to good preservation, the greater value of these materials is to play their role in the Hodge conjecture.
The mathematicians who originally 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 spread and used, it has no value to the knowledge.
After handling 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 Rockefeller Residential College began to glow with a hint of golden color. 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 covered with sycamore fruits.
The early morning sunrise shines brightly in the dark blue clouds. The golden and dark green leaves outside the window are intertwined, and heavy plane fruits are embedded in them.
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 sailing as he expected, he was always full of confidence in the final result.
Two months later, in the unknown ocean that Hodge guessed, he finally found a coastline that appeared in front of him.
That's the New World!
Looking at the scenery outside the window, Xu Chuan turned back to the table with a smile.
Although Hodge's conjecture has not yet been perfectly solved, he has already seen the horizon where the coasts intersect and the new continent towering in the sky.
All that's left is to row your boat across.
Picking up the ballpoint pen on the table, Xu Chuan picked up the pen where he had not finished writing before and continued:
".Suppose V is an algebraic variety in a complex projective space, and Vˊ is a set of regular points of V. The L2-deRham cohomology group on Vˊ relative to the Fubini-Study metric is isomorphic to the cross-cohomology group of V .”
“If Y is a closed subalgebraic variety of Times TatetwistQ`(nj)
This mapping is the same as the restriction mapping: H2(nj)(Xkk, Q`)(nj)→H2(nj)(Y, Q`)(nj)”
"."
"According to Poincare's duality theorem: Hom(H2(nj)(Xkk, Q`)(nj), Q`)=H2j(Xkk, Q`)(j)"
Time passed by his writing bit by bit, and Xu Chuan devoted himself wholeheartedly to the final breakthrough.
Finally, the pen in his hand suddenly changed.
".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(Xkk, Q`)(j) generated by the cohomology class of the closed subalgebraic variety defined on k whose codimension of X is j"
"When i≤n/2, the quadratic form x→(1)iLr2i(x.x) on Ai(X)∩ker(Ln2i+1) is positive definite."
"From this, it can be concluded that on non-singular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed-chain classes."
"That is, the Hodge conjecture is established!"
The ballpoint pen in his hand marked the last dot on the white manuscript paper. Xu Chuan breathed a long sigh of relief, 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, nor satisfaction and achievement.
But with some incredible 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 perfection of algebraic varieties and group mapping tools, and finally Solution to the Hodge Conjecture.
On this road, he has experienced too much.
After staring at the ceiling for a long time, Xu Chuan finally came 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 to make sure that it was really his own work, a bright smile finally appeared on his face, as bright as the sunshine coming through the window.
If there were no accidents, he succeeded.
Successfully solved the Hodge conjecture, a century-old problem.
This is the most important breakthrough in issues related to the Hodge conjecture since the mathematician Lefshetz proved the Hodge conjecture of the (1,1) class in 1924.
Although he doesn't know yet whether it will stand the test of other mathematicians and time.
But in any case, he once again took a big step forward in mathematics. After completing the paper that proved Hodge's conjecture, Xu Chuan spent some time going over the things on the manuscript paper again and perfected some other details.
After processing these, he began to organize them into a notebook.
Then prepare to make it public.
For the proof of any mathematical conjecture, the prover is not qualified to evaluate whether it is correct.
Only full disclosure, peer review and the test of time can we determine whether it has truly been successful.
After spending a whole week, Xu Chuan finally entered all the nearly 100 pages of manuscript paper into the computer.
More than one-third of the hundreds of pages of proofs are devoted to explanations and demonstrations of algebraic varieties and group mapping tools that solve Hodge's conjecture, and another one-third is devoted to Hodge's conjecture. A theoretical framework built with algebraic variety and group mapping tools.
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 and publish them as independent papers.
Just like Peter Schulz’s ‘p-adic perfect space theory’.
If these things are finally accepted by the mathematical community, they will be enough for him to win a Fields Medal.
This is not the cheapness of the Fields Medal, but the importance of mathematical tools to mathematics.
A great mathematical tool that can solve more than just one problem.
Just like an ax, it can not only be used to cut down trees, but can also be used as a woodworking tool, process items, and can also be used as a weapon for fighting.
In the same way, the algebraic variety and group mapping tools he constructed are not limited to Hodge conjecture.
It can be used to try many algebraic varieties, differential forms, 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 of 'the Hodge theory of algebraic surfaces should determine whether the Chow group of zero cycles is finite-dimensional', and some motivations for finite coefficients. The homology group isomorphism is mapped to etale cohomology problem guessing and so on.
These conjectures and problems supported each other, and mathematicians continued to make progress on one or the other, trying to show that they led to huge advances in number theory, algebra, and algebraic geometry.
Algebraic variety and group mapping tools can solve Hodge's conjecture, so it cannot be said that it can fully adapt to the same type of conjecture, but it can at least play a partial role.
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 the establishment of a basic connection between the three disciplines of algebraic geometry, analysis and topology.
To solve this problem, the prover required has a deep understanding of mathematics in these three fields.
For most mathematicians, it is quite difficult to conduct in-depth research in any one of the three major fields of algebraic geometry, analysis, and topology, let alone be proficient in all three major fields.
For Xu Chuan, analysis and topology were the areas of mathematics he was proficient in in his previous life, but algebraic geometry was not within the scope of his research.
But he followed Deligne to study mathematics in depth throughout his life. With such a mentor, his progress in algebraic geometry was 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 his two mentors, 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 for computers to take over, there are still a large number of mathematicians and physicists on it.
By posting your unpublished papers, you can not only take advantage of them in advance to prevent plagiarism, but also expand the influence of your papers in advance.
For proof papers on issues such as Hodge's conjecture, it will undoubtedly take a long time to completely complete the verification.
For example, the three-dimensional case of the Poincaré Conjecture was previously 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 is also related to the fact that Perelman refused almost any award awarded to him and lived in seclusion.
After all, if a prover of a conjecture does not promote his own proof method and process, it will be 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 it and answer the confusion of other colleagues, it will be difficult for other mathematicians to fully understand the paper.
In addition, for major conjectures such as the Millennium Mathematics Problem, the mathematical community generally takes a long time to accept them.
After all, the relationship between whether it is correct or not is extremely important.
Just like the Riemann Hypothesis, since it was proposed by the mathematician Bornhard Riemann in 1859, there have been more than thousands of mathematical propositions in the literature of the mathematical community to the establishment of the Riemann Hypothesis (or its generalized form). as a premise.
If once the Riemann Hypothesis is proven false, let alone the collapse of the mathematics building, 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 have 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 human mathematics will usher in an extremely vigorous development.
In fact, the review speed of the proof of a problem or conjecture depends to a large extent on the popularity of the problem or conjecture, and how far the research work on this problem or conjecture has progressed in the mathematical community.
In addition, there are methods, theories and tools used to prove this problem or conjecture.
For example, when he previously proved the weak Weyl_Berry conjecture, he only made some innovations in the two fields of Banach space symmetry structure theory and spectral asymptotics on connected regions with fractal boundaries, using the counting functions associated with fractal drum pairs. Made an opening.
Therefore, the proof process of the weak Weyl_Berry conjecture was quickly accepted by Professor Gowers.
When proving the Weyl_Berry conjecture process, he made a breakthrough in the previous method. He used the Dirichlet domain to limit the fractal dimension of Ω and the spectrum of the fractal measure, and then supplemented it with the expansion of the domain and the conversion of the function into sub-dimensions. groups and establish connections with intermediate fields and collections.
The mathematical community has been much slower to accept this approach.
Even though his paper was eventually reviewed by six top experts, four of whom were Fields Medal winners, and he was on hand to answer questions throughout the entire 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 proof process of the Weyl_Berry conjecture.
Even though he later extended this method to the astronomical community, raising its importance.
As for the process of proving the Hodge conjecture in his hands now, there is no need to mention it.
God knows how long it will take for the mathematical community to fully accept this paper.
One year? Three years? Five years? Or longer?
During this long period of time, Xu Chuan was not willing to see his thesis shelved.
He hopes that more mathematicians and even physicists will participate in expanding and applying it to more and wider fields.
(End of chapter)