MTL - The Science Fiction World of Xueba-Chapter 422 Ask and solve problems

Pang Xuelin smiled and said, "This doll is actually ..."

"Cough ..." Zuo Yiqiu on the side suddenly coughed twice.

"In fact, I bought it myself. I saw that the matryoshka you gave me was pretty, so I bought another ..."

After that, Pang Xuelin blinked at Zuo Yiqiu.

Zuo Yiqiu's fair face quickly became flushed, and even her ears became red.

Ai Ai didn't respond at all, wondering, "But didn't you go out yesterday?"

"I asked Professor Mochizuki Shinichi to buy it for me."

"Oh!"

Ai Aiyinyin felt that her master hadn't told the truth to herself, but she was not Pang Xuelin and she didn't ask much for a while.

She did not doubt Zuo Yiqiu. Zuo Yiqiu's attitude towards Pang Xuelin was always a public affairs office, and basically he did not show anything special.

More importantly, she could feel that Zuo Yiqiu had a bit of contempt for her master's chaotic emotional life.

Bang bang ——

At this moment, the door knocked again, and the three looked around, and saw a photographer and a female reporter in charge of the interview standing at the door.

"Professor Pang, I'm CCTV reporter Liu Xiaolin. Is it convenient to interview now?"

Pang Xuelin was a little hesitant, and then she remembered that she won the award this time, and I am afraid it will make a lot of waves in the domestic media.

He smiled and said, "Okay, come in."

At this time, Zuo Yiqiu said, "Well, Ai Ai, Professor Pang is still busy. Let's not disturb him."

"Well, Master, let's go back to the room first."

Pang Xuelin smiled and said, "Go and go."

Ai Ai took the lead, and Zuo Yiqiu followed behind her.

When he went out, Zuo Yiqiu couldn't help but look back at Pang Xuelin. The two eyes met, and when he saw Pang Xuelin, he just looked at her with a smile. Zuo Yiqiu quickly withdrew his gaze, but his heart could not help but throbbing violently.

Pang Xuelin smiled and shook her head. I really didn't understand. One of the two girls was a bit dull, the other looked a little careful, but their skin was too thin, and I didn't know how the two of them would become girlfriends.

Later, Pang Xuelin turned his attention to Liu Xiaolin: "Reporter Liu, it is 12:30 at noon. There will be a briefing in the afternoon. It will take about an hour to prepare, so I can only give you 10 minutes."

Liu Xiaolin said with a smile: "Professor Pang, ten minutes is fine. There is a five-hour time difference between St. Petersburg and China. It is 5:30 pm capital time. Our interview will be broadcast directly in the news web tonight. "

"Okay, let's start now."

Pang Xuelin sat down on the sofa in the living room, and the photographer pointed his camera at Pang Xuelin, and the interview officially started.

"Professor Pang, can you tell us how you felt when you won?"

Pang Xuelin laughed and said, "It is not an accident to be awarded. The only thing that surprised me a bit is that the International Mathematical Federation will come up with a Fields Special Award, which makes me feel honored and happy."

Liu Xiaolin said, "Professor Pang, you are the first scientist to win both the Fields Prize and the Nobel Prize, and have won a great honor for our country. How do you balance the research work of mathematics and other disciplines? "

Pang Xuelin smiled and said, "There is a saying in the academic world that mathematics is the queen of science and the servant of science. Many physical, chemical and even biological problems can be solved mathematically. You have to obey the guidance of mathematics. This is why mathematics is the queen of science. But at the same time, mathematics serves the natural sciences and is a tool we use to understand the objective world, so it is a servant of science. To me Mathematics is fundamental and interest, so there is no such thing as a balance, at least for me, when studying problems in other fields, it will not affect my research on mathematics ... "

"Professor Pang, after your news of winning the award returned to China, it caused a huge sensation on the Internet. Many young college students have taken you as an example and used you as their spiritual mentor. Do you have anything to say to them? ? "

Pang Xuelin groaned for a moment and said with a smile: "Thank you very much for your support. The future development of our country and the improvement of people's living standards depend on the improvement of productivity. Science and technology are precisely the primary productive forces. I hope we can have more and more The more young people enter the field of scientific research, and the more people can deal with the chores of life, and at the same time have some time to look up at the sky above me ... "

...

Next, Pang Xuelin answered Liu Xiaolin's several questions before ending this short interview.

After Liu Xiaolin left, Pang Xuelin didn't directly start preparing the report for the afternoon, but took out his mobile phone and watched the response on the Internet.

Naturally, the reports of the major media need not be said, but basically they are all cheers.

People's Daily: "Today, the opening ceremony of the 29th International Congress of Mathematicians was successfully held in St. Petersburg, Russia. Professor Pang Xuelin won the Fields Special Award and became the first Chinese mathematician to win this award in history."

Sina.com: "The Fields Special Award is tailor-made for Professor Pang Xuelin, and its status is much higher than the ordinary Fields Award. Professor Pang has won the throne of the first person in contemporary mathematics."

Observer: "The Fields Special Award? Pang Xuelin Award? Anyway, Professor Pang Xuelin has put his name in the history of mathematics."

Tencent News: "It is both a winner and an awarder. The International Mathematical Federation has tailor-made a new award for Professor Pang Xuelin, and named after Professor Pang. Professor Pang Xuelin has been respected and loved by mathematicians worldwide."

...

Compared to major news media, reports on social platforms are much more exaggerated.

Hot search on Weibo, from the first to the fifth, was once again dominated by Pang Xuelin, the Fields Special Award, the Pang Xuelin Award, and the award ceremony almost rolled over, Pang Xuelin won the Fields Special Award, and Robert Langlan I highly appreciate Professor Pang's achievements and so on.

At the same time, Pang Xuelin's personal Weibo has long been occupied by various sand sculpture netizens.

Among them, the most liked one comment is like this.

"Professor Pang can take photos of the Fields Medal and let us see?"

"Haha, I watched the live broadcast of the awards ceremony all the way. I almost thought that Professor Pang's Fields Award was cool. Fortunately, I continued to watch, Professor Pang ..."

"Professor Pang's speech was very interesting, but unfortunately I didn't understand a word about the academic part."

"I don't know if everyone found it. At the beginning, Robert Langlands announced that there was no Professor Pang Xuelin's name in the award list. The entire conference hall was almost blown up. From this, you can imagine the recognition that Professor Pang Xuelin has gained in the international mathematical community How tall it is. "

...

Pang Xuelin probably flipped through the comments on Weibo, thought about it, then got up and found his own Fields special prize-quality medal, took a picture of each side, and posted it on Weibo.

Then, Pang Xuelin stopped paying attention to the personal Weibo that he had been boiling, and began to prepare for the report meeting one hour later.

If it were an ordinary person, such an ultra-high-level mathematical report would be prepared in as short as ten days and a half months, and as long as several months.

However, after the system reform, Pang Xuelin has greatly improved both his memory, thinking ability and neural response speed.

Therefore, he doesn't need to make such detailed preparations, he just needs to make an outline of what he wants to talk about.

An hour later, at 1.40 pm, Pang Xuelin came out of the room and went to the conference venue.

When Pang Xuelin arrived, the entire lecture hall was crowded with mathematicians from all over the world.

In the warm applause at the scene, Pang Xuelin stepped onto the stage, and everyone focused on him.

Looking at the audience, Pang Xuelin said, "Hello everyone! One hundred and two years ago, German mathematician David Hilbert gave a famous speech at the International Congress of Mathematicians in Paris. The 23 questions asked by Hilbert in the book guide the development of mathematics throughout the twentieth century. Some problems have not yet been solved, such as the famous Riemann conjecture. These have become the focus of our exhaustion. History teaches us that science The development of China has continuity, and each era has its own problems. These issues will provide a new direction for the latecomers. After more than a hundred years have passed, I think it is time for some of the problems we face, It ’s time for a formal review. The end of a great era has not only prompted us to look back, but also to adapt our minds to the unknown future. "

"In mathematics, asking questions is often more important than solving them. We are now facing the question, what exactly is the subject of mathematics as the source of the problem? Among those branches of mathematics, those oldest questions, Definitely originated from experience, put forward by analysis and analysis of external phenomena, the integer algorithm was discovered in this way in the early stage of human civilization. Just as today's children learn and operate through empirical methods, these rules are the same for the original geometry This is also the case with problems such as the double cube problem that has been passed down since ancient times, the problem of turning circles into squares, etc. There are also solutions to numerical equations, curve theory calculus, Fourier series, and those original problems in Wei ’s theory, Not to mention, there are a lot of problems in chemistry, physics, astronomy, biology, etc. "

"However, with the further development and refinement of the branch of mathematics. We have begun to approach logical combination, generalization and specialization, and cleverly analyze and synthesize concepts to raise fruitful questions. This has resulted in prime numbers and polynomials. Problems such as the efficient solution of equations, the solution of discrete logarithms, the existence of one-way functions, etc. "

"As for what general requirements should be put forward for the answer to a mathematical problem, I think that we must first be able to prove the correctness of the problem by reasoning in finite steps based on finite premises, which are included in the problem's In the statement, and must have a precise definition of each problem. The requirement of logical deduction by means of finite reasoning is, in short, the requirement of the rigor of the proof process, which is already required in mathematics like The motto has become well-known. On the other hand, only if the requirements are met, the content of the problem and its rich meaning can be fully reflected. A new problem, especially when it comes from the world of external experience, is like a young plant The saplings only need to be carefully transplanted to the existing old stems in accordance with strict horticultural rules, and it will thrive and bloom. "

"So today I will use my shallow knowledge to talk about some of the problems we are facing in the development of our mathematics."

Pang Xuelin's voice dropped, and a buzzing sound could not help but sound at the scene.

Almost everyone looked at Pang Xuelin in shock.

No one expected that Pang Xuelin made such a speech at this report.

Does he follow the example of David Hilbert more than a hundred years ago and point the way for the future development of mathematics?

There was a buzzing noise at the scene.

Everyone's face was excited.

No one thinks Pang Xuelin does not have this qualification.

In fact, although mathematics has evolved to this day, the branches are being refined step by step.

But almost all progress in the field of mathematics is accompanied by the raising and solving of problems.

From David Hilbert's twenty-three questions to Hilbert more than a hundred years ago, to the Langlands program proposed by Robert Longlands more than sixty years ago, and to the United States Seven Millennial Conjectures from the Ray Institute of Mathematics.

Each time the solution of a problem points the direction for the development of mathematics, it provides a new impetus.

Especially in recent years, with the emergence and rapid development of Ponzi geometry theory, BSD conjecture, ABC conjecture, Polignac conjecture, Hodge conjecture, etc. have been solved one after another. The mathematical community needs a leading figure to stand out for the future. Development points the way.

As the creator of Ponzi's geometric theory, Pang Xuelin is undoubtedly a suitable candidate.

Off stage.

Deligne said to Faltins sitting next to him: "Faltings, I have a hunch."

"What hunch?"

"This young man may far exceed my teacher in the future."

Faltings couldn't help but be taken aback.

Although Pang Xuelin is highly appraised by the current mathematics community, he still basically treats him equal to Grothendieck of the last century.

Even in the eyes of Faltings, Pang Xuelin was a younger version of Grothendieck.

"Pierre, why do you say that?"

Faltins curious.

Deligne turned to look at Faltins and smiled, "I saw passion and ambition in his eyes. He is only twenty-five years old now, and at least twenty years of its peak, you can imagine, twenty How much can he accomplish this year? Even if he completely unified the two basic disciplines of algebra and geometry, it didn't surprise me. "

Pang Xuelin ignored the noise of the audience and smiled slightly, saying: "I think in the next 100 years, the following problems will be some problems that our mathematics community needs to solve urgently. First, the main conjecture of Iwasawa theory."

"In number theory, Iwasawa theory is the Galois model theory of ideal groups. It is a set of mathematical properties developed by Japanese mathematician Kenichi Iwasawa in the late 1950s to study the arithmetic properties of the Zp extension of a number field In theory, the most common Zp expansion is the so-called bipartite Zp expansion. This type of field was first studied by the German mathematician Kummer to prove Fermat's Theorem. In fact, if the integer ring Z [C?] Is the only factorization Ring, then you will not encounter so many difficulties in your journey to prove Fermat's Last Theorem.

Cyclic Zp expansion is the expansion of the following cycloid domain:

K = Q (CP) C ... CKn = Q (C; +1) ?? CXoo = Q (CP ~),

Among them, the Galois group Gn of KJK is a cyclic group for any aZ / pnZ, aa (CP) = CpV. According to Galois theory, the Galois group G of K / K is the projective limit of G ?, that is, p enters the integer ring Zp. .

...

Iwasawa's main conjecture (or main conjecture of Iwasawa theory) is: ch (A) = ch (s / C). It can be seen that A illustrates the ideal class of a number field, which is a pure algebraic object. The circle unit is essentially an analytical object. In fact, let ((P, s) = C (s). (1-p ~ s) = ∑1 / n ^ s, this function is called the V-to-C function, which is a continuous function and its The value at a negative integer can be represented by a first polynomial interpolation.

The P-in function is an example of the p-in function, which reflects the analytical nature of the corresponding number field.

The work of Coates-Wiles and Coleman in the apparent reciprocity law shows that the above polynomial and ch (f / C) differ only by a fixed polynomial. So we know that the main conjecture is a conjecture about the profound connection between the algebraic and analytic properties of the bisecting domain.

Iwasawa theory has been an important tool for number theory research since its inception. In 1972, Mazur established the Iwasawa theory of elliptic curves and proposed the principal conjecture on the imaginary quadratic domain. Later, many other forms of principal conjectures were proposed, including the principal conjecture on motivation. The study of Iwasawa theory on p-in Galois representation is very important for p-in BSD conjecture, Serre conjecture, etc.

In 1983, Mazur and Wiles used deep algebraic geometric methods to prove the Iwasawa master conjecture. Using Kollivagin's Euler system, Rubin proved the principal conjecture on the imaginary quadratic domain, and gave a new proof of the principal conjecture on the bifurcated domain.

But other forms of master conjecture are still hot topics in the study of number theory and arithmetic algebraic geometry. "

...

"Second question, HOPF guesses."

"One of the core problems of global differential geometry is studying the relationship between local invariants and global invariants, and studying the relationship between curvature and topology.

Let's examine the surface S. There is a metric on it, which is also the Gauss curvature K. If the surface is compact and boundless, the Gauss curvature K can be integrated over the entire surface. A surface does not necessarily have only one metric. It can have another metric. After the metric is changed, the corresponding Gauss curvature K also changes, but the integral value has nothing to do with the metric of the surface, but only the Euler's invariant number x of the surface. (* 5) Related.

This is the profound meaning revealed by the Gauss-formula.

For the high-dimensional Riemannian manifold M, the Gauss curvature can be generalized as the section curvature. It is determined by the Riemann curvature tensor. The integrand is a very complex algebraic expression composed of the curvature tensor. It is called Gauss-integrand. The integral over the entire manifold should be determined by the Euler explicit number of this manifold. Its internal proof proved to be obtained by Chen Sheng, and it was later called the Gauss_-Chen formula.

For a compact and boundless even-dimensional manifold M2 ", if it contains a Riemannian metric with non-orthogonal curvature, then its Euler explicit number satisfies

(-l) nX (M2n) 0 (1) (When the cross-sectional curvature is negative, the above formula is a strict inequality).

This is the famous Hopf conjecture.

So far, Hopf's conjecture has only been verified under some additional conditions, such as the work of the curvature of the section between two negative constants: Bourguignon-KarcherPl, Donnelly-Xavier, and Jost-Xin.

Borel confirmed the conjecture for non-compact rank 1 symmetric spaces.

If the manifold has a KShler metric, in the case of negative cross-section curvature, the conjecture has been confirmed by Gromov, and in the case of non-positive cross-section curvature, it has been confirmed by Jost-Zuc and Cao-Xavier. "

...

"The third question, Kaplanski's sixth conjecture."

"Kaplanski's sixth conjecture is one of the ten conjectures about Hopf algebra put forward by Kaplansky in 1975, and it is also one of the frontier problems in the research of Hopf algebra and even algebra. Algebra originated in the 1940s, and is mainly an algebraic system established by Hopf's axiomatic study of the topological properties of Lie groups.

In the 1960s, Hochschild-Mostow developed and enriched Hopf's theory of algebraic systems in the application of Lie groups and subsequent research, and laid the basic framework of Hopf's algebra theory.

In the 1980s, with the rise of quantum group theory established by mathematicians such as Drinfeld and Jimbo, people discovered that quantum groups are a special type of Hopf algebra. Quantum group theory is closely related to many other mathematical fields, such as low-dimensional topology, representation theory, and non-commutative geometry and accurate solvable model theory of statistical mechanics, two-dimensional conformal field theory, and angular momentum quantum theory.

The rise of quantum group theory has also promoted the rapid development of Hopf algebra theory. Many wonderful research results have been obtained around Kaplansky's ten conjectures, leading to the solution or partial solution of some of them.

Kaplanski's sixth conjecture Let H be a finite-dimensional semi-simple Hopf algebra on the algebraic closed field, then any irreducible dimension of H divides the dimension of H by.

This conjecture is closely related to the classification of finite-dimensional semi-simple Hopf algebras ~ www.novelbuddy.com ~ which has attracted the interest of many algebraic mathematicians.

In 1993, Zhu used the feature theory to study Kaplansky's sixth and eighth conjectures, and obtained some results.

He proved that if char⑷ = 0, H is semi-single and R (H) is in the center of H's dual algebra, where R (H) is the sub-algebra of JI * formed by H's irreducible feature, then the card Plan's sixth conjecture holds.

In 1996, Nichols and Richmond analyzed the ring structure of the Grothendieck group of H to prove that if H is more than semisimple and has a 2-dimensional single comodule, then H is even-dimensional.

In 1998, Etingof and his research on the structure and promotion of quasi-triangular semi-simple and semi-simple Hopf algebras proved that W: if Ug is a semi-simple and semi-simple Hopf algebra, and D {H) is H's Drinfelddouble, then D ( The dimension of the irreducible representation of H) divides the dimension of H by one.

From this they proved that if H is a quasi-triangular semi-simple co-simple Hopf algebra, then the dimension of the irreducible representation of H is divisible. "

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