The Science Fiction World of Xueba

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

"Ahem..." Zuo Yiqiu who was beside him suddenly coughed dryly twice.

"Actually, I bought it myself. I thought the doll you gave me was pretty, so I bought another one..."

After finishing speaking, Pang Xuelin blinked at Zuo Yiqiu.

A blush quickly appeared on Zuo Yiqiu's fair face, even the roots of his ears were red.

Ai Ai didn't react at all, and asked suspiciously: "But didn't you go out yesterday?"

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

"oh!"

Ai Ai faintly felt that her master was not telling the truth to herself, but she was not someone like Pang Xuelin, so it was not easy to ask more questions for a while.

She didn't doubt Zuo Yiqiu at all, Zuo Yiqiu's attitude towards Pang Xuelin was always business-like, and basically he didn't show any special abnormalities.

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

bang bang bang—

At this time, there was another knock on the door, and the three of them followed the prestige and saw a photojournalist and a female reporter in charge of the interview standing at the door.

"Professor Pang, I'm CCTV reporter Liu Xiaolin, is it convenient for me to accept an interview now?"

Pang Xuelin was taken aback for a moment, and then he remembered that this time he won the award, I am afraid it will cause quite a stir in the domestic media.

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

At this time, Zuo Yiqiu said: "Okay, Ai Ai, Professor Pang still has something to do, so let's not bother him."

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

Pang Xuelin said with a smile: "Go, go."

Ai Ai went out first, and Zuo Yiqiu followed behind her.

When it was time to go out, Zuo Yiqiu couldn't help but look back at Pang Xuelin, their eyes met, seeing that Pang Xuelin just looked at her with a slight smile. Zuo Yiqiu quickly retracted his gaze,

The heart couldn't help beating violently.

Pang Xuelin shook his head with a smile, he really couldn't understand, one of these two girls was a bit cute, the other looked a little scheming, but his skin was extremely thin, and he didn't know how the two of them became girlfriends.

Afterwards, Pang Xuelin turned his gaze to Liu Xiaolin and said, "Reporter Liu, it's 12:30 noon, and there is a report meeting in the afternoon, and 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 enough. The time difference between St. Petersburg and China is five hours, and it is now 5:30 p.m. capital time. Our interview will be broadcast directly on tonight's news broadcast later. "

"Okay, let's start now."

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

"Professor Pang, can you tell me how you felt when you won the award?"

Pang Xuelin laughed and said, "It's not a surprise to win the award. The only thing that surprises me is that the International Mathematical Union will come up with a Fields Special Award. This makes me feel very honored and very happy."

Liu Xiaolin said: "Professor Pang, you are the first scientist in history to win the Fields Medal and the Nobel Prize at the same time, and you have won great honors 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 academia that mathematics is the queen of science and the servant of science. Many problems in physics, chemistry and even biology can be solved by mathematical methods. You have to follow the guidance of mathematics. This is why mathematics is the queen of science. But at the same time, mathematics serves natural science and is a tool we use to understand the objective world, so it is a servant of science. To me Mathematics is fundamental to me, and it is also my interest, so there is no such thing as a balance, at least for me, when I study problems in other fields, it will not affect my research on mathematics..."

"Professor Pang, after the news of your award came back to China, it caused a huge sensation on the Internet. Many young college students followed your example and regarded you as their spiritual mentor. Do you have anything to say to them?" ?”

Pang Xuelin pondered 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 all rely on the improvement of productivity. Science and technology are precisely the primary productivity. I hope we can have more and more The more young people enter the field of scientific research, and at the same time, more people are able to deal with the trivialities of life, and at the same time, have some time to look up at the sky above my head..."

...

Next, Pang Xuelin answered several questions from Liu Xiaolin before concluding this brief interview.

After Liu Xiaolin and the others left, Pang Xuelin did not directly start preparing for the afternoon report, but took out his mobile phone and checked the reactions on the Internet.

Needless to say, the reports of the major media are basically full of cheers.

People's Daily Online: "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 in history to win this award."

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

Observer: "Fields Special Award? Pang Xuelin Award? No matter what, Professor Pang Xuelin has already recorded his name in the annals of mathematics."

Tencent News: "As both a winner and an award presenter, the International Mathematical Union tailor-made a new award for Professor Pang Xuelin, and named it after Professor Pang. Professor Pang Xuelin has won the respect and love of mathematicians around the world."

...

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

Hot searches on Weibo, from the first to the fifth article, were once again occupied by Pang Xuelin alone, namely Fields Special Award, Pang Xuelin Award, the award ceremony almost overturned, Pang Xuelin won the Fields Special Award, Robert Langland Hereby speak highly of 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 comment with the highest praise is like this.

"Professor Pang, can you post a photo of the Fields Medal for us to see?"

"Haha, after watching the live broadcast of the awards ceremony, I almost thought that Professor Pang's Fields Award was lost. Fortunately, I insisted on watching, Professor Pang is amazing..."

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

"I don't know if you have noticed. At first, Professor Pang Xuelin's name was not included in the list of winners announced by Robert Langlands, and the entire conference hall was almost blown up. From this, one can imagine the recognition of Professor Pang Xuelin in the international mathematics community." How high is it?"

...

Pang Xuelin probably flipped through the comments on Weibo, thought for a while, then got up and found his Fields Special Prize Medal, took a photo of the front and back, and posted it on Weibo.

Then, Pang Xuelin ignored his personal Weibo, which had been completely boiling, and began to prepare for the report meeting in an hour.

If it were an ordinary person, the preparation time for such a super-high-standard mathematics report meeting could be as short as ten days and a half month, or as long as several months.

It's just that after the system transformation, Pang Xuelin's memory, thinking ability and neural response speed have been greatly improved.

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 in the afternoon, Pang Xuelin came out of the room and went to the report meeting venue.

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

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

Looking at the audience, Pang Xuelin said: "Hi everyone! One hundred and twenty-two years ago, the German mathematician David Hilbert delivered a famous speech at the International Congress of Mathematicians in Paris. Hilbert's twenty-three questions raised 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 devotion. History teaches us that science development, with continuity, and each era has its own problems. These problems will provide a new direction for those who come after. More than a hundred years have passed, and I think it is time for some of the problems we are facing, A formal review is under way. The end of a great era not only prompts us to look back to the past, but also to adapt our thoughts to the unknown future."

"In mathematics, asking questions is often more important than solving them. We are now faced with the question, what is the source of questions in the discipline of mathematics? In those branches of mathematics, the earliest and oldest questions, Affirmation originates from experience and is put forward by the analysis of external phenomena. In this way, the rules of integer arithmetic were discovered in the early days of human civilization. Just as today's children learn and operate through experience, these rules are the same as for the original geometry. The same is true for problems such as the traditional double cube problem, the problem of squaring a circle, etc. The same applies to the solution of numerical equations, the calculus of curve theory, Fourier series and those original problems in Wei's theory, Not to mention, a large number of problems pertaining to chemistry, physics, astronomy, biology, etc.”

"However, with the further development and refinement of the branch of mathematics, we began to come into contact with methods such as logical combination, generalization and specialization, cleverly analyzed and synthesized concepts, and raised fruitful questions. In this way, prime number problems, polynomial problems, etc. Efficient solutions of systems of equations, solutions of discrete logarithms, 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 should firstly have the possibility to prove the correctness of the problem by reasoning in finite steps based on finite premises contained in the problem's statement, and must have an exact definition for each problem. This requirement of logical deduction with the help of limited reasoning, in short, is the requirement for the rigor of the proof process, which has already been used in mathematics as On the other hand, only when such a requirement is fulfilled can the intellectual content of the question and its richness of meaning be fully realized. A new question, especially when it originates from the external world of experience, is like a young plant As long as we carefully transplant it to the existing old trunk according to strict horticultural rules, it will grow vigorously and bloom and bear fruit."

"Therefore, today I will use my shallow knowledge to talk about some problems that we will face in the current development of mathematics."

After Pang Xuelin's words fell, there was a buzzing sound at the scene.

Almost everyone looked at Pang Xuelin in shock.

No one expected that Pang Xuelin would make such a speech at this report meeting.

Is he trying to emulate David Hilbert more than a hundred years ago and point out the direction for the future development of mathematics?

There was a buzzing sound at the scene.

Everyone had excited expressions on their faces.

No one thinks that Pang Xuelin is not qualified.

In fact, although mathematics has developed to this day, each branch is being refined step by step.

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

From Hilbert's 23 Questions proposed by David Hilbert more than a hundred years ago, to the Langlands Program proposed by Robert Langlands more than 60 years ago, to the The seven conjectures of the millennium proposed by the Ray Institute of Mathematics.

Every time the problem is solved, it points out the direction for the development of mathematics and 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 mathematics community needs a leading figure to stand up for the future. Development points the way.

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

offstage.

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

"What premonition?"

"This young man may far surpass my teacher in future achievements,"

Faltings couldn't help being taken aback.

Although the current mathematics community speaks highly of Pang Xuelin, they basically regard him as the same as Grothendieck in the last century.

Even in the eyes of Faltings, Pang Xuelin is a young version of Grothendieck.

"Pierre, why do you say that?"

Faltings asked curiously.

Deligne turned his head and glanced at Faltings, and smiled: "I saw enthusiasm and ambition in his eyes. He is only twenty-five years old now, and he still has at least twenty years of his peak period. As you can imagine, twenty years How many achievements can he make within a year? Even if he completely unifies the two basic subjects of algebra and geometry, it will not surprise me."

Pang Xuelin ignored the noise from the audience, smiled slightly, and said, "I think that in the next hundred years, the following problems will be urgently needed to be solved by our mathematics community. First, the main conjecture of Yan Ze's theory."

"In number theory, Iwasawa's theory is the Galois module theory of ideal groups. It is a set of arithmetic properties of the Zp expansion of the number field (that is, the finite expansion of Q) developed by the Japanese mathematician Kenji Iwasawa in the late 1950s. theory, the most common Zp expansion is the so-called fractional circle Zp expansion. This type of field was first studied by the German mathematician Kummer to prove Fermat's last theorem. In fact, if the integer ring Z[C?] is the only decomposition ring, then there will not be so many difficulties in the journey to prove Fermat's last theorem.

The expansion of subcircle Zp is the expansion of the following subcircle domain:

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

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

...

Yan Ze's main conjecture (or called the main conjecture, that is, the main conjecture of Yan Ze's theory) is: ch(A)=ch(s/C). It can be seen that what A describes is the ideal group of number fields, and is a pure algebraic object. The circular unit is essentially an analytic object. In fact, let ((P, s)=C(s).(1-p~s)=∑1/n^s, this function is called the V-advance C function, it is a continuous function, and its Values ​​at negative integers can be represented by a prime polynomial interpolation.

P-adic function is an example of p-adic i-function, which embodies the analytic property of the corresponding number field.

Coates-Wiles and Coleman's work on the law of apparent reciprocity 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 cyclic field.

Iwasawa's 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 main conjecture on the imaginary quadratic field. Later, many other forms of the main conjecture were proposed, including the main conjecture on the motive. The study of Iwasawa theory on p-adic Galois representation is very important for p-adic BSD conjecture, Serre conjecture, etc.

In 1983, Mazur and Wiles proved the Iwasawa main conjecture using profound algebraic geometry methods. Using the method of Kolivagin's Euler system, Rubin proved the main conjecture on the imaginary quadratic field, and gave a new proof of the main conjecture on the subcircle field.

Other forms of master conjectures are still hot topics in number theory and arithmetic algebraic geometry. "

...

"Second question, Hopf (HOPF) conjecture."

"One of the core issues in global differential geometry is to study the relationship between local invariants and global invariants, and to study the relationship between curvature and topology.

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

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

For the high-dimensional Riemannian manifold M, the Gauss curvature can be extended to the section curvature, which is determined by the Riemann curvature tensor, and the integrand is a very complex algebraic formula composed of curvature tensor, called the Gauss-integrand, which The integral on the whole manifold should be determined by the Euler characteristic number of this manifold. Its intrinsic proof was obtained by Chen Shengshen, and it was later called the Gauss_-Chen formula.

For a compact and infinite even-dimensional manifold M2", if it accommodates a Riemannian measure of non-positive section curvature, then its Euler characteristic satisfies

(-l)nX(M2n)0(1) (when the section curvature is negative, the above formula is a strict inequality).

This is the famous Hopf conjecture.

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

Borel confirmed the conjecture for noncompact rank-1 symmetric spaces.

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

...

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

"Kaplansky's sixth conjecture is one of the ten conjectures about Hopf algebras proposed by Kaplansky in 1975, and it is also one of the frontier issues in the research of Hopf algebras and algebras. Hopf Algebra originated in the 1940s. It is mainly an algebraic system established by Hopf's axiomatic research on the topological properties of Lie groups.

In the 1960s, Hochschild-Mostow developed and enriched Hopf's theory of this algebraic system in the application and follow-up research of Lie groups, and established the basic framework of Hopf's algebraic theory.

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

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

Kaplansky's sixth conjecture Suppose H is a finite-dimensional semi-simple Hopf algebra on a closed algebraic field, then the dimension of any irreducible representation of H divides the dimension of H.

This conjecture is closely related to the classification of finite-dimensional semisimple Hopf algebras, and has attracted the interest of many algebraists.

In 1993, Zhu studied Kaplansky's sixth and eighth conjectures by using the characteristic mark theory, and obtained some results.

He proved that: if char⑷=0, H is semi-simple and R(H) is in the center of the dual algebra of H, where R(H) is the subalgebra of JI* spanned by the irreducible characters of H, then the card Poulein's sixth conjecture was established.

Nichols and Richmond proved in 1996 by analyzing the ring structure of the Grothendieck group of H that if H is cosemisimple and has a 2-dimensional simple comodule, then H is even-dimensional.

In 1998, Etingof and He proved W when studying the structure and promotion of quasi-triangular semi-simple and semi-simple Hopf algebras: If Ugly is a semi-simple and semi-simple Hopf algebra, D{H) is a Drinfelddouble of H, then D( The dimension of the irreducible representation of H) is divisible by the dimension of H.

From this they prove that: if H is a quasi-triangular semi-simple remainder semi-simple Hopf algebra, then the dimensions of the irreducible representation of H are divisible. "

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