Chapter 142 New Inspiration

After the opening ceremony and award ceremony on the first day, the exchange party was held.

The exchange party was held at Wenjin International Hotel. As the university with the most abundant scientific research funding in China and Shuimu University, which has always been ranked first in the university funding rankings, was also very generous.

During the World Conference of Chinese Mathematicians, it directly booked the right to use Wenjin International Hotel for a week, which can be said to be rich and powerful.

After dinner and a short rest, Xu Chuan went downstairs and followed Rong Zhizhuan to the auditorium of the exchange party.

It was only about 7:30 in the evening, and there was still half an hour before the exchange party started, but the auditorium was already crowded.

As soon as the two entered the venue, they bumped into Zhang Weiping and Hu Xingjian from Sun Moon University.

After greeting each other, Zhang Weiping's eyes moved, and he threw his students to Xu Chuan, and he followed Rong Zhizhuan away.

In the name of letting young people communicate more, the two old men would not join in the fun together.

Looking at the two teachers who disappeared in the vast crowd in the blink of an eye, and then looking at Hu Xingjian standing beside him, Xu Chuan's mouth twitched.

"Master Chuan, thank you for your trouble."

Hu Xingjian smiled shyly and a little embarrassed, probably because he also felt that his mentor was a little unreliable.

"You're welcome, this kind of exchange will be more relaxed and casual." Xu Chuan said with a smile, and walked towards the center of the venue with Hu Xingjian.

"Is this the first time for Brother Hu to attend this kind of exchange party?" Xu Chuan asked as he walked.

Hu Xingjian nodded, looking around and replied: "This is indeed the first time for me to attend such a formal event. Has Master Chuan attended it before?"

"I attended it once when I went to Princeton to give a report before."

Xu Chuan nodded, took a few grapes from the white table in the venue and stuffed them into his mouth, and gave two to Hu Xingjian.

Hu Xingjian took the grapes, his eyes revealing a longing look: "Princeton, the legendary holy land of mathematics, I heard that anyone you bump into on the street there is a top math expert."

Xu Chuan smiled and said, "How can it be so exaggerated, but there are still many math experts in Princeton. Brother Hu, do you want to apply for a PhD at Princeton?"

"Of course I have the idea, after all, I study mathematics, and there is no one who doesn't want to go to Princeton, but I probably don't have the chance."

Hu Xingjian shrugged and said that for students studying mathematics, Princeton is almost the ultimate goal, and no one doesn't want to go.

But they can only recruit a few students in the whole of China in a year, and the competition is so fierce that it is countless times more difficult than getting into Tsinghua and Peking University through the college entrance examination.

If you want to apply for Princeton in China, you have to stand at the top of the pyramid. Although he won the gold medal in the Qiu Chengtong College Mathematics Competition, it is still far from enough.

Xu Chuan smiled, and was about to say a few polite words to cheer him up, but another voice came to his ear.

"Hi, Xu, I finally found you."

A somewhat rough voice sounded, and then a man with a full beard came over.

Seeing the person coming, Xu Chuan was stunned for a moment, and then a smile appeared on his face.

"Long time no see, Professor Avila."

The person who came was Professor Artur Avila, a Brazilian mathematics professor who had met at the Princeton Mathematical Society before. The two had chatted about mathematics for a whole morning.

After shaking hands with the Brazilian mathematician enthusiastically, Xu Chuan asked curiously: "Why did you come to attend this World Chinese Mathematician Conference?"

Although this World Chinese Mathematician Conference was held quite grandly, and many foreign mathematicians came to attend the conference, it was mainly for Chinese and Chinese-American mathematicians.

Xu Chuan really didn't expect Artur Avila to appear at this mathematics exchange meeting.

Artur Ávila said with a smile: "I have a cooperative project with P University in your country on mathematical dynamical systems. I happened to be here during this period and received an invitation letter, so I came to see it and see if I could meet you by the way."

Xu Chuan nodded suddenly and said: "So that's the case. Welcome."

Aside, Hu Xingjian watched the two people's communication with curiosity, waiting eagerly for Xu Chuan to introduce.

After noticing Hu Xingjian's gaze, Xu Chuan introduced with a smile: "Brother Hu, this is Professor Artur Ávila, a Brazilian mathematics professor. He has won top mathematics awards such as the European Mathematical Society Award and the Fields Medal. He has profound research in chaos theory, dynamical systems and spectral theory."

"Professor Ávila, this is a student of one of my teachers, Hu Xingjian. His research field is also spectral theory and Hilbert space."

After listening to Xu Chuan's introduction, Hu Xingjian was dumbfounded.

He didn't expect Xu Chuan to know such a super boss and chat with him.

As a graduate student studying spectral theory, he naturally heard of the name Artur Ávila.

This is one of the most famous mathematicians in Brazil, a Fields Medal winner, a top expert in chaos theory, dynamical systems and spectral theory. I didn't expect to meet him here today.

"Hello, nice to meet you." Artur Ávila smiled and extended his hand.

Hu Xingjian was stunned for a moment, then quickly shook hands with him: "Hello, Professor Ávila, I have heard of your name for a long time. I have read your book on spectral theory, "Spectral Fractal Regional Reconstruction."

Artur Ávila smiled and said: "In terms of spectral theory and spectral fractals, Xu is the real expert. My achievements are nothing compared to him."

Hearing this, Hu Xingjian couldn't help swallowing his saliva.

Good guy, I knew Xu Chuan was awesome, but I didn't expect him to be so awesome.

A Fields Medal winner said that his area of ​​expertise was far inferior to his. Even if there was a component of mutual flattery, it was incredible.

He was determined to hold on to this thigh!

After shaking hands with Hu Xingjian and saying hello, Artur Avila turned his attention back to Xu Chuan.

"Xu, when we met in Princeton last time, we talked about spectral theory and the spectrum of fractal dimension and fractal measure, which gave me a lot of inspiration."

"Inspired by you, I have some new things I want to talk to you about recently. I wonder if you have time and whether it is convenient?"

Xu Chuan asked curiously: "What aspect?"

Artur Avila: "Non-commutative harmonic analysis and automorphic form theory in the Langlands program. I wonder if you are familiar with or have some knowledge of these two fields."

Hearing this, Xu Chuan frowned slightly and said: "There are too many things in these two fields. I don't know which specific aspect you are referring to."

"In addition, why did you suddenly study number theory and harmonic analysis?"

Xu Chuan looked at Artur Avila curiously. The most famous research of this Fields Medal winner is in the fields of chaos theory and dynamical systems. Spectral theory is also involved, but he seems to have never heard of this professor's research on number theory and harmonic analysis.

Artur Avila smiled and said, "I am not familiar with number theory and harmonic analysis, but I have done some research on spectral theory and Laplace operators."

"After I talked to you last time, I looked through your proof of the Weyl-Berry conjecture."

"In your paper, there is a very essential method that uses the Dirichlet function field to transform the Laplace operator and the Laplace hyperbolic equation, as well as the expansion of the field and how to transform the function into a subgroup and establish a connection with the intermediate field and the collection."

"I am studying how this method seems to be applicable to some problems in the Langlands program."

"For example, some conjectures derived from automorphic L functions."

"You know, automorphic functions are generalized from circular functions, hyperbolic functions, elliptic functions, and other functions in elementary analysis, and automorphic L functions are defined by automorphic group representations."

"In the Langlands conjecture, automorphic L functions satisfy certain harmonious relations and have a unique factorization, which is reflected in the automorphic group representation. This is the functor between automorphic group representations."

"And this functor conjecture can be completely determined by L groups are determined by mappings between them. Given a linear algebraic group, the relationship between its automorphic group representation on the basis field and its group representation on a certain extension field is called basis transformation, which is a special case of functor property."

". Can the method in your paper also be applied to basic automorphic functions? Or, in some way, to further generalize Gauss's reciprocity law?"

After listening to Professor Artur Avila's words, Xu Chuan frowned and thought. He seemed to have caught something from these words.

On the side, Hu Xingjian, who was listening to the whole process, looked at Xu Chuan and then at Professor Artur Avila with a confused face.

What the hell is this?

He knows the Weyl-Berry conjecture, he also knows the Dirichlet function, and he has heard of the famous Langlands conjecture.

But when these nouns are put together, why can't he understand a word?

Is his English not good?

But he obviously has an IELTS A score of 7.9.

Suddenly, Hu Xingjian felt a surge of despair. He had thought about standing on the stage like Xu Chuan before, but now it seems that the gap between the two is really huge.

Catch up? What can he catch up with? He began to not understand what others said.

Xu Chuan was thinking, Hu Xingjian was confused and desperate, and Professor Avila looked at Xu Chuan expectantly.

The three stood in the venue until another voice came.

"Professor Avila, you are here."

At the same time, a thin middle-aged man came over.

"Professor Mo." Seeing the person coming, Professor Artur Avila smiled and nodded and greeted him.

"Professor Avila, what's wrong?"

Seeing Xu Chuan who was thinking on the side, Professor Mo asked in confusion. Xu Chuan just won the Morning Star Mathematics Award in the auditorium of Shuimu University in the afternoon. It is impossible for him not to know him.

Artur Avila smiled and said, "We just talked about some mathematics. He may have some ideas? He is thinking."

"Oh? Can I hear what you talked about?" Hearing this, Professor Mo raised his eyebrows and asked with interest.

"Of course, it is mainly non-commutative harmonic analysis and automorphic form theory in the Langlands program."

Avila would naturally not refuse someone to communicate about mathematics, and Professor Mo is a researcher in number theory, maybe they can communicate together, so he smiled and simply repeated the theory and problem just now.

"Langlands program, automorphic form theory, do you have any ideas?"

Hearing this, Mo Kun's eyes lit up and asked quickly, which happened to be one of the academic fields he studied.

Avila shook his head and looked at Xu Chuan and said, "I am not very familiar with this area. If you have any ideas, I am afraid you have to ask him. Maybe you can communicate?"

At this moment, Xu Chuan came back from his thoughts, opened his mouth and was about to speak, when his eyes fell on Mo Kun beside him.

"Who is this?"

Xu Chuan looked curiously at the fourth person who suddenly appeared. The person who appeared here should be a mathematician, but this math parent is quite special. His face is very distinctive. Not to mention he has a horse face, and the bridge of his nose is exactly where his head is. There are still some protrusions, and the appearance is indeed average.

"This is Professor Mo Kun from P University. He is an expert in number theory and number fields. He is also a member of the project I am cooperating with P University."

"This is Xu Chuan, my good friend."

On the side, Artur Avila introduced each other with a smile.

"Oh, hello, Professor Mo." Xu Chuan reached out his hand and said hello.

Mo Kun shook hands with a smile and said: "Hello, classmate Xu Chuan, I have heard of you earlier. When I saw you today, you are indeed well-deserved. You are young and promising."

"I just heard Professor Avila say that you are talking about non-commutative harmonic analysis and automorphic formal theory? Do you have any ideas? Let's talk about it together?"

Xu Chuan didn't think much about it and said with a smile: "I have a little idea, but I don't know if it will work."

Hearing this, Mo Kun quickly asked: "What do you think?"

Xu Chuan: "I just talked with Professor Avila about automorphic forms and automorphic L-functions. Generally speaking, the sources of L-functions are composed of two types, arithmetic L-functions and automorphic L-functions. These two The two are closely related. According to Robert Langlands’ conjecture, all meaningful L-functions come from automorphic L-functions.”

"The so-called automorphic form is a special type of complex variable function that satisfies the transformation law described by the automorphic factor under a certain discrete transformation group. Strictly speaking, the automorphic representation is not a group representation in the ordinary sense, but is the modulus of the global Heckian algebra.”

"When I was studying the Weyl-Berry conjecture before, I took advantage of the fact that every boundary point in the Dirichlet function domain is regular and is a type of boundary point, and used it to complete the transformation Laplacian operator and Laplace’s hyperbolic equation.”

"If we can build a group domain out of this, we might be able to discover something."

"For example, the study of functor properties can provide a driving force for the Langlands program. Such studies can generally be unified under a framework called Howe duality or Theta correspondence."

"But now it seems possible to achieve partial functority by studying the Dirichlet function to construct a number field that transforms the Laplace operator, and then using its curve equations to restrict to the dual reduction group."

"That might be a viable path forward."

Xu Chuan simply stated some of the ideas he had just conceived.

Under normal circumstances, unless they are in the same project team or have a very friendly relationship, ordinary researchers will not tell their inner thoughts and research in detail to other colleagues, especially unfamiliar colleagues.

Because this may lead to your ideas being used by others first, or being published first.

But since Professor Mo knew Avila and was a member of his project team, there was no harm in chatting.

I have to say that Artur Avila really brought him a huge surprise this time. He had never thought of applying this method to the fields of number theory and group theory before.

But now, it seems possible to give it a try.

The Dirichlet function is indeed a super treasure among functions, and there are too many things that can be mined in it.