Chapter 112: Graduation Thesis that can be published in the annual journal [Please subscribe to more]

The group defense ended at noon. When Xu Yuan came out, he saw Deng Yanna who was still there. He turned around and said something to Qi Tongxiang and others and then walked over.

When I approached and stopped, I still heard the familiar name.

"Monitor."

When Deng Yanna heard these two words, she subconsciously raised her eyes. Seeing that it was Xu Yuan, she quickly stood up and greeted him: "I really didn't expect that we would meet in this way after graduating from high school. How do you feel about Qinghua?" She chatted openly with a smile on her face.

"I'm also surprised that you, class monitor, will participate in the Higher Education Society Cup Mathematical Modeling Competition."

Xu Yuan smiled and said, "As for Qinghua, she's OK."

"You are very famous now. The newspaper says that you are a rare genius in mathematics." Deng Yanna continued to talk to Xu Yuan in a relaxed manner.

"You're not bad, squad leader."

"I'm far behind you." Deng Yanna modestly asked, "By the way, why did you come to our university to be a preliminary judge for the competition? I was shocked when I saw you in the defense room just now."

"Professor Qi from the Department of Mathematics at your University of Science and Technology invited me here. He has an open class in the afternoon." Xu Yuan looked a little helpless.

Deng Yanna smiled and said, "So that's how it is. I'll have to go and listen to it then."

After a few simple exchanges, the two seemed to have found the feeling of getting along with each other again when they were in Guyuan No. 1 High School.

Considering that it was getting late, Xu Yuan took the initiative to suggest having dinner.

"How about we have lunch together at noon? I also asked Yang Bin to come over. I wanted to wait for him to arrive and then talk with you, class leader, but I didn't expect to run into you at the defense site."

"Sure." Deng Yanna agreed readily.

So, at Deng Yanna's recommendation, they went to a restaurant, and soon Yang Bin also arrived.

-

"Brother Yuan, I miss you so much."

When Yang Bin saw Xu Yuan, he immediately rushed up to him and gave him a tight bear hug. After he let go, he noticed that Deng Yanna was sitting next to him. A hint of embarrassment appeared on his face but quickly disappeared.

"The squad leader is here too."

Keeping smiling, he greeted people actively, pulled out a chair and sat down.

There are only three of my former classmates now, but I don’t feel strange when chatting with them.

While waiting for the food, Xu Yuan told Yang Bin about what happened to him and Deng Yanna. Yang Bin was not too surprised to learn that Xu Yuan came to Luyang to be a judge. After all, he usually paid attention to the news related to Xu Yuan.

After listening to it, I just felt a little regretful.

"I thought you were here specifically to look for the class monitor, Yuan. If I had known you were going to be a judge, I would have signed up for the Higher Education Society Cup."

The last sentence was said in a joking tone.

However, Xu Yuan knew Yang Bin very well and knew that he lacked self-control.

When he was in his senior year of high school, it was because of his supervision of his studies and his help in reviewing mathematics that he was able to perform well in the college entrance examination and be admitted to the University of Technology.

Now that he is entering his sophomore year, he thinks it is very good that Yang Bin can guarantee not to fail any course.

Participating in the Higher Education Society Cup is just talk.

But what is certain is that Xu Yuan will not mention these things at this moment. After all, everyone has their own life, and he cannot help plan everything.

Yang Bin didn't discuss this matter too much and took the initiative to change the subject.

"Brother Yuan."

"You have a lot of fans in our school. You looked so handsome when you gave a speech at the international conference."

"It's a pity that Wang Yongjie is not here today."

In just one year, Xu Yuan has published a paper in the Annals of Mathematics and delivered a speech at an international cryptography conference. In contrast, his classmates are still ordinary college students, who either think about eating or not failing their courses every day.

The gap is simply huge.

If you ask me whether I am envious or not, the answer is definitely yes.

But more people were happy for Xu Yuan from the bottom of their hearts. After all, everyone could see Xu Yuan's talent and hard work.

Today's achievements are what we deserve.

Although Deng Yanna didn't say anything at this time, she felt the same way as Yang Bin.

She still remembered how shocked she was when she first saw the news reports about Xu Yuan. She couldn't imagine that Xu Yuan could achieve such great results just by attending Qinghua.

Back to his senior year in high school, Xu Yuan had some talent in mathematics and was successfully selected for the national team for the Olympic Mathematics.

Who could have guessed that he would soar into the sky right after graduation.

This sometimes makes her feel a little regretful. If she had not listened to her parents' conservative advice and chosen to apply to Qinghua University when estimating her scores and filling out her application form, perhaps the situation would have been different.

But time cannot go back.

Since it has become a foregone conclusion, she can only look forward and live her own life well.

Relying on his keen observation skills, Xu Yuan naturally felt the strange atmosphere that permeated the surroundings at this time.

The next second he immediately took over Yang Bin's words and said.

"It's ok."

"When we have a chance, let's call Lao Wang to come over and have dinner together. Let's eat first."

After hearing this, Yang Bin's smile became even brighter. No matter what position Xu Yuan achieved in the academic world in the future, they were still high school classmates and desk mates.

Upon learning that Xu Yuan would be giving an open lecture at the Department of Mathematics at USTC in the afternoon, Yang Bin did not go home after dinner, but planned to attend the lecture to show his support.

To be honest, he hadn't listened to Xu Yuan's lectures for a long time.

Since the news that Xu Yuan was going to give a lecture was announced on the official website of USTC, and combined with what happened in the morning's defense, everyone knew that Xu Yuan had come to USTC and had served as a preliminary judge for the Higher Education Press Cup Competition. Many people were discussing this at lunch in the cafeteria.

In addition, some people expressed surprise that they were finally able to see this mathematical genius in the news.

But there is also no shortage of questioning voices.

I feel that Xu Yuan’s qualifications are not enough to serve as a judge.

Especially considering that Xu Yuan is only a sophomore, I don’t think he can give a very good public lecture.

Moreover, Xu Yuan's mathematical academic achievements were in the field of number theory. As a mathematical content with a relatively low research threshold, number theory has always been popular among amateur scientists, so naturally its value is discounted.

Therefore, many people are looking forward to this open class and want to see what will happen on the spot.

So in addition to the sophomores and juniors from the Department of Mathematics who signed up, many graduate students and students from other departments also came to join in the fun.

Before the open class officially started, the classroom was already crowded with people.

There are more than two hundred students gathered here.

It can be said to be quite lively.

While Xu Yuan and the professor were not here yet, everyone was whispering to each other, discussing their own understanding of Xu Yuan.

"Xu Yuan is actually one of the judges of this year's competition organizing committee. I thought they were just joking. Isn't it a bit too much to let a sophomore student be a judge?"

"It seems that Professor Qi specially invited him here."

"I really can't imagine how much content a sophomore can use to teach."

"It's probably number theory again."

"There are many amateur scientists studying number theory in China. I want to see if he has any talent in other fields."

"We will all know whether the title of genius is worthy of it soon."

……

Duan Le also took over the conversation in the classroom. After being refused by Deng Yanna to have lunch together at noon, he found out Xu Yuan's identity. He came here in the afternoon and specially prepared a few math problems for junior year students, hoping to find an opportunity to ask Xu Yuan for advice to expose his genius personality.

On the other hand, Deng Yanna and Yang Bin sat together, looking forward to what Xu Yuan was going to say.

"I didn't expect that Brother Yuan has so many fans in your university. There are more people coming to listen to his lectures than the most popular professor in our school."

At this time, Yang Bin looked around the classroom and turned to express his admiration to Deng Yanna.

Deng Yanna replied calmly without looking at him: "Maybe they all want to see what Xu Yuan's actual math level is even though he is called a math genius." Then she remained focused and said no more.

Seeing this, Yang Bin didn't say much and could only wait patiently for Xu Yuan to come.

But if he looked more closely, he would find other familiar people sitting in the classroom.

"It seems like they are talking about Xiaoyuan. There should be no problem with this open class." Xu Ying listened to the discussions around her, lowered her head and whispered to Zhao Lingling beside her.

Yes.

Xu Ying knew that Xu Yuan had come to USTC to give lectures.

As an older sister, she would definitely come over to take a look, since she had never seen her younger brother standing on the podium.

So after telling her best friend Zhao Lingling about it, the two of them immediately hit it off and ran over.

"You still don't have confidence in your brother." Zhao Lingling rolled her eyes and said, "I don't think there's anyone else of his age who has a better math talent than him."

"That's true." Xu Ying nodded in agreement.

Xu Yuan had no idea what was happening in the classroom at the moment. He was rushing here with Qi Tongxiang and several other mathematics professors.

"Have you thought about the content of this class? Is it about Carmichael numbers?"

Qi Tongxiang considered the effect of the lecture and suddenly said this while walking. He wanted to know the specific content that Xu Yuan and others would talk about in the open class.

This matter also attracted the interest of several other professors.

Without waiting for Xu Yuan to answer, they all echoed his thoughts.

"I think this is fine, especially the sieve method you used to prove the Carmichael number spacing problem, Xiao Xu."

"It seems that I have to pay attention to the class today."

"Then let's be students again this time."

After listening to the professors' words, Xu Yuan showed a mysterious smile and told Qi Tongxiang about his plan.

"Professor Qi."

"I'm not going to talk about number theory today."

"Don't talk about number theory?" Qi Tongxiang was a little stunned when he heard that.

As we all know, Xu Yuan's mathematical achievements are currently limited to number theory. He only needs to talk about the content of the report he gave at the number theory academic conference. After all, cryptography involving hash functions is not appropriate here.

It turns out Xu Yuan had other plans.

The thought stopped here and I immediately asked, "So what are you going to talk about?"

"You'll find out in a moment."

Xu Yuan did not directly answer Qi Tongxiang's question, but kept it a secret for the time being.

Then, under the expectation of several professors including Qi Tongxiang, he walked into the large classroom which was already full of people. When everyone saw Professor Xu Yuanhe, they immediately became quiet.

Qi Tongxiang and others were playing the role of students at this time. They entered the classroom and walked towards the stage.

There was no intention to stay on the stage.

A student from the Department of Mathematics saw this and was very considerate enough to give up his seat and choose to stand nearby and listen.

Xu Yuan walked to the podium and glanced around the classroom. He immediately noticed his sisters Xu Ying and Zhao Lingling, who even winked at him.

Although he was a little surprised that his sister would come to HKUST without saying hello, it didn't affect his pace.

"Thank you all for coming to this class. Today I will talk about nonlinear partial differential equations."

With a warm expression on his face, he picked up the chalk, turned to the blackboard and wrote a series of math formulas.

"detDu=f(x,u,Du)"

It was the Monge-Ampere equation that he studied in his graduate thesis.

He thought that since he had made no progress in studying the Monge-Ampere equation, he might as well take this opportunity to try to prove it on the podium.

Perhaps being in a different atmosphere can spark inspiration and generate new ideas to drive progress.

As Xu Yuan wrote out this formula, everyone's reactions in the classroom were completely different.

Juniors and sophomores, including Duan Le, were completely confused. They didn't expect that Xu Yuan was going to talk about not number theory, but nonlinear partial differential equations, which are basically learned in juniors and seniors.

It makes people feel dumbfounded.

What happened to number theory, which was said to be easy for even amateur scientists to get started with?
When did he learn partial differential equations as a sophomore...

A small number of graduate students in mathematics were an exception.

As the saying goes, if you study real analysis ten times, you will feel cold in the face of functional analysis, and partial differential equations are even more difficult.

Although they have studied partial differential equations and passed the exam successfully, they do not dare to use the knowledge to give public lectures. It would be very embarrassing if they were pointed out for making mistakes.

Besides, there were several professors sitting in the audience.

He ignored the subject of number theory, which he was good at, and chose to talk about the more difficult partial differential equations.

I really don't know if I'm too conceited or just too naive.

Mainly in their opinion, no matter how talented Xu Yuan was, he was just a sophomore in college. He could only be called a senior by the freshmen. Without enough time to accumulate knowledge, how much knowledge could he master?

It's just some basic knowledge of partial differential equations taught by myself.

Thinking of this, I felt somewhat sorry for Xu Yuan. I was afraid that this open class would not go very smoothly.

It can only be said that Zhang Jundong is not here, otherwise, after hearing the thoughts of the graduate students from USTC, he would definitely clear Xu Yuan's name at the first opportunity.

How can a sparrow understand the world in the eyes of a swan?

However, several mathematics professors including Qi Tongxiang became interested and began to look forward to it upon seeing this.

"No wonder that guy was keeping me in suspense just now. It turns out he wanted to talk about the Monge-Ampere equation."

“This is too fast a learning process.”

Qi Tongxiang murmured to himself. He found that he was increasingly unable to understand Xu Yuan at this moment.

I remember that when he was invited to give lectures to the provincial team, Xu Yuan was only working on algebraic sets and geometry problems. But in just one year, he was able to teach others partial differential equations.

He knew something about Xu Yuan's character. Since he dared to choose such content in such an occasion, it meant that he was fully confident.

The thought stopped here, and he immediately stopped thinking and focused all his attention on Xu Yuan.

Xu Yuan didn't care about the reactions of the audience, and continued to speak and write on the blackboard according to the process he had prepared.

He first started with the existence and uniqueness of the proven solutions to the Monge-Ampere equation.

Then it is extended to the smoothness of free boundary problems.

List the two conditions for two regions to be uniformly convex and for the density function to be smooth.

And try to prove that these two conditions can be removed.

"Given that a convex open set in a Euclidean space is a second-order smooth function by means of a gradient map, then the gradient map is a diffeomorphism."

"Convexity of domain and positive definiteness of function defined by implicit function theorem."

"It is possible to derive a global homeomorphism."

"It might be a good idea to use the Legendre Transform."

"df = ydx + ydx + ... + yndxn"

……

"(q)+β`(p)=(1 +β)`(0)=(1 +β)h"

A large number of ideas emerged and collided in Xu Yuan's mind. After being rejected, new proof methods were regenerated, so that the speed at which his hands wrote became faster and faster, making it difficult for people to keep up.

It's like playing a beautiful piece of music on the blackboard, and those mathematical symbols are the piano keys.

At the beginning, the graduate students could still understand some of the content and were amazed at Xu Yuan's understanding of the Monge-Ampere equation, but later on, all they had left were various question marks in their minds.

I am gradually unable to understand the equations and formulas written by Xu Yuan, and I don’t even know why they are used there.

As for the other undergraduates and people from other departments who came to join in the fun, their expressions froze on their faces the moment Xu Yuan started speaking.

It was basically confusing from beginning to end.

Not falling asleep is the greatest respect for Xu Yuan.

"Brother Yuan is still the same Brother Yuan I know, but unfortunately I am no longer qualified to listen to Brother Yuan's lectures."

Yang Bin, who had been listening carefully throughout the whole process, said in a low voice with a look of emotion on his face.

Thinking of the scene when Xu Yuan explained the questions to him in the senior year of high school, and then looking at today, I can only say that Xu Yuan's growth rate is too terrifying. On the other hand, after so much time has passed, he has been standing still for the same period of time.

Although Deng Yanna, who was sitting next to him, did not say anything after hearing this, the complicated expression on her face was enough to explain the problem.

If Qi Tongxiang was mostly curious just now, then at this moment his pupils were as wide as an earthquake. He could naturally see that Xu Yuan wanted to solve the free boundary problem of the Monge-Ampere equation.

Just last year, we proved the intervals between Carmichael numbers, and now we are working on the free boundary problem of the Monge-Ampere equation.

It is true that the higher the difficulty, the more challenges you take. Not only him, but even the professors next to him cannot have such courage.

"How many surprises does this kid have hidden? I didn't expect that his understanding of partial differential equations has reached this level."

Qi Tongxiang said this in a low voice with a smile, which was immediately echoed by several professors around him.

"Today's class was definitely not in vain."

"The future is terrifying!"

“Although it is very difficult to completely solve the free boundary problem of the Monge-Ampere equation, it is rare for a sophomore to dare to challenge the research.”

Xu Yuan naturally couldn't hear what the professors said while standing on the stage, but as they said, it was not that easy to prove.

Seeing that two hours had passed and the open class had been delayed for a few minutes, Xu Yuan temporarily stopped his proof.

He knew very well that it was impossible to prove something in class in such a short period of time, and it was already a rare opportunity to come up with new ideas.

When you move your eyes to the panel in front of you, you can clearly see that the progress bar has grown significantly.

-

Task: Proof of Monge-Ampere equation

Subject: Mathematics

Progress: 30%
Result: Not completed
The rapid increase from the previous single-digit progress to 30% showed that his idea was correct. All he had to do next was to continue the calculation according to his original idea, and sooner or later he would be able to thoroughly prove the problem of the smoothness of the free boundary of the Monge-Ampere equation.

"It turns out that we should change the environment more often to generate more inspiration."

After the panel gradually disintegrated from his sight, Xu Yuan threw the chalk in his hand on the desk in front of him, murmured something happily to himself, and slowly raised his eyes to look at the students in the classroom again.

"That's all I have to say for this class. If anyone has any questions, please raise them now."

Although he said this at the end according to the procedure, none of the students sitting in the audience said a word.

There were more than two hundred people gathered here, but it was even quieter than usual exams.

Everyone looked around and exchanged glances with each other, but ultimately they all chose to remain silent tacitly.

The main thing is that after listening for two hours I didn't understand a single word. It's embarrassing enough. Who would take the initiative to ask questions?

The key is that I don’t know what to ask.

However, the roommate who came with Duan Le lowered his head and whispered something.

"Duan Le, you have prepared a few difficult questions, why not take them out and let the other party answer them?"

Duan Le, who was still immersed in shock, was awakened by these words and immediately rolled his eyes after coming to his senses. He rejected the proposal decisively without even having time to think.

He just glared at the other person and retorted.

"Why don't you go up and ask?"

He really didn't have much respect for Xu Yuan at the beginning, and felt that the school thought too highly of him and let him serve as a judge.

So I thought of using some difficult questions to embarrass the other party.

But this situation only applies when the other party is really a sophomore student, not when he has already studied partial differential equations.

It would be foolish to bring up those questions now, since he was not a fool.

All I can say is that Xu Yuan's performance exceeded the expectations of everyone present.

"No wonder Professor Qi attaches so much importance to him. This kind of learning efficiency and talent is really daunting!"

Finally, he sighed silently with a gloomy look on his face, and then remained silent again.

The whole atmosphere lasted for about half a minute, and Qi Tongxiang suddenly stood up from his seat, put on an appreciative smile on his face, and applauded actively.

Then several professors around him did the same.

Led by Qi Tongxiang, a chain reaction soon took place, causing warm applause to break out in the entire classroom.

It lasted for nearly a full minute before the applause gradually subsided and finally stopped.

After that, the open class ended and everyone stood up and left the classroom, still with confused looks on their faces.

Deng Yanna, Yang Bin, Xu Ying, Zhao Lingling and others came over and started talking to Xu Yuan.

When they had held the top scholar banquet in their hometown of Xuzhai Village, they had eaten at the same table, so they were not strangers to each other.

Although Xu Ying didn't understand what Xu Yuan was saying at all, since several professors took the initiative to applaud, it showed that the quality of the content was guaranteed.

We should praise when it is due.

"Xiao Yuan, you gave a really good lecture this time."

Xu Yuan didn't expose his sister after hearing this, but just asked normally: "Sister, why are you here too?"

"I know you won't be here for long, so I thought I'd come over to see you." Xu Ying said happily as she handed over the things she brought. "I picked out two pieces of clothing for you last time I went shopping with Lingling, and today I'm bringing them back to school for you to wear."

"Yingying and I picked it together. It suits you very well." Zhao Lingling next to her took over the conversation and added.

Xu Yuan reached out and took the things: "Thank you both sisters."

As soon as he finished saying this, he saw the smile on Zhao Lingling's face become even brighter.

It was obvious that she was very happy with Xu Yuan's words.

After all, she had said a long time ago that she hoped to have a younger brother as outstanding and handsome as Xu Yuan.

Since Yang Bin had to go back to his school, he did not stay any longer at HKUST.

Then he said something to Xu Yuan and left.

Anyway, there will be many opportunities for the two to meet in the future, so there is no need to worry too much about this moment.

When most of the students in the classroom had left, Xu Yuan walked up to Qi Tongxiang who was still sitting in his seat.

"Professor Qi."

"It was the most correct decision for me to let you serve as a judge for the preliminary evaluation of the competition area. Otherwise, I would not have had the opportunity to see such a wonderful open class." Qi Tongxiang stood up and gave Xu Yuan a thumbs up and said, "What you said is even more exciting than the academic conference in our department."

He did not hide his love and praise for Xu Yuan at all, and gave a very high evaluation of what he had just said.

Even the professors next to them seemed a little excited.

"This class is really worth it. It's hard to imagine that a sophomore like you can have such a deep understanding of partial differential equations."

"You want to completely prove the free boundary problem of Monge-Ampere's equation and remove the two existing necessary conditions. Even I can't help but admire your courage and determination."

"The last line of thought is very brilliant, I hope you can keep it up."

After the other professors finished speaking, Qi Tongxiang took the initiative to ask a question.

"Xu Yuan."

"Why did you suddenly want to study the Monge-Ampere equation?"

Faced with Qi Tongxiang's question, Xu Yuan did not keep him in suspense and answered truthfully.

"The study of Monge-Ampere equation was not sudden. It was the content of my graduation thesis."

"Graduation thesis?"

"You are about to graduate?" Qi Tongxiang looked surprised when he heard that.

But after about two seconds of reaction, he felt that this was a very normal thing.

It is not surprising that Xu Yuan could even master partial differential equations and graduated early with full credits.

But Xu Yuan actually took the free boundary problem of Monge-Ampere equation as his graduation thesis, which really surprised him. He had never seen anyone who would increase the intensity and pressure on himself like this.

Originally, he thought Xu Yuan was just trying it out of interest and didn't have to solve the problem.

"I have already completed my undergraduate professional courses and obtained all the credits. The department arranged for me to graduate early and pursue a doctorate in science."

"Graduating early is indeed a good thing for you, but do you really intend to use the free boundary problem of the Monge-Ampere equation as your graduation thesis?" Qi Tongxiang hesitated for a moment but still asked more.

Xu Yuan naturally knew that Qi Tongxiang was thinking about him, so he thought for a while and organized his words before giving his answer.

"I came across the Monge-Ampere equation by chance. At that time, I was having a headache about choosing a topic for my graduation thesis, so I just studied the theory of the Monge-Ampere equation directly."

When Qi Tongxiang heard this answer, he subconsciously wanted to persuade her, but when the words came to his lips, they turned into agreement.

"Most mathematicians complete their academic achievements when they are young. Now that you have made up your mind, stick to it. I am waiting for your news on the complete proof of the free boundary problem of the Monge-Ampere equation."

"I will definitely tell you the good news when the time comes." Xu Yuan replied with a smile on his face.

……

As the matter came to a temporary end, Xu Yuan did not stay in the campus of USTC for too long.

After the open class ended, I returned to Yenching that evening.

The burst of inspiration in class gave him new ideas for the problem of free boundary smoothness of the Monge-Ampere equation. He must have wanted to immerse himself completely in the proof of the formula as quickly as possible so as to complete the proof as soon as possible so as not to affect his undergraduate thesis defense next year.

It is also worth mentioning that he also received the list of participating teams sent to the national organizing committee after the preliminary review and defense of the competition area.

This includes the team that Deng Yanna is in.

However, for the final selection of the National Award and the Higher Education Press Cup, they will still need to go to Yenching to participate in expert interviews.

This time, the Higher Education Press Cup expert interview was held at Yenching University, so Xu Yuan naturally didn't have time to join in the fun.

Time flies by like a fleeting moment.

Two months passed quickly.

At seven o'clock that evening, Xu Yuan came to the discussion room alone after dinner.

Faced with the large amount of draft paper filled with mathematical formulas on the table, he sat down on the chair next to him as usual, picked up a pen and continued to deduce and prove the steps from the afternoon calculations.

A lot of snack packaging can be seen in the trash can next to him, indicating that he has been staying here for a long time.

This is indeed the case.

After returning from HKUST, he devoted himself to proving the problem of free boundary smoothness of the Monge-Ampere equation.

Except for eating and sleeping, I spend most of the day in the discussion room.

And in order to ensure that I can always maintain a deep learning state, I put a lot of snacks in it.

Whenever the body's energy was consumed too quickly, he would pick up the food and continue to deduce and prove while eating.

Plus, Chen Yuran would come over to feed him often, so he didn't have to worry about starving during deep learning.

It's because I haven't paid attention to my personal image management for a long time, and my overall appearance looks a little sharp.

But Xu Yuan no longer had time to care about this.

Because his gains in the past two months can be said to be huge. In addition to the progress reaching almost 90%, he also successfully proved that he had eliminated one of the necessary conditions.

In other words, he took the study of Monge-Ampere equation a step further.

The current theory of Monge-Ampere equation can be newly expressed as the optimal transmission solution is smooth when the two regions are uniformly convex.

It is no exaggeration to say that just by removing one condition, this achievement is enough to attract everyone's attention to partial differential equations.

It can also be used as your graduation thesis.

But since he is a person who pursues perfection, and he has made up his mind to study the Monge-Ampere equation, he must thoroughly prove the complete smoothness of the Monge-Ampere equation under natural boundary conditions.

It's totally not his style to only solve half of the problem and leave a hole there.

For this reason, every time he went to Tang Shihong to discuss something, he did not mention that he had already proved one of the conditions.

Otherwise, Tang Shihong might start the graduation defense early, forcing him to give up the subsequent certification.

"It is indeed much more difficult to solve the condition of uniform convexity of the region than the other condition."

"If the calculation is slightly off, it will cause progress to be slowed down."

At this time, Xu Yuan tore off the piece of draft paper in his hand, crumpled it into a ball, and threw it into the trash can while muttering to himself.

On the most important condition of uniform convexity, his proof encountered a bottleneck, causing the progress to slow down from time to time. Every time this happened, it meant that there was a problem with the steps and they were wrong.

So much so that the trash can is full of discarded drafts every day.

Soon Xu Yuan stopped writing, stood up and walked around the discussion room, thinking again about the proof process.

This is also the method he usually uses.

When his body was in motion, he found that his mind would be more agile and it would be easier to come up with new ideas.

So every time the direction went astray and the progress bar went down, he would analyze it in detail to find out where the problem lies, and then continue to prove it.

Otherwise, if you know that your thinking is wrong and you still try to calculate, you will only make more mistakes.

"It is now clear that the original proof method cannot solve the necessary condition of uniform convexity."

"We must rethink our thinking..."

……

"Partial differential equations are mathematical physics equations. If you want to complete the mathematical proof, you might try using physical methods."

“That should be doable.”

Suddenly, a new idea popped into his mind and Xu Yuan stopped walking. His eyes became brighter as he spoke, as if he had finally found a way out after being trapped in some strange place.

The next second he almost ran to the chair and sat down, quickly picking up the pen on the table and starting to calculate on the new draft paper.

It can be said that it enters the deep learning state instantly.

It means that you are very focused at the moment.

This time he continued to use Legendre transformation and tried to prove it with the help of Lagrange method.

As the pen tip slid quickly across the draft paper, symbols appeared one after another on the paper.

“det D2u(x)= f(x)”

……

“|D2u(z) D2u(0)|≤ C|z|α”

"|D2u(x) D2u(y)|≤|D2u(x) D2u(……"

……

As time passed, more and more draft papers were filled with mathematical formulas written by Xu Yuan.

More than half an hour had passed when he suddenly stopped writing the last mathematical symbol, with a look of surprise in his eyes.

"Success."

He was stunned for half a minute, then slowly straightened up and uttered three words.

Because in his field of vision, the progress bar on the panel directly grew to 100%.

This shows that he successfully eliminated two necessary conditions and proved the overall smoothness of the Monge-Ampere equation under natural boundary conditions.

In fact, he has studied the equation thoroughly in the past two months, and successfully solving one of the conditions is the best explanation.

In a way, he was just one wall away from complete proof.

But I still couldn't find a way to break the wall.

Today, he tried to use physical methods to prove a mathematical problem, which allowed him to successfully find the opportunity, and completing the entire proof was a natural thing.

At this moment, Xu Yuan clenched his fists tightly, feeling excited as he enjoyed the spiritual satisfaction of solving a difficult mathematical problem.

He sat there for a few minutes, and only stood up and walked towards the writing board after his emotions had completely calmed down.

Use a marker to verify your proof.

Although there is a 100% progress bar on the panel, there will be no problem with his proof process, but I still want to verify it myself to see if there is room for optimization of the steps.

After all, it was a proof process that he completed himself, so Xu Yuan verified it very quickly.

Soon the entire writing board was filled up.

"There is nothing wrong with my proof method and steps. From now on, the solution of Monge-Ampere's equation will not require any conditions to determine its overall smoothness."

Xu Yuan looked at the contents on the writing board and slowly spoke to himself, saying something that would be enough to surprise many mathematicians.

Then he took out his cell phone from his pocket and called Tang Shihong.

These days, Tang Shihong has been very concerned about his graduation thesis and is always willing to discuss with him.

Now that we have finally achieved results, we naturally have to inform the other party immediately so that we don’t have to worry about him anymore.

The phone rang for more than ten seconds before a very familiar voice came from the receiver.

"Xu Yuan, what's the matter with you contacting me so late?"

"I just got out of school."

When Xu Yuan learned that Tang Shihong had returned home, he planned to let him see his proof process tomorrow and just gave a casual explanation for it.

"It's nothing serious, Professor Tang. You can go back and rest first. I'll take the manuscript to you tomorrow morning."

Just as he was about to end the call, he suddenly heard a few high-pitched shouts coming from the microphone.

"What did you say?"

“Paper manuscript.”

"You proved the Monge-Ampere equation?!"

"The entire proof process was completed just now." Xu Yuan answered, moving the phone a little further away.

Sure enough, the sound doubled in the next second.

"Don't leave. I'm going back now."

"Then Professor Tang, I'll be waiting at the door of your office." Xu Yuan had just finished speaking when he saw that the call had been hung up.

His discussion room had not been cleaned up for some time. He thought it would be better to go to Professor Tang's place. He put his cell phone in his trouser pocket, tidied up the draft papers on the table, and pushed the door open to go out.

……

Not long after.

In the office, Tang Shihong took the draft paper as if he had found a treasure, took out his glasses from his pocket, put them on and looked at them carefully.

The clock on the desk next to him ticked away regularly.

five minutes.

ten minutes.

Tang Shihong still maintained his original posture and did not even turn to the next page of the manuscript.

Xu Yuan stood nearby and remained silent, fearing to disturb the other party.

A few minutes passed like this, and Tang Shihong's expression became more and more serious. He silently walked to the desk, sat down, and picked up a pen to verify the process.

Seeing this, Xu Yuan also found a chair and sat down. He knew that Tang Shihong's verification would definitely not be too fast.

Another long hour passed, and just when Xu Yuan was wondering whether he would have to stay in the discussion room all night, he finally saw Tang Shihong stop what he was doing.

He straightened his chest and knocked his back with his fist, then raised his eyes and looked at Xu Yuan.

"You know what, Xu Yuan? These past few days I have been trying to persuade you to give up your research on the smoothness of the Monge-Ampere equation and choose a new research topic for your graduation thesis."

"I didn't expect you to actually do it."

"Your proof thoroughly proves the overall smoothness of the Monge-Ampere equation under natural boundary conditions."

"This is a huge step forward in the study of Monge-Ampere equations, which is equivalent to expanding the scope of a theorem to a wider field. It is a paper in the field of partial differential equations that can be published in the Annals of Mathematics."

……

(End of this chapter)