Technology invades the modern world

Chapter 132 International Congress of Mathematicians

Who dares to say that such a tradition is fake?
Lin Ran always mentions his time in Göttingen in his public speeches, as he was born in Göttingen. Siegel also publicly acknowledges that Lin Ran was a student he "personally" mentored.

Even though Berlin newspapers, especially Pravda in East Berlin, frequently mocked Göttingen for its lack of appreciation, and local Göttingen newspapers questioned him, he never wavered.

My roots are in Göttingen, and I'll be able to return to Göttingen sooner or later.

Siegel held this simple idea.

Neither he nor Lin Ran denied it; who would dare say that Randolph wasn't from Göttingen?

Since this is true, it is also an indisputable "fact" that the Göttingen mathematical master handed over the reins from the first half to the second half!
Lin Ran smiled and said, "Thank you for your guidance, Professor. Without Göttingen, there would be no Randolph today."

This is what tacit understanding is.

An unspoken understanding among everyone.

The conference was held in the auditorium of the Royal Institute of Technology, where the Swedish flag and the flag of the International Mathematical Union were displayed. Mathematicians from both the free world and the Soviet bloc were present.

André, whom Lin Ran had met in Geneva, was also present and was to give a one-hour academic report at the Mathematicians' Congress.

Of course, Lin Ran also wanted to go, and Lin Ran was the one who took the lead.

After Lennart Carlson's opening speech, Lin Ran was to deliver the opening academic report.

Distinguished mathematicians, scholars, ladies and gentlemen:

Welcome to beautiful Stockholm for the 1962 International Congress of Mathematicians. I am Lennart Carlson, President of the International Mathematical Union, and it is a great honor to address you here to open this grand event in the mathematical community.

After Lennart Carlson finished speaking, the area behind him quickly transformed from a dark velvet curtain into several large blackboards.

"Hello everyone, this is the first time I'm giving a presentation in front of so many mathematicians. Some of you are doing analysis, some are doing geometry, some are doing number theory, and some of you are doing something I don't know."

Modern mathematics has developed to such an extent that even within the same subfield, it can take mathematicians a considerable amount of time to understand what the other is saying about two different problems.

Just like a tree growing upwards, it keeps growing and becoming more and more lush and leafy, but the branches also branch out more and more.

I once said that mathematicians are divided into birds and frogs, but each of us is also searching for our own fruit.

Today, I hope to talk about something interesting.

I know many of you are expecting me to talk about the Randolph program, hoping I can discuss the connection between automorphic forms and Gavarro representations, and how the complete establishment of Randolph correspondences can be verified in higher dimensions and in general.

Although you don't know if I've proven it or not, you still hope I'll share my thoughts.

Of course, I would love to share this with you all, but this might not be very friendly to mathematicians who haven't studied the Randolph program.

Not every mathematician is familiar with harmonic analysis and automorphic forms, and not every mathematician is interested in my research area.

Today, I am fortunate to have the opportunity to lecture in front of all the mathematicians attending the conference in the Great Hall of the People. I feel that I should return to the essence of mathematics and talk to you about some basic and interesting content.

So let's set aside those complex mathematical theories and return to the most basic, primal joy.

Lin Ran walked to the blackboard, and his words undoubtedly piqued the interest of the mathematicians present.

Indeed, as Lin Ran said, not everyone can understand what he is saying, and not everyone will be interested in the Randolph Program.

The audience erupted in chatter, everyone curious about what Lin Ran was going to say, and also discussing what their most primal joy was.

Doyle, who was sitting next to Siegel, asked, "Professor, what does Randolph want to talk about?"

Siegel shook his head: "I don't know, but you can think about what your initial joy was in mathematics."

Doilyn hesitated for a moment, "Is it the joy that comes from solving problems?"

Before the mathematicians in the audience could reach a conclusion, Lin Ran's voice rang out:
"Initially, we learn mathematics by solving real-world problems."

For example, how many apples are there if you add one apple to another? And how many apples are there if you put ten fingers together, with or without a few more fingers?

Initially, mathematics was meant to provide guidance for the real world, but it gradually became more and more abstract, to the point that we could no longer find corresponding real-world problems in the actual world.

It became a purely logical thinking game.

Regardless of its practical significance, I just have to find the answer.

That's good, that's certainly good. Mathematics represents the limits of human intelligence.

All of you here are explorers of the limits of human capability.

But I still want to talk about real-world issues and introduce some new concepts to everyone.

My topic today is the Four Color Theorem.

Lin Ran drew an irregular circle behind him, then divided it into four irregular sections, and filled the four sections with chalk of different colors.

"The four-color problem refers to whether any flat map can be colored with no more than four colors so that adjacent areas have different colors?" Lin Ran said.

"The theoretical framework of the four-color problem is based on graph theory and combinatorics, which fall under the category of elementary mathematics. I believe everyone here can understand it."

Let's begin.

We treat each region on the map as a vertex in the graph.

If two regions share a common boundary, then connect these two vertices in the graph with an edge.

Thus, the map coloring problem is equivalent to coloring the vertices of a graph such that adjacent vertices have different colors, and the total number of colors does not exceed four.

In other words, it proves that any planar graph necessarily contains certain specific subgraph structures, and these structures cannot be avoided.

For each unavoidable configuration, we prove that if a large graph contains such a configuration, it can be simplified, for example by removing or merging some vertices or edges, to a smaller graph without affecting the validity of the Four Color Theorem.

This simplifies the problem.

Lin Ran continued, "Of course, the four-color problem is not limited to these."

We also need to introduce a graph theory technique called the discharge method. It's a new method I devised based on Professor Kempe's chain method and Professor Heathwood's method of analyzing vertex degree and face degree in the proof of the five-color map theorem.

After briefly introducing the chain method and the proof of the five-color theorem, Lin Ran continued:
The core idea of ​​the discharge method can be divided into three steps:

The first is the initial charge assignment, where we assign an initial charge to each vertex or face in the graph.

The value of a charge is usually related to the degree of a vertex or the degree of a face.

(Degree refers to the number of edges connected to that vertex, and the number of edges refers to the number of edges on the boundary of a face.)

"For example, a common way to assign a charge of 6deg(v) to each vertex v, where deg(v) is the degree of the vertex."

The second is the discharge rules, which are a set of rules designed to allow charge to transfer between vertices or faces.

If a vertex has a lower degree, it can borrow charge from adjacent vertices with higher degrees; the higher-degree face distributes charge to its adjacent lower-degree face.

"Finally, there is the analysis after charge adjustment."

After applying the discharge rules, examine the final charge of each vertex or face. By analyzing the charge distribution, it can be proven that certain specific configurations in the graph, such as certain subgraphs or loops, necessarily exist, or that certain properties necessarily hold.

Lin Ran concluded, "In the end, all we need to do is apply the discharge method to the four-color problem."

First, using Euler's formula for a planar graph, V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces, we can deduce that the average facets must be less than 6.
Therefore, we can assign an initial charge of def(f)-6 to each face f, where def(f) is the degree of the face.

The discharge rules then allow charge transfer between surfaces or between a point and a surface. Through the discharge process, we can demonstrate that certain specific configurations lead to the appearance of negative charges. These configurations constitute an unavoidable set, meaning that any planar diagram contains at least one of these configurations.

Therefore, in proving the four-color theorem, we only need to find a set containing a finite number of configurations using the discharge method, and then further verify the reducibility of these configurations, ultimately proving the four-color theorem.

After Lin Ran finished speaking, everyone understood what he was saying, but like Lin Ran, they felt that the job was too tedious.

It's something where you can find a method, but you might never figure it out in your entire life.

"I know that people will think that the method I proposed is nonsense because the amount of computation is too huge. Human mathematicians may not be able to find the result even after a lifetime of research."

But I want to remind everyone that we now have tools like computers.

I believe that with the help of computers, we can solve this problem in a very short time, perhaps one or two years.

The Four Color Problem was originally supposed to be fully proven in 1976 by mathematicians Kenneth Appel and Wolfgang Haken using a computer.

The method they used was the same one Lin Ran mentioned – the discharge method.

However, compared to Lin Ran, these two are clearly far less famous.

Therefore, after Lin Ran made the suggestion, no one questioned it. Those who had heard of computers were thinking about how to use them to solve the problem, while those who hadn't were asking what computers were.

To elaborate further, Appel and Haken used an IBM 370-168 computer, released in 1972, to solve the Four Color Problem, and it took them a total of 1200 hours.

However, this does not mean that the current IBM 7090 cannot solve the problem.

The IBM 7090's 128KB of memory was insufficient to store all configurations and intermediate results simultaneously, so data was processed in batches and relied on magnetic tape for storage.

Configuration data and verification results can consume a lot of storage space. Intermediate results can be stored on magnetic tape to ensure the integrity of the data during the calculation process.

"I hope that at the Mathematicians' Congress four years from now, we will hear the good news that the Four Color Theorem has been solved," Lin Ran concluded.

Lin Ran's academic report was like music to the ears for mathematicians who understand computers; it was like clearing away the fog and seeing the results directly.

The more I learn about computers, the more I want to rush back to research institutes or schools to start proving the four-color problem.

You don't even need to think of a method yourself; Lin Ran has already written it out very clearly.

He didn't even want to attend subsequent mathematicians' congresses anymore.

Whoever gets the result first will prove the Four Color Theorem, which has puzzled mathematicians for over a hundred years.

This is Lin Ran giving out perks.

To a mathematician unfamiliar with the Four Color Theorem, what you're saying is anything but basic.

Doyle could understand what Lin Ran was saying, but he was already dumbfounded. Before Lin Ran returned to his seat, he turned to Siegel and said, "Professor, aren't you going to remind Randolph that after you finish your work, you don't necessarily have to explain your thought process at the Mathematicians' Congress?"
Furthermore, even when sharing one's own ideas, one shouldn't say that one's thinking isn't thorough enough, that there might be problems, or that there are some interesting ideas that need improvement, and ask everyone to help think about them and see if they can be perfected.

Instead of having already come up with a solution, contributing the solution so that others can directly solve the problem?
This is a four-color problem!

The Four Color Theorem is a very easy problem to understand, even for laymen. There are very few problems that are both interesting and valuable.

Solving one problem means one less problem to solve.

Moreover, the four-color problem has been around for over a hundred years.

Such a mature solution to the problem was not used by himself. Even if he didn't use it, he could have left it for students or provided it to other collaborators in the school. Instead, Lin Ran made it public.

"This is what it means to have the demeanor of a master," the young mathematicians present thought to themselves.

Doiling, the head of the mathematics department, was heartbroken; such a solution had been given away for nothing.

After all, when it comes to the application of computers, Göttingen can't compare to universities like America.

Universities like NYU and Columbia have collaborative labs with IBM. What do they have to compare with?

Siegel said, "Randolph was thinking about the entire mathematics community, not just Göttingen."

You need to broaden your perspective. Randolph helped mathematicians prove the Four Color Theorem using computers, and Göttingen also contributed to that!

Indeed, Siegel has become increasingly adept at dealing with Doilyn's complaints.

It's useless to criticize me. As long as Randolph is still my student, the more successful he is, the more Göttingen will benefit.

As long as Doilyn cannot break this logic, Siegel remains invincible.

"Stockholm, August 1962 — At the 8th International Congress of Mathematicians (ICM) in Stockholm, Sweden, Randolph Lin, a brilliant mathematician of Chinese descent and one of the most prominent figures at the congress, did not disappoint the attendees."

Randolph was invited to give an academic presentation, where he announced his solution to the Four Color Problem, a famous unsolved problem in the history of mathematics. This groundbreaking achievement not only opened a new chapter in the field of graph theory but also won him widespread praise from the mathematicians present.

The Four Color Theorem, originating in 1852, proposes that any planar map can be colored with only four colors, ensuring that adjacent regions are different colors. This conjecture has perplexed the mathematical community for over a century; despite numerous attempts, a convincing and rigorous proof has remained elusive. Randolph's new method presented at this conference details his innovative mathematical approach, combined with profound insights from graph theory and computing. His proof, with its simplicity and innovation, astonished the mathematicians present.

Conference Chairman and renowned mathematician Lennart Carlson commented: "Randoff's work embodies a perfect combination of mathematical creativity and rigor, demonstrating that mathematics must keep pace with the times and integrate with new tools."

Lin Ran's academic report at the opening ceremony quickly spread globally through newspapers.

Unlike Appel and Haken, whose solution to the Four Color Problem was met with widespread skepticism in the mathematical community because it was proven by computer and was not accepted by mathematicians until more than a decade later when it was finally compiled and published, Lin Ran's solution quickly gained unanimous approval, with everyone agreeing that his method was indeed a solution to the Four Color Problem.

This illustrates the different reactions within the mathematical community to the solutions proposed by masters and non-masters to the same problem.

Just like the ABC conjecture, if Shinichi Mochizuki claims to have proven it, the mathematics community, unable to understand his paper, will say it's questionable, but won't directly reject it. If another unknown mathematician claims to have proven it and then pulls out a bunch of things that are completely incomprehensible, the academic community will simply reject your paper.

Having a reputation and not having a reputation are two completely different things.

That's just how realistic the mathematics community is.

On August 22, 1962, the final day of the 8th International Congress of Mathematicians, the award ceremony was solemnly held at the Stockholm Concert Hall.

The building, known for its blue exterior and elegant design, welcomed hundreds of mathematicians, scholars, and guests from around the world to witness the awarding of the highest honor in mathematics.

At 3 p.m. sharp, the ceremony commenced with melodious orchestral music. The orchestra played a section from Hugo Alfvén's Symphony No. 1, a solemn and powerful melody that added a touch of ceremony to the upcoming awards presentation.

As the music faded, the conference chairman, Lennart Carlson, slowly walked onto the stage. He was dressed in a black tailcoat, smiling, and waving to the audience.

Lennart Carlson addressed the audience in a deep and clear voice: “Ladies and gentlemen, welcome to the award ceremony of the 1962 International Congress of Mathematicians. Today, we not only celebrate the brilliant achievements of mathematics, but also witness the pinnacle of human wisdom.”

His opening remarks ignited the enthusiasm of the entire audience, who erupted in enthusiastic applause.

Everyone gave Lin Ran a lot of face, which was out of respect for his wisdom.

After a brief review of the conference's academic highlights, the awards ceremony officially began. Lennart Carlson announced that this year's Fields Medal would be awarded to a "young scholar who has changed the face of mathematics through extraordinary talent and innovation."

After he finished speaking, the same name flashed through everyone's minds:

Randolph Lin

As Lin Ran's name was called, thunderous applause erupted throughout the venue.
A slender young man with a calm expression stood up. He wore a dark gray suit with a slightly loose tie, exuding a casual air.

"Professor, congratulations." Jenny, who was sitting next to Lin Ran, promptly offered a cheek kiss.

Lin Ran walked slowly towards the stage, each step accompanied by the gaze of the audience.

After Lin Ran stepped onto the podium, Lennart Carlson shook his hand and presented him with a gold medal. The medal was engraved with "ICM 1962," and the reverse side featured Euclid's portrait, symbolizing the enduring legacy of mathematics. Following this, a gold-embossed certificate was handed to him, inscribed in Latin and English: "Awarded to Randolph Lin for his outstanding contributions to Fermat's Theorem."

(End of this chapter)