转自：Birkar教授讲座回顾：一文读懂数学之美
7月24日，菲尔兹奖得主、刚入职清华的Caucher Birkar教授为丘成桐中学生数学夏令营做题为"数学之美”的讲座。讲稿经整理并由Caucher Birkar博士后曲三太加以修正，特此刊出，与数学爱好者分享。这是一篇值得打印出来，放在案头，细细品读的文章。
I'm proud to be in China and teach Chinese students because I think education is something universal and teaching in China is as valuable and noble as teaching anywhere else in the world. So it's my pleasure to bein China and work in higher education.
— Caucher Birkar
ART OF MATHEMATICS
Today I'll talk about the art of mathematics. Mathematics has played a very fundamental role in shaping civilization. If you go back 5,000 years ago when written history started, of course there was history before that, but essentially around 5,000 years ago when civilization really took off, you can see the role of mathematics already at that very early stage of civilization.
Maths Shaping Civilization
For example, it was used in architectures. In the picture above, the Egyptian Pyramids were built about 4,500 years ago, almost at the beginning of civilization. You can see from this huge monument that one cannot build without using mathematics. It means that the Egyptians already had advanced mathematics, advanced for that age. Looking at the measures of these huge pyramids, there are only very tiny errors in it, like a few centimeters. This cannot be done without some kind of very precise mathematics. Mathematics is also used in taxes. In fact, one of the very earliest examples, the mathematical writing is about tax as tax is always about numbers and about mathematics.
Mathematics also played a very important role in how people see the world, about philosophy of viewing ourselves, as well as our interactions with nature.
If you go back, for example, to the Greece times and visit the school by Pythagoras, he tried to explain music and also the whole world using mathematics. They already understood that mathematics was at the very basis of music, and they tried to generalize that to everything in the world.
So mathematics has been very important throughout history. But in more recent times, in the last few years of human history, mathematics has been much more influential to the human society. One of the reasons is thatthe whole scientific revolution is very much based on mathematics.
During the renaissance period, scholars like Galileo made the point that mathematics was the language of the universe of the world. They already understood, of course before them there were examples using mathematics to explain natural phenomena, but around this period mathematics was used much more systematically to explain the world.
After Galileo, there was Newton who proposed the law of physics and tried to explain the solar system. But that was all using mathematics. That was one of the reasons that calculus was invented to explain how planets move.
We have the industrial revolution about some 200 years ago. Again, mathematics played a very important role there. To make machines, to manage a huge number of people and to produce products, you need mathematics because you always have a lot of numbers to look at. You have to use statistics; you have to make very precise measurements for making machines and for mechanical engineering. Naturally you need a lot of mathematics.
Maths is Everywhere
If you get closer to our time right now, you see that, for example, computers play a role in almost everything in our life. And computer itself is based very much on mathematics. In both making the computers, the hardware, and also running the computer, the software, mathematics is everywhere.
Almost every aspect of our life today is related to mathematics somehow.
Just look around ourselves, this building is not possible without mathematics because you have to make measurements about the shape of this building. You have to make measurements about the material that goes into the building. You have to know how strong they should be in a bid to make such a big building. There are a lot of mathematics already involved. If you look at this projector and the screen, you cannot make them without mathematics. Making these involves a lot of understanding of physics，chemistry and so on. And that all involves a lot of mathematics. Pretty much about everything in this room somehow has to do with mathematics.
In general, mathematics is the very base of science, especially physics. That might be the closest kind of science to mathematics, but also biology, chemistry, computer science, engineering, economics, and the health care, communication. Really almost anything today that you look at，there is mathematics somehow appearing.
Now, mathematical sciences have been used to improve our lives, the quality of life. For instance, this building allows us a better way of living. The architecture is one of the examples. In many other aspects, for example, we have high speed trains in China. Only two or three decades ago, it was much more difficult to travel around, especially to the rural areas. That's an amazing change of the quality of life.
Mathematics is also used much more in medical science and health care, for example, making vaccines. When you try to make a vaccine, you have to process a lot of data. Understanding data involves mathematics. As well as equipments, complicated machines in the hospital, you need a lot of mathematics.
Maths Solving Social Issues
Unfortunately, mathematics is quite often misused. What is a prime example of that misuse? A lot of mathematics goes into making machines that kill people. Sadly, that has been the case for a long time and still goes on.
Humansociety faces a lot of questions and problems in our time right now, for instance, the climate change. It’s one of the hottest topics. We also face a lot of health issues, for example, millions of people die every year in Africa because of malaria. Mathematics can play a fundamental role in solving this kind of questions. But, mathematics alone cannot do it. It needs a lot of humans to get involved. It needs supports, investments and so on.So it's really a shame that millions of people die because of the disease, which is solvable. It is something that mathematics can play a big role, if there are supports, if there are enough people using the mathematics.
Human needs to tackle the climate change and prevent the catastrophe. You have to understand the nature of weather and how the earth changes. You have to understand its history and how this has happened before and so on. That will also involve naturally a lot of mathematics. In general, when you have a problem in nature, in society, you try to make a model and translate that problem into a mathematical problem. That’s how mathematics works.
You face a problem, make a mathematical model, and then that generates mathematical problems. People could try to think about these problems and solve the original issues. That’s how mathematics play a big role in the human history and in human society.
Language of the Universe
If we ignore the human aspect, nature itself is very closely related to mathematics. Galileo said that the language of the universe was mathematics, and there is a good reason for that. When you try to understand what happens and understand the phenomena in nature,you always involve some types of mathematics.
That is a good reason to say that mathematics is the language of the universe. In fact, to understand life itself, how living beings emerge and evolve from the very beginning in the origin of life, involves a lot of mathematics. Especially these days, a lot of biological problems can be translated into mathematical problems, and then you can understand the very basics of life by solving these problems.
For instance, in this picture above, what you see, on the right side, is the picture of a protein. Understanding the behavior, the protein itself, and especially the structure of the protein, is very important for medical sciences.
In biochemistry, you can use mathematics, for example, algebraic geometry which is the field that I work in. Some people have tried to use that to understand the shape of the protein. Because understanding the shape of protein allows you, for examp, to design medicines in the right way. And that's why it's important to know their shapes and you can use mathematics to do that.
Now, we are experiencing the global pandemics. It’s not the first pandemics. Pandemics have been around as long as people have been around. But about 100 years ago, mathematicians and biologists started to understand pandemics in terms of mathematics. And they came up with some equations, which, in this case, happen to bedifferential equations. You have some functions, derivatives and so on. That naturally gives you a differential equation.
In general, in nature, many problems have to do with change. When you start the change, you often use differential equations because they reflect how things change. So, not surprisingly, applied mathematics uses a lot of differential equations.
Pure & Applied Mathematics
That brings us to applied mathematics. So far, I'm just trying to say how mathematics plays a role in our society and in our history. Now, there is a part of mathematics, which is called applied mathematics.
We generally divide mathematics into, in a superficial way, to applied mathematics and pure mathematics.Applied mathematics is basically driven by trying to solve problems which come from outside mathematics, from physics, economics, engineering and so on.
When you try to solve these problems, you could get into very abstract areas of mathematics. People sometimes started the specific problem for decades, and then they might even forget about the origin of the problem, why the problem was asked in the first place. Starting the problem itself leads people to new ones, new theories, new techniques and ideas, and then that itself becomes a field, which does not maybe look like what people started in the beginning. So that will take you to pure mathematics.
Pure mathematics is, in general, mathematical research, which is either driven by curiosity.It means you don't care all about applications. You forget about the origin of that problem, but just consider them as a mathematical problem.Or it can be motivated by applications to some other areas of mathematics or to outside mathematics.You have all kinds of researches in pure mathematics and you have all kinds of mathematicians. Some people are directly related to applications. Even if it is pure mathematics, some are still kind of geared toward applications. There are also people like myself who don't consider applications at all.
When I work, I don't think about application. I just think about the mathematical structure as something abstract. I have enough reasons to think about this thing, even though I don't consider the application. There are just different types of mathematicians.
There is a good reason for doing mathematics even if you don't care about applications. History proves that good mathematics, by ‘good’ I mean deep and fundamental mathematics, always find applications, sometimes maybe decades later or maybe even a few hundred years later. There are many examples.
As far as I know,there is no fundamental area of mathematics where humans regret for doing research in that area. So it always happen to be useful in some way.
Maths=Arts
Because mathematics is related to philosophy, the way we view ourselves and also the world, mathematics can be considered just as an intellectual activity, or even some kind of art.
If you go to the museums and look at some art works, paintings or sculptures and so on, you don't ask them what application of this is. It's generally something they created for its beauty or for whatever. They don't even have to tell you what it means. In most cases, they just say you will find your own interpretation. When you listen to a piece of music, you may find the meaning out of that piece of music. You don't ask the composer to explain. I can view mathematics in the same way. I create the piece of mathematics and that makes myself happy. And some time at least, it makes other people happy. That's an enough reason for doing that type of mathematics.
Important research in pure mathematics leads to new ideas, new techniques, and also new problems. But,this evolution can take time. You cannot expect it to be very fast. And creativity is one of the most important factors for being a successful mathematician. And I think that's where you get connected with arts. In arts, a really important piece of work has to have some kind of creativity. If you just copy from other people, that's not art any more. It’s the same in mathematics when you create new ideas, and then the research becomes more important. Quite often, to solve a problem which is very fundamental and hard, you need new ideas. Otherwise, other people would have solved it long before.
Even the most abstract theories can find applications. There are many examples. Here I just list two examples.
One is elliptic curve.There are many ways to define elliptic curves. For example, you can define them as Riemann surfaces of genus one, or you can consider them more algebraically as geometric spaces defined by the degree three equations inside a two-dimensional space. Anyway, these are some kinds of geometric spaces, and they happen to have applications. There are so many different areas of mathematics, including also cryptography, where elliptic curves are used very often.
Another example is the derived category.These are usually in algebraic geometry and elsewhere. They are abstract even for algebraic geometers. Algebraic geometry is considered to be very abstract inside mathematics, but these derived categories are even abstract for a typical algebraic geometer. But, they happen to have applications to explain physics. When they were created in the beginning in the 1960s, the creators did not care about physics at all.
In fact, it was created by Grothendieck and his school. They did not do physics; they just did pure mathematics. But it happened to have applications in physics. So this happens very often in mathematics. And it's kind of not very surprising.
Beauty of Maths
Mathematics itself has a lot of very nice characteristics. I want to talk about the nature of mathematics itself.
One of these characteristics is being precise.This is important. If you look at any other subjects outside mathematics, you see a lot of people arguing, fighting and so on. Look at philosophy, for example, it is supposed to be very precise. If you make any kind of claims, you can find some people supporting your claim or against it, but mathematics is very precise. When you prove something, I bet everyone agrees, or they find mistakes if enough people read the work. This is something really nice.
When something is proved in mathematics, it stays forever.Theorems of the Babylonians, written 4,000 years ago, are still true today, and we don't have to replace them with something else. But in a subject like physics, which is the closest science to mathematics, that's not true. The theory evolves always. Something is replaced by something else, like Newton’s laws are not applicable anymore, except for some cases where you don't have to be very, very precise. They are replaced by Einstein’s theory, and there are other people come to propose other theories. So it tries to evolve.
But mathematics is not like that，it stays immortal. And mathematics is beautiful and universal. It unites humanity. Mathematics was found in Mesopotamia, in Middle East，in China, in India, in Egypt and everywhere.It's a language that we all can understand and appreciate. That's important for peace.
If I can work with mathematicians from a particular country, then I don't have to fight with them because I understand there is another person in that country who thinks just like me. That's important. It can lead people to make peace.
Also, mathematics is challenging and exciting. All these things together makes mathematics something that you can enjoy.
Before high school, I liked mathematics and started mathematics. But especially in high school,I found mathematics was something that I could use to enjoy my life. There are many other things like eating and so on that everyone can enjoy. But mathematics is another thing that I can enjoy. I can spend a lot of time on doing mathematics, and then try to make my life more meaningful and more enjoyable. So why not?
A lot of people hate mathematics, but at least some people can enjoy it. That's important for us as human.
I know many rich people. I'm not rich myself yet, maybe later. The biggest problem they have in their life is that they, kind of, feel life become meaningless. They can buy a lot but they can't buy happiness. So there's always some kind of vacuum in their life.
But if you were a mathematician, you don't have to have that vacuum. Because you can fill that with doing something meaningful which is mathematics.
Mathematical structure and formula, they have some kinds of beauty, not just one kind, but some. One of the most obvious ones is just a visual beauty.
In this picture, on the right side, you have something which is a purely geometric shape. It's not even a physical thing. I think you can make it physical by drawing it.
Almost any human that looks at this picture, they find some kind of beauty. Whether you can explain that beauty or not, it's another issue. In fact, this kind of shapes are very common in middle eastern art. For religious reasons and so on, they couldn't make sculpture or paint, people and so on. Instead, they use geometric shapes to make very beautiful buildings and paintings. You can find a very large number of examples of this kind of geometric beauty.
On the left side, also, you see an architecture. This is also something almost purely mathematical.
In fact, if you know manifolds and so on, you see this is a kind of triangulation of some facets. If you know topology, then you understand it better. But, any way, you just see some pyramids, triangles, and so on. Thare is some beauty to it. Maybe not everyone like it, but at least a portion of the human society likes it. And this is just mathematical.
There is also the inner beauty to mathematics.And that's even more important to us, mathematicians. We get a visual beauty of what everyone can see, but this inner beauty takes time to understand. But by the end, when you understand it, then you see the beauty.
One example is Euler formula.The formula is a very short, very small formula. However, it's very rich. It has a lot of meanings. One of the reasons is that these numbers, e、π、i and in fact, the - 1 itself, have historical significances in human history.
For example, the number e was invented for trying to multiply very large numbers around 400 years ago. People, especially those who worked in astronomy, they had very large numbers, for say, numbers of 20 digits. They tried to multiply them. Multiplying numbers takes a lot of time. It's not easy. To make it easy, they invented this number e, which is the base of the natural logarithm. The idea is that instead of directly multiplying the two numbers you can first take the logarithm. After taking logarithm, you just need to add the two numbers. In the end, you also raise it to the power e. This is a much quicker way of calculating multiplication than just direct calculation when there was no calculator or computer.
The number π has a much longer history. It goes back at least to the Babylonians where they tried to calculate the area of the circle, and they related the area of the circle or its circumference to the radius of the circle. And then there is a direct relation between the area and the radius which is expressed in terms of π: the area is πr², and r is the radius, when they just considered the approximation of the number π. Throughout the history, people get more and more precise. Now we can express π in many digits, but we know that it's not a rational number, so you cannot write it all down. That also has a very important history.
And then there is a number i, the complex number. This was invented in order to solve cubic equations，equations of degree three. People played with something like imaginary number, i, even though they did not believe that there was such a number, but somehow, they still played with it in a very formal sense. It helped to solve equations. But later on, it appears in other areas of mathematics, and it became very common. They call it imaginary because people didn't believe they existed. In fact, even right now, most people in the world, they don't believe there is such a number simply because they don't understand how you define this kind of numbers.
Why is this equation beautiful? Because it brings the number e, π, i and also -1, which were discovered throughout the history of humans for completely different reasons at different times. You have just one very short formula bringing all of them together. That's where the beauty lies.
That's why I called it the inner beauty because it's not just visual but something when you go deeper to the equation and you see why it's beautiful.
Another example of beauty of mathematics is when you take polynomial equations they define geometric spaces by taking the roots of this polynomial.In this picture on the left side, you have a cone and then intersection of the cone with a plane. These intersections will give you several types of shapes: circle, ellipse, parabola, hyperbola, and so on. This is ancient mathematics. It goes back to at least 2,000 years ago. But later, people realize that you can write down these shapes in terms of equations.
You can describe them by polynomial equations, which are written in the picture on the right side. These shapes are defined inside the plane. There are two-dimensional spaces inside a two-dimensional space. But when you're going to higher dimension, for example, if you go to this usual three-dimensional space and you take polynomial equations, you can also define geometric spaces. And one of these spaces, a very special example of this kind of space, is called Kummer surfaces.
There's one example here,a very long equation.I have three variables. If I put numbers in place of x, y and z, I can consider these numbers as points in the three-dimensional space, and that will give me a surface, a two-dimensional shape. why it isbeautiful? One reason is that people discovered these were related to physics that were related to how light goes through a crystal.
Understanding that, people realize it is related to these purely mathematical objects which are just solutions of a polynomial equation. Much more recently, people have also used this in cryptography to make computer system secure. Below is a picture of Kummer surfaces,drawn by computer.
The shape itself has a kind of visual beauty. You just make a sculpture of this. In fact, people have already done that. In some museums, you could find the exact pictures or sculptures made out of glass or something else. There is a visual beauty that you can just consider it as a form of artistic product. But there is a much deeper and inner beauty here, which is mathematics.
Why is this mathematically interesting? One reason is that they contain 16 singular points. By ‘singular’, I mean in this shape you have points, the tiniest points where the shape shrinks just to a point. Those are singularities. This shape, this space, has exactly 16 of these singular points.
There are a lot of symmetries here, which I'm not going into details. But if you're trying to remove these singularities, you'll get a new space. These new spaces are called K3 surfaces. Why is this important? Because these K3 surfaces are also examples of another fundamental class. They are two-dimensional examples of what we call Calabi-Yau. They were essentially invented by professor Yau and are closely related to physics. There are two-dimensional examples of that kind of spaces. But on the other hand, this Kummer surface can be constructed as the quotient of something which called abelian surface.
Abelian surface is another very important class of spaces in mathematics.They are related to elliptic curves, which I mentioned before. If you take two elliptic curves and you take their products, this gives you an abelian surface. And out of that, you can also make a Kummer surface, and a K3 surface. You see that there are many fundamental types of spaces involved here together with the singularity, the resolution and so on.
This was beautiful because there are so many different notions coming together in one picture and that is what I call inner beauty. When you understand all these things, then you think that this is something beautiful and you can enjoy it.
Solving Problems
When you do math, you have to solve problems. The very act of solving problem is the kind of heart of mathematics.We learn how to solve problems already in primary school. That's a very standard way of learning mathematics. It’s just very usual and there is nothing new about it. If you go back to 4,000 years ago, you can find original text in Mesopotamia where they gave you a list of problems and people were supposed to solve these problems in order to learn mathematics. Learning mathematics, it's quite often done by solving problems. But that's the kind of easy problems.
If you consider more challenging kind of problems, then there are two scenarios. Either you cannot solve it and you get annoyed, or you can solve and enjoy it. There are two possibilities.
In this picture here, you see what we call Pascal’s zeros, which essentially says that if you have a hexagon inside a circle of ellipse, and then you have those opposite edges (you draw lines that intersect), and there are those three red points that are on one line.It's a theorem, but this can be also stated as a problem. People then ask you, for example, in a book to solve it.When you solve this, you enjoy it and feel excited.
That's very natural as it's erected in our DNA. It is a part of the human evolution that we like to solve problems. That's how we have survived as human. That's why we are very different from anything else because we have the ability of solving problems. Kids, already from the age 2 or 3, start to look around to understand their environment and to try to solve problems. It's just written in our genetics.But, we, mathematicians, take it to the extreme. We just like to solve difficult problems.
In fact, let me tell you a little story about me solving exciting and challenging problem. When I was in the first year of the graduate, we had calculus. And then the professor one day he stated a problem on the board, and I thought he was supposed to give us a solution but he could not solve it. Then he said, “okay, if anyone can solve this problem, not on the spot, you go home and you solve the problem, anyone who can solve the problem will get 50% of the marks of the final exam.” That was a very good bargain. When I went home, I immediately started thinking about this problem. I don’t remember the precise statement, but it was something about integration, not calculating an integral, but something theoretical: why integration behaves in a certain way. That's why it was more difficult because it didn't ask you to calculate integration from, for example, a to b.
I spent a few days on this problem, and then I solve it. The teacher, he kept his promise; he gave me the 50% of the mark. I just remember that it was really something very exciting just to think about this challenging problem and solving it.
Forget about the benefit of getting the 50% mark. The act, the doing itself, solving the problem, was much more enjoyable.
When you go into more advanced mathematics, you have to solve research problems, not just textbooks problems. They are usually much harder. There are many different types of research problems, but the fundamental problems usually happen to be very hard.
Other people have tried to solve and they couldn't do it for some reasons. You'renotsupposed to try to solve. It can take a very long time. In the picture above, you see two objects. The right picture is referring to resolve singularities which already appeared in an example of Kummer surfaces whichhave singularities. And by some alteration, you can remove them. This is called resolution of singularities. Now, we know that we can do that in any dimension. The prove of that is very complicated. But when you go to positive characteristics, if you know something about fields, fields of positive characteristics like finite fields, then this is still an open unsolved question.
And, on the other side, we have the picture of a cubic surface. This is a surface defined by a degree three equation. In the Kummer surface, we have degree four, but it doesn't have degree three. It is one of the very first examples people started in algebra geometry, and it's also an example of something that we call a Fano variety, one of the simple examples of two-dimensional Fano varieties. And this kind of variety is similar to Calabi-Yau. They have played a fundamental role in the evolution of mathematics in the last one hundred, even maybe two hundred years.
In my own case, one problem that I solved was related to Fano varieties. I learned about this problem of Fano variety when I was a PhD student, around and . But I solved this 13 years later in . That doesn't mean I spent all of my time, 13 years, thinking about this problem. No, but I always had it in the back of my mind. And I thought about it. I went to do some other things. I came back to think about it. And along the way, I learned more techniques. I also invented more of my own techniques. And, finally, I could solve it.
It's a very long journey, but that's where the beauty lies. That's why it is exciting because you achieve something after so many years. It's like raising a child.
You have a small child. You help them survive and grow. When they are at the age of 15 or 20, they become mature. That makes you feel proud. It's a similar feeling when you solve a very hard mathematical problem.
There is another type of mathematicians. Some people don't maybe like to solve problems or they don't spend too much time on their job, or not good at solving problems. They are much better in proposing problems.
There are some mathematicians whose main contributions are actually not following the problems but to create or invent new problems and just tell other people go and solve those.That's another way. But you cannot do that if you don't have a deep understanding of mathematics.You cannot just make up problems for no reason.
Birkar教授讲座回顾：一文读懂数学之美
