Mathematics in the 20th Century

Michael Atiyah (translated by Bai chengming, translated by Zhou wewei and Feng huitao)

Thank you for inviting me to join this activity. of course, if someone wants to talk about the end of a century and the beginning of the next century, he has two very difficult options: one is to review the mathematics of the past century; the other is the prediction of the Development of mathematics over the next century. I chose this difficult task. Anyone can predict the future and we cannot determine whether it is right or wrong. however, everyone can raise an objection to any previous review.
What I am talking about here is my personal point of view. this report cannot contain all the content, especially some important content that I don't plan to cover, partly because I am not an expert in those fields, this is partly because they have been commented on elsewhere. for example, I won't talk about famous events that occur in the field of logic and computing. These events are often related to great names such as Hi1bert, Gödel, And Turing, in addition to the application of mathematics in basic physics, I will not talk too much about other applications of mathematics. This is because the application of mathematics is too broad and requires special discussion. A dedicated report is required for each aspect. maybe everyone else at this meeting
The report will hear many speeches about the content. in addition, trying to list some theorems, even the names of famous mathematicians in the past one hundred years, is meaningless. It is just a boring exercise. so, instead of them, I tried to choose theme that I think is important in many aspects to discuss and emphasize what happened around these theme.
First, I have a general description. century is an approximate digital concept. we don't really think that during the entire one hundred years, some things will suddenly stop and start over again. So when I describe the mathematics of the 20th century, some content may actually be cross-century, if an incident occurred in 1890s and lasted until the beginning of the 20th century, I would not care about the time details. what I do is like an astronomy working in an approximate digital environment. in fact, many things began in the 19th century, but in the 20th century, they were fruitful.
One of the difficulties in this report is that it is difficult to put ourselves back as a mathematician in 1900, this is because the mathematics of the last century has been absorbed by our culture and ourselves. it is hard to imagine what the Times look like without our terms. in fact, if someone has a really important discovery in mathematics, then he will be ignored with it! He will be fully integrated into the background, so in order to review the past, we must try our best to imagine the situations when people think about problems in different ways in different times.

From local to overall
At the beginning, I prepared to list some topics and discuss them around them. the first theme I am talking about is the transition from local to overall. in the classical period, people have basically studied things using local coordinates in a small scope. in this century, the focus has shifted to trying to understand the nature of things as a whole and in a wide range. because the overall nature is more difficult to study, most of them only have qualitative results, and the idea of topology becomes very important. it was Poincaré who not only made a pioneering contribution to the development of topology, but also predicted that topology would be an important part of mathematics in the 20th century. By the way, let me mention it, the well-known issue, Hilbert, was not aware of this. topology is difficult to find specific representations in his questions. but for Poincaré, he can clearly see that topology will become an important part.
Let me try to list some fields and then everyone will know what I'm thinking. for example, let's take a look at the complex analysis (also known as "function theory"), which was the center of mathematics in the 19th century and a great character like Weierstrass.
Center. for them, a function is a complex variable function; For Weierstrass, a function is a power series. they are things or formulas that can be used to write down and clearly depict. functions are some formulas: they can be explicitly written down. however, next Abe1, Riann and many people's subsequent work left us away from this, so that functions can be defined without explicit formulas, but more by their overall nature: through the distribution of their singular points, through their fixed domain location, through their value range. these general properties are unique in a specific function. partial expansion is just a way to look at them.
A similar thing occurs in a differential equation. At first, people need to find a definite local solution to solve a differential equation! Something that can be written down. with the development of things, the solution does not have to be an explicit function. People do not have to use good formulas to describe them. the singularity of the solution is something that really determines its overall nature. this spirit is similar to what is happening in semi-analysis, but it is a little different in detail.
In the differential ry, the classic work of Gauss and others describes the space of small pieces, the curvature of small pieces, and the local equation used to describe local ry. as long as people want to understand the overall image of the surface and the topology that is accompanied by them, it is natural to change from these classical results to a wide range. when people are from small to large, the most meaningful nature is the nature of the topology.
Number theory also has a similar development, although it is not obviously applicable to this framework. in this way, number theory scientists differentiate what they call "Local theory" and "Overall Theory": the former is when they discuss a single prime number, a prime number at a time, and a finite prime number; the latter is when they discuss all prime numbers at the same time. the similarity between the prime number and the point, between the local and the whole, plays a very important role in the development of number theory, and those ideas generated in the Development of topology have a deep influence on number theory.
Of course, this situation also occurs in physics. Classical physics involves local theory. At this time, we write a differential equation that can fully describe the small-range nature. Next we must study the large-scale nature of a physical system. all physics involves is that when we start from an elementary perspective, we can know what is happening in a large scope, predict what will happen, and move forward along these conclusions.

Increase in dimension

My second topic is somewhat different. I call it the increase of dimension. let's start from the classic theory of complex variable functions again: the classic theory of complex variable functions is mainly to discuss and refine a complex variable theory. the promotion of two or more variables basically takes place in this century and is in the field where new phenomena occur. not all phenomena are the same as a variable. Here there are completely new features, and the theoretical study of n variables is becoming increasingly dominant. This is also one of the main achievements of this century.
On the other hand, in the past, differential physicists mainly studied curves and surfaces. Now we are studying the ry of the n-dimensional manifold. If you think about it carefully, we can realize that this is an important change. in the early days, curves and curves were what people could see in space. the high dimension has a little fictional component, which people can imagine through mathematical thinking, but at that time people may not take them seriously. take them seriously and study them with the same degree of attention
This idea is actually a product of the 20th century. similarly, there is no clear evidence that our pioneers in the 19th century have considered the increase in the number of functions. The research is not just a single but a few functions, or vector-valued function ). so we can see that there is a problem with the increase in the number of independent and non-independent variables. linear Algebra always involves multiple variables, but its increase in dimensions is more dramatic. Its increase is from finite dimension to infinite dimension, from linear space to the Hilbert space with infinite variables. of course, this involves the analysis. After multiple variable functions, we have function, that is, function. they are functions in function space. essentially, they have an infinite number of variables. This is what we call the theory of Variational. A similar thing occurs in the general (non-linear) function.
The development of the number theory is an old topic, but the remarkable achievements have been made in the 20th century. This is my second topic.

From swap to non-swap

The third topic is the transition from exchange to non-exchange. this may be one of the most important features of mathematics in the 20th century, especially on behalf of mathematics. the non-exchange aspects of algebra have become extremely important. Of course, it originated from the 19th century. it has several different origins. hamilton's work on the Quaternary element may be the most amazing and has a huge impact. In fact, this is inspired by the idea used to handle physical problems. there is also Grassmann's work on external algebra. This is another algebra system and has now been incorporated into our theory of differential forms. of course, there are also work on the matrix based on linear algebra and Galois work on group theory.

All of these forms form the cornerstone of introducing non-exchange multiplication into the algebra theory in different ways, and I vividly describe them as the "bread and butter" on which the algebra machines of the 20th century depend ". we can not think about this now, but in the 19th century, all the above examples have made major breakthroughs in different ways. Of course, these ideas have developed dramatically in different fields. the Application of matrix and non-exchange multiplication in physics produces quantum theory. heisenberg is one of the most important application examples of non-exchange algebra in physics for non-exchange relations.
N is extended to his theory of operator algebra.

Group theory is also an important theory in the 20th century. I will come back to discuss it later.

From Linear to nonlinear

My next topic is a transformation from linear to nonlinear. most or basically linear of classical mathematics, or even if it is not very accurate, it is also the approximate linearity that can be studied through certain disturbance expansion, it is very difficult to deal with the real non-linear phenomenon, and only in this century, we have done a real research on it within a large scope.
Let's start with ry: Euclid ry, plane ry, space ry, straight line ry, all of which are linear. from the different stages of non-European ry to the more general emann ry of Riann, the discussions are basically non-linear. in the differential equations, the real research on non-linear phenomena has processed many new phenomena that we cannot see through classical methods. here I will only give two examples, isolated child and chaos, which are two very different aspects of the theory of differential equations. This century has become an extremely important and well-known research topic. they represent different extremes. an isolated child represents an unpredictable and Organized behavior of a non-linear differential equation, while chaos represents an unpredictable ungrouped behavior (disorganized behavior ). the two are very interesting and important in different fields, but they are basically non-linear. we can also trace the early history of some isolated jobs to the lower part of the 19th century, but that is only a small part.
Of course, in physics, the Maxwell equation (basic equations of electromagnetism) is a linear partial differential equation. the famous Yang-Mills equations, which are non-linear equations and are assumed to be used to regulate the forces related to the material structure. these equations are non-linear because the Yang-Mills equation is essentially a matrix of the Maxwell equation, and the fact that the matrix cannot be exchanged leads to a nonlinear term in the equation. so here we see an interesting connection between non-linearity and non-swapping. non-exchange produces a special type of non-linearity, which is indeed very interesting and important.

Ry and Algebra

Now I am talking about some general theme. Now I want to talk about a binary cross in mathematics, but it is always with us, this gives me the opportunity to come up with some philosophical thoughts and explanations. I am referring to the division between ry and algebra. ry and algebra are two pillars of mathematics and have a long history. ry can be traced back to ancient Greece or even earlier, while modern mathematics originated from the ancient Arabic and ancient Indians. therefore, they have all become the foundation of mathematics, but there is a natural relationship between them.

Let me start with the history of this issue. euc1id ry is one of the earliest examples in mathematical theory until Descartes introduces algebraic coordinates in what we now call the flute plane, it has always been a Pure Geometric Descartes approach is an attempt to convert geometric thinking into algebraic operations. from the perspective of modern mathematicians, this is of course a major breakthrough or a major impact on the geometry. If we compare the analysis work of Newton and Leibniz, we will find that they belong to different traditions. Newton is basically a ry, and Le1bniz is a typical mathematician, which has profound principles. for Newton, ry, or calculus developed by him, is a mathematical attempt to describe natural laws. he cares about physics in a wide range and physics in the geometric world. in his opinion, if someone wants to understand things, he will have to think about it from the perspective of the physical world and look at it from the perspective of geometric images. when he develops calculus, he wants to develop a manifestation of calculus that is as close as possible to the physical meaning hidden behind it. therefore, he uses geometric arguments, because this can maintain a close relationship with the actual meaning. On the other hand, Leibniz has a goal, an ambitious goal, that is, formalizing the entire mathematics, turn it into a huge algebra machine. this is totally different from Newton's path, and they have many different marks. as we know, in this big debate between Newton and Leibniz, the mark of Leibniz finally wins. we still use his mark to write partial derivatives. newton's spirit remains, but it has been buried for a long time.

In the late 19th century, one hundred years ago, Poincaré and Hilbert were two main figures. I have mentioned them before, and I can roughly say that they are the descendant of Newton and Leibniz, respectively. Poincaré's thought is more about the spirit of ry and topology, he uses these ideas as his basic insight tools. hilbert is more of a formalist. He wants to be public, formal, and give a strict, formal description. although none of the great mathematicians can easily be attributed to any category, it is clear that they belong to different traditions.

When preparing this report, I think I should write down the names of representative people who can inherit these traditions in our current generation. it is very difficult to talk about healthy people. Who should put them on this list? Then I secretly thought about it: Who cares which side of the famous list will be put on? So I chose two names: Arnold Bourbaki, who is the successor of the Poincaré-Newton tradition, and the latter, which I think is the most famous successor of Hilbert. arnold uncertain believes that his viewpoint of mechanics and physics is basically geometric and originated from Newton. He thinks that there is something between them, it is a misunderstanding except for a few people like Riann (he does deviate from both. bourbaki strives to continue his research on the form of Hilbert, which has pushed mathematics to a remarkable scope and achieved some success. each viewpoint has its advantages, but it is difficult to reconcile them.

Let me explain how I think about the difference between ry and algebra. of course, ry is about space, which is beyond doubt. if I face the audience in this room, I can see a lot in a second or within a microsecond and receive a lot of information. Of course this is not an accidental event. the construction of our brain has an extremely important relationship with vision. I learned from some neurophysiology friends that vision occupies 80% or 90 of the cerebral cortex. there are about 17 centers in the brain, each of which is specifically used to take charge of different parts of visual activity: Some part involves vertical and some part is related to the horizontal direction, some are about color and perspective, and the last part involves the specific meaning and explanation of what you see. understanding and perceiving the world we see is a very important part of our human evolution. therefore, spatial intuition or spatial perception is a very powerful tool, and it is also the reason why geometry occupies an important position in mathematics, it can be used not only for things with obvious geometric properties, but even Things without obvious geometric properties can also be used. we strive to attribute them to geometric forms, because this allows us to use our intuition. our intuition is our most powerful weapon. especially when explaining a data type to students or colleagues, you can see it clearly. when you explain a long and difficult argument, the students finally understand it. what Will students say at this moment? He will say, "I saw it (I understand it )!" We can also use the word "Perception" to describe them at the same time. At least this is correct in English, comparing this phenomenon with other languages is equally interesting. I think one thing is very basic: humans get a lot of information through this huge capability and visual instantaneous activity, so as to develop, and the teaching is involved and perfected.

On the other hand (maybe some people do not think so), algebra essentially involves time. no matter which type of algebra we are doing now, a series of operations are listed one by one. Here "one by one" means we must have the concept of time. in a static universe, we cannot imagine algebra, but the essence of ry is static: I can sit here and observe without any changes, but I can continue to observe. however, algebra is related to time because we have a series of operations. Here, when I talk about "Algebra", I don't just refer to modern algebra. any algorithm or any computing process is followed by a series of steps. The development of modern computers makes it clear. modern computers use a series of 0 and 1 to reflect their information and give the answer to the question.

Algebra involves time operations, while ry involves space. they are two vertical aspects of the world, and they represent two different ideas in mathematics. therefore, the arguments or dialogs between mathematicians about the relative importance of algebra and ry in the past represent some very basic things.

Of course, it is not worthwhile to argue Which side has lost and which side has won. when I think about this, there is an image analogy: "Would you like to be a mathematician or a physicist?" The question is like: "Are you willing to be a blind man or a blind man ?" Same. if a person's eyes are blind, the space will be invisible. If the person's ears are deaf, he will not be able to hear it. Hearing occurs in time. In general, we would rather have both.

In physics, there is also a similar, roughly parallel division between physical concepts and physical experiments. physics has two parts: Theory-concepts, ideas, words, laws-and experimental instruments. I think concepts are geometric in a broad sense, because they involve things that happen in the real world. on the other hand, the experiment is more like an algebraic computation. it takes people time to do things, determine some numbers, and substitute them into the formula. however, the basic concept behind the experiment is part of the geometric tradition.

In a more philosophical or literary language, algebra is the so-called "dedication of Fushi" for physicists ". as we all know, in the story of Goethe, the devil can get what he wants (the love of a beautiful woman) at the cost of selling his soul, algebra is the offering provided by the devil to mathematicians. the devil will say, "I will give you this powerful machine that can answer any of your questions. what you need to do is to give your soul to me: Give up the ry and you will have this powerful machine "(now you can think of it as a computer !). Of course we want to have them at the same time. We may be able to cheat the devil and pretend that we sell our souls, but don't really give it. however, the threat to our soul still exists, because when we transfer it to algebraic computing, We will essentially stop thinking, stop thinking about the problem with the concept of ry, and stop thinking about its meaning.

Here I am talking about the power of mathematicians, but the goal of basic soil and algebra is always to create a formula, put it in a machine, and turn the hand to get the answer. that is to say, take a meaningful thing, convert it into a formula, and then get the answer. in such a process, people no longer need to think about the ry corresponding to these different stages of algebra. in this way, insights are lost, and this is very important in those different stages. we must never give up on these insights! In the end, we still need to go back to this point. This is what I 've talked about.
I'm sure this lecture is sharp.

The choice of ry and algebra leads to the emergence of cross topics that can combine the two, and the difference between algebra and ry is not as straightforward and unpretentious as I have mentioned. for example, modern mathematicians often use digraphs ). what can a schema be except Geometric Intuition?

General Technology

Now I don't want to talk too much about content-based topics, but I want to talk about theme based on the technologies and common methods I have used, that is to say, I want to describe some common methods that have been widely used in many fields. the first one is:

Same tone theory

In history, the same tone theory is developed as a branch of topology. it involves the following situations. there is a complicated topology space. We want to obtain some simple information about it, such as the number of its holes or similar things, and obtain some linear constants associated with it. this is a construction of a dry linear invariant under nonlinear conditions. from the geometric point of view, the closed chain can be added or subtracted, so that we can get a space group. in the same tone theory, as a basic algebra tool for retrieving certain information from the topological space, it was found in the first half of this century. this is an algebra that benefits a lot from ry.

The same tone concept also appears in other aspects. another source of the family can be traced back to the study of Hilbert and its Multi-number. polynomials are non-linear functions, and they can be multiplied to obtain a polynomial of a higher number. it was Hilbert's great insight that prompted him to discuss the "ideal", linear combination of Polynomials with common zero points. he is looking for these ideal products. there may be many generated yuan. he examines the relationships between them and their relationships. so he got a hierarchy of these relationships, which is called the "Hilbert system ". the theory of Hilbert is a very complicated method. He tries to convert a non-linear case (polynomial Study) into a linear case. in essence, Hilbert constructs a complex system with linear relationships. some information about non-linear things such as polynomials can be included.

This algebra theory is actually parallel to the above topology theory, and now they have been integrated to form the so-called "same-tone Algebra ". in algebra, one of the greatest achievements of the 1950s s was the development of the layer-based simultaneous adjustment theory and its extension in analytic ry, which was developed by Leray, Cartan, the French school of Serre and Grothendieck. from this, we can feel a topological idea that has both Riemann-Poincaré and a Hilbert algebra, coupled with the fusion of some analysis methods,

This indicates that the same tone theory is also widely used in other branches of algebra. we can introduce the concept of simultaneous group, which is usually a linear thing related to non-linear things. we can apply it to group theory, such as finite groups and Lie algebra: they all have the same group. in terms of number theory, the Galois group produces very important applications. therefore, in a wide range of circumstances, simultaneous discussion is one of the powerful tools. It is also a typical feature of mathematics in the 20th century.

K-Theory

Another technology I want to talk about is K-theory ". it is similar in many aspects to the same tone theory, and its history is not long (it did not appear until the mid-20th century, although some aspects of its origins may be traced back to earlier ), however, it is widely used and has penetrated into many parts of mathematics. in fact, K-theory is closely related to the expression theory. The expression theory of finite group can be explained, which originated in the 19th century. however, its modern form-K-theory has only a relatively short history. k-theory can be understood in the following way: it can be considered an attempt to apply matrix theory. we know that the multiplication of a matrix is not interchangeable, so we want to construct an Interchangeable Matrix or linear invariant. trace, dimension, and determinant are interchangeable constants in matrix theory, while K-theory is a systematic method that attempts to process them. It is also called "Stable linear algebra ". the idea is that if we have many matrices, we can place two unchangeable matrices A and B on the orthogonal positions of different blocks, and they can be changed, because in a large space, we can move objects at will. therefore, in some approximation cases, this is very beneficial, which is enough for us to get some information. This is the cornerstone of K-theory as a technology. this is similar to the same theory. Both of them obtain linear information from complex nonlinear situations.

In algebraic ry, K-theory is first introduced by Grothendieck and has achieved great success. Some of them are closely related to the layer theory we just talked about, it is also closely related to his work on the Riemann-Roch theorem.

In terms of topology, Hirzebruch and I copied these ideas and applied them to a pure topology. in a sense, if the work of Grothendieck is related to the work of Hilbert in combination
Our work is closer to Riemann-Poincaré's work on simultaneous calls. We use continuous functions, and he
Polynomial. K-theory plays an important role in the study of elliptic operator index theory and linear analysis.

From another different perspective, Milnor, Quillen and others have developed K-theory algebra, which has great potential in the study of number theory. the development in this direction has led to many interesting problems.

In terms of functional analysis, the work of many people, including Kasparov, extends the continuous K-theory to non-exchange C *-algebra. A spatial continuous function forms an exchange algebra in the product of a function. however, in other circumstances, similar discussions about non-exchange situations are naturally generated. At this time, functional analysis is naturally a hotbed of these problems.

Therefore, K-theory is another field that can be used to process a wide range of different aspects of mathematics with this simple formula, although in every situation, there are many very difficult and skillful problems that are specific to this aspect and can be connected to other parts. k-theory is not a unified tool. It is more like a unified framework with analogy and similarity between different parts.

Much of this work has been promoted by Alain Connes to "non-exchange differential ry ".

Interestingly, Witten recently found that many interesting methods are related to K-theory through his work on string theory (the latest idea of basic physics, and K-theory seems to provide a natural "home" for the so-called "constant ". although the same tone theory was regarded as the natural framework of these theories in the past, it seems that K-1 theory can provide better answers.

Li Qun

The other concept is not only a technology, but also uniformity. now speaking of Li Qun, we basically refer to Orthogonal Groups, you groups, Xin groups, and some exception groups. They play a very important role in the history of mathematics in the 20th century. they also originated in the 19th century. sophusLie is a Norwegian mathematician in the 19th century. as many people have said, he and Fleix Klein, among others, have promoted the development of "continuous group theory. for Klein, at the beginning, this was an attempt to uniformly process Euclid and non-euro ry. although this subject originated from the 19th century, it started in the 20th century.
The problem is integrated into the unified framework of research. Li Qun's theory has a profound influence on the 20th century.

Let me talk about the geometric importance of Klein's thoughts. for Klein, ry is homogeneous space where objects can move freely and keep shapes unchanged. Therefore, they are controlled by a related symmetric group. the Euclid group gives the Euclid ry, while the hyperbolic ry originates from another Li Qun. therefore, each homogeneous ry corresponds to a different Li Qun. but later, with the further development of the geometric work of Riann, people are more concerned about those non-homogeneous ry. At this time, the curvature changes with the change of location, and the space does not have the overall symmetry, however, Li Qun still plays an important role, because we have Euclid coordinates in the cut space, so that Li Qun can appear on an infinitely small level. as a result, from an infinitely small perspective, Li Qun appeared again. However, to differentiate the differences between different positions, we need to use a certain method to process different Li Qun to move objects. this
The theory is developed by Eile Cartan and becomes the cornerstone of modern differential ry.
The theory of relativity of ein also plays a fundamental role. Of course, the theory of Stein has greatly promoted the full development of differential ry.

In the 20th century, the general nature I mentioned above involves the Li Qun and differential ry at the overall level. A major development is to give the so-called "indicative class" information. This symbolic work is provided by Borel and Hirzebruch. The indicative class is a topological invariant and integrates three key parts: Li Qun, differential ry and flutter, of course, also contain algebra related to the group itself.

In terms of analytical taste, we have obtained the theory that is now called non-exchange harmonic analysis. this is an extension of the Fourier theory. For the latter, the Fourier series or the Fourier Integral essentially correspond to the switching Li Qun of the circumference straight line. When we replace them with more complex Li Qun, we can get a very beautiful and exquisite theory that integrates Li Qun's representation theory and analysis. this is essentially the work of Harish-Chandra throughout his life.

In number theory, the entire "Lang1ands program" is now called by many people and is closely related to Harish.
-The Chandra theory is originated from the Li Qun theory. for every Li Qun, we can give the corresponding number theory and implement the Langlands program to some extent. in the second half of this century, a large number of work in algebraic number theory was greatly affected. formal research is a good example. This also includes Andrew Wiles's work on the Fermat theorem.

Some people may think that Li Qun is only important in the geometric field, because it is out of the need of continuous variables. however, this is not the case. Similar discussion of Li Qun in a finite field can give a finite group, and most of them are produced in this way. therefore, some techniques of the Li Qun theory can even be applied to discrete situations such as finite fields or local fields. there is a lot of work in this field of pure algebra, such as the work associated with the name of George Lusztig. in these work, the Representation Theory of finite groups has been discussed, and many of the technologies I have mentioned can also be used here.

Finite Group

The above discussion has brought us to the topic of Finite Groups, which reminds me that the classification of finite single groups is a task that I must admit. many years ago, when the classification of a limited single group was just to be completed, I was interviewed and asked my views on the classification of a limited single group, I rashly said that I didn't think it was so important. my reason is that the results of the limited single group classification tell us that most single groups are known, and there is a table related to several exceptions. in a sense, this is just the end of a field. I am not very excited when things are replaced by the end. however, many of my friends who work in this field will naturally feel very unhappy when I hear this. Since then, I have to wear bullet-proof clothing.

In this study, there is an advantage that can make up for shortcomings. I actually refer to the one with the largest "Magic group" name in all the so-called "sporadic groups. I think the discovery of magic itself is the most exciting result of the limited single group classification. it can be seen that the magic group is extremely interesting.
And are still being understood. it has an unexpected relationship with a large part of many branches of mathematics, such as the relationship with elliptical model functions or even theoretical physics and quantum field theory. this is an interesting by-product of classification. as I said, the limited single group category closed the door, but the magic Group opened a door.

Physical impact

Now let me turn the topic to a different topic, that is, to talk about the impact of physics. throughout history, physics and mathematics have a long connection, and most mathematics, such as calculus, have developed to solve physical problems. in the middle of the 20th century, with most pure mathematics still developing well independently from physics, this influence or connection may become less obvious. but in the last 1/4 of this century, things have undergone dramatic changes. Let me try to briefly review the interactions between physics and mathematics, especially with ry.

In the 19th century, Hamilton developed the classical mechanics and introduced the formalization of the Hamilton quantity. classical Mechanics is now known as the "Xin ry ". this is a branch of ry. Although some people have studied it for a long time, this topic has not actually been studied until the last two decades. this is already a very rich part of ry. ry, which I use here means that it has three branches: the Riann ry, the complex ry, And the Xin ry, and corresponds to three different types of Li Qun respectively. xin ry is the latest of them, and in a sense it may be the most interesting. Of course it is also closely related to physics, this is mainly because its historical origins are related to Hamilton mechanics and Its Relationship with quantum mechanics in recent years. as mentioned above, the Maxwell equation, which serves as the basic Linear Equations of electromagnetism, is the source power of Hodge's work in harmonic form and in algebraic ry. this is a very fruitful theory, and has become the basis of a lot of work in ry since the 1930s S.

I have mentioned the general theory of relativity and the work of Stein. quantum Mechanics also provides an important example. this is not only reflected in the relationship with ease, but also in the emphasis on the Hilbert space and spectral theory.

In a more specific and obvious way, the classical form of crystallization is related to the symmetry of the crystal structure. the first example to be studied is the finite symmetry group that occurs around the point, which is in view of their application in crystallization. in this century, the more profound application of group theory has shifted to the relationship with physics. It is assumed that the basic particles used to form matter seem to have hidden symmetry on the smallest layer. At this layer, some Li Qun is here, and we can't see this, but when we study the actual behavior of particles, their symmetry is revealed completely. therefore, we assume that a model in which symmetry is an essential element, and currently many different theories, such as SU (2) and SU (3) such basic Li Qun is integrated into it and forms the basic symmetry group. Therefore, these Li Qun appear to be the bricks and stones for Building Material buildings.

Not only do tight Li Qun appear in physics, but some non-tight Li Qun also appear in physics, such as Lorentz. it was the first time the physicist began to study the Representation Theory of Non-tight Li Qun. they are the representations that can occur in the Hilbert space. This is because, for a tight group, all non-relational representations are finite dimensions, rather than those of a tight group, they need an infinite dimension representation, this was first realized by physicists.

In the last 25 years of the 20th century, as I have just explained, there is a huge penetration from the new thinking of physics to the mathematics, which may be one of the most striking events of the Whole century, A complete report may be required for this issue. However, Basically speaking, quantum field theory and chord theory have influenced many branches of mathematics in a striking way, many new results, new ideas, and new technologies have been obtained. here, I mean that physicists have been able to predict something that is correct in mathematics through their understanding of physical theory. of course, this is not an accurate proof, but it does have strong intuition, some special cases and analogy support. mathematicians often examine the results predicted by physicists and find that they are basically correct, although it is difficult to give proof and many of them have not yet been fully proved.

So in this direction, great results have been made in the past 25 years. these results are extremely detailed. this is not like what physicists say: "This is something that should be right ". they said: "There is a clear formula, and there are the first 10 instances (involving more than 12 digits )". they will give accurate answers to complex questions. These are by no means the things that can be obtained by speculation but need to be computed by machines. quantum field theory provides an important tool, although it is difficult to understand it in mathematics, from the perspective of application, it has an unexpected return. this is an exciting event in the last 25 years.

Here I will list some important achievements: SimonDona1dson's work on Four-Dimensional manifold; VaughanJones's work on Torsion invariant; mirror symmetry and quantum group; in addition, the "magic group" I just mentioned"

What is this topic about? As I have mentioned earlier, the 20th century witnessed a transformation of dimensions and ended in an infinite dimension. physicists have gone beyond this. In terms of quantum field theory, they really try to conduct a detailed research on a wide range of infinite dimensional spaces. The infinite dimensional space they process is a typical function space. They are very complicated, not only because they are infinite dimensional, they also have complex algebra, ry, and topology, and a large Li Qun around them, that is, the infinite-dimensional Li Qun, therefore, most of the mathematics in the 20th century involves the development of ry, topology, algebra, finite-dimensional Li Qun, and manifold analysis. This part of physics involves similar processing in the infinite dimension. of course, this is a very different thing, but it is indeed a huge success.

Let me explain in more detail that quantum field theory exists in space and time. the real meaning of space is three-dimensional, but a simplified model allows us to take space into one dimension. in one-dimensional space and one-dimensional time, the typical things that physicists encounter are in mathematical language, it is a group composed of the peripheral differential and the same embryo, or a group composed of the differential ing from the circumference to a tight Li group. they are two very basic infinite dimensional Li Qun examples in quantum field theory that appear in these dimensions. They are also a mathematical thing of course and have been studied by mathematicians for a while.

In such a 1 + 1 dimension theory, we take time and space as a Riann surface and obtain new results. for example, studying the model space of a given number of Riann surfaces is a classical topic that can be traced back to the last century. however, quantum field theory has produced many new results about the simultaneous tuning of these modulo spaces. another very similar modulo space is a flat g-Confluence modulo space on the Riann surface with a deficient number G. these spaces are very interesting and quantum field theory provides some precise results about them. in particular, we can get some very
Nice formula, which involves the value of the Gini function.

Another application is related to the Count curve. if we look at the number and type of algebraic curves, what we want to know is, for example, how many curves are there after so many points, so we have to face the ry counting problem, these problems have been classic in the last century. it is also very difficult. now they have been solved by modern technology known as "quantum simultaneous tuning", which is completely obtained from quantum field theory. or we also want to get in touch with more difficult problems about curves not on the plane but on the bending family, in this way, we obtain another beautiful theory called mirror symmetry with clear results, all of which are produced in 1 + 1-dimension quantum field theory.

If we increase the dimension, that is, the 2-dimension space and the 1-dimension time, we can obtain the theory of Twist-knot invariant of Vaughan-Jone. this theory has been explained and analyzed in terms of quantum field theory.

Another result of quantum field theory is the so-called "quantum group". Now the best thing about quantum groups is their names. Clearly, they are not groups! If someone asks me about the definition of a quantum group, it may take me half an hour to explain that they are complex things, but there is no doubt that they have a deep connection with quantum theory. They originate from physics, and now they are used by down-to-earth mathematicians who actually use them for definite computation.

If we increase the number of dimensions to a four-dimensional theory (three-plus-one), this is the four-dimensional manifold theory of Donaldso, where quantum field theory has a major impact. in particular, this also led Seiberg And Witten to establish their corresponding theory, which is based on physical intuition and provides many extraordinary mathematical results. all these are some outstanding examples. there are more examples.

Next is the string theory and it is outdated! What we are talking about now is M 1 theory, which is a rich theory with a large number of mathematics, the results obtained from the research on it still need to be further digested and can keep mathematicians busy for a long time.

Summary of History

Now I will make a brief summary. Let me briefly talk about history: What happened to mathematics? I casually put the 18th and 19th centuries together and regard them as the era of classical mathematics, which is associated with people like Euler and Gauss, all the great classical mathematical results were also discovered and developed in this age. some people may think that is almost the end of mathematics, but on the contrary, the 20th century is actually very fruitful, which I have been talking about.

The 20th century can be divided into two parts. I think that the first half of the 20th century was called the "era of specialization". This is an era in which Hilbert's solutions are widely used. That is, efforts are made to normalize and define various things carefully, and always implement it in every field. as I have mentioned, the name of Bourbaki is associated with this trend. under this trend, people focus on what they can get from a particular algebra system or other systems at a specific period. in the second half of the 20th century, ye was more referred to as the "unified era". In this era, the boundaries of various fields have been broken, and various technologies can be applied from one field to another, in addition, things become more and more cross-cutting to a large extent. I think this is too simple, but I think it simply sums up some aspects of mathematics we have seen in the 20th century.

What will the 21st century be? I have already said that the 21st century is the age of quantum mathematics, or if you like it, it can be called the age of infinite-dimensional mathematics. What does this mean? The meaning of quantum mathematics is that we can properly understand the algebra of analysis, ry, topology, and various non-linear function spaces. Here, we can "properly understand ", I mean being able to give a more accurate proof of the wonderful things that physicists have inferred in some way.

Some people want to say that if we use a naive method (naive way) to study infinity and ask naive questions, we can only get the wrong answer or the answer is meaningless, the application, insights, and motivations of physics enable physicists to ask wise questions about infinity, and to make some meticulous work when there are reasonable answers, therefore, it is no easy task to analyze infinite dimensions in this way. we must follow this correct path. we have got a lot of clues, and the map has been spread out: Our goal is there, but there is still a long way to go.

What will happen in the 21st century? I want to emphasize the non-exchange differential s of Connes. alainConnes has this magnificent unified theory. similarly, it integrates everything. it integrates analysis, algebra, ry, topology, physics, and number theory, all of which are part of it. this is a framework theory, which allows us to engage in the work that differential physicists usually do in the scope of non-exchange analysis, including the relationship with topology. there is a good reason for this, because it has applications in number theory, ry, discrete groups, and Physics (with great potential or special. an interesting physical connection has just been discovered. how far can this theory go and what results can be obtained remains to be further observed. it is of course the subject that I expected to achieve significant development in at least the first decade of the next century, and find it with immature (precise) the connection between quantum field theory is completely possible.

We go to another aspect, that is, the so-called "arithmetic ry" or the Arakelov ry. It tries to unify the algebraic ry and number theory as much as possible. this is a very successful theory. it has a wonderful start, but there is still a long way to go. who knows this?

Of course, all of these have something in common. I expect physics to influence it across all places, even number theory: Andrew Wiles doesn't agree with me. In this case, only time will explain everything.

These are several aspects that I can see in the next decade, but there are also some elusive things: Return to low dimensional ry. with all the imaginary infinite-dimensional things, the low-dimensional geometric situation is a bit embarrassing. in many aspects, the dimensions we discussed at the beginning, or the dimensions at the beginning of our ancestor, left some unsolved mysteries. objects with dimensions of 2, 3 and 4 are called "low" dimensions. for example, Thurston's work in three-dimensional ry aims to give a geometric classification on a three-dimensional manifold, which is much more profound than the two-dimensional theory. thurston's program is far from being completed. It will certainly be an important challenge to complete it.

Another striking event in 3D is that the ideas of Vaughan-Jones are essentially physical work. this gives us more information about 3D, and they are almost not included in Thurston's program. connecting these two aspects is still a huge challenge, but the recent results suggest that there may be a bridge between the two. Therefore, the entire low-dimensional field is physically related, but there are too many things that are hard to understand.

Finally, I would like to mention the very important "dual" in physics ". in general, these duals come from two different implementations when a quantum theory is regarded as a classic theory. A simple example is the dual of positions and momentum in classical mechanics. in this way, the dual space replaces the original space. In linear theory, the dual space is the Fourier transformation. however, in nonlinear theory, how to replace Fourier transformation is one of the great challenges. most of mathematics is related to promoting dual in non-linear situations. physicists seem to be able to do this in an extraordinary way in their string theory and M theory. they have constructed an amazing dual instance. In a broad sense, they are the infinite-dimensional nonlinear embodiment of the Fourier transformation and seem to be able to solve the problem, however, understanding these non-linear parity seems to be one of the great challenges of the next century.

I want to talk about it here. there is a lot of work here, and I think it is a good thing for an old man like me to talk to so many of you, and I can also say to you: there will be a lot of work waiting for you to complete in the next century.

(Translated by Bai chengming, translated by Zhou xingwei and Feng huitao)