Basic mathematics:

Number theory: classical number theory, analytic number theory, algebraic number theory, transcendental number theory, model and modular function theory.

Algebra: linear algebraic group theory, group representation theory, Lie group, Lie algebra, algebraic group, typical group, homology algebra, algebraic K theory, Kac-Moody algebra, ring theory, algebra, body, lattice, ordered structure, domain theory and polynomial topological group matrix theory vector algebra tensor algebra.

Geometry: (global and local) differential geometry, algebraic geometry, manifold analysis, Riemannian manifold and Lorenz manifold, homogeneous space and symmetric space, harmonic mapping, submanifold theory, Yang-Mills field and fiber bundle theory, symplectic manifold, convex geometry and discrete geometry Euclidean geometry non-Euclidean geometry analytic geometry.

Topology: differential topology, algebraic topology, low-dimensional manifold, homoethics, singularity and catastrophe theory, point set topology, large-scale analysis of manifold and cavity complex, differential topological homology theory complex manifold.

Function theory: function approximation theory.

Functional analysis: (nonlinear) functional analysis, operator theory, operator algebra, difference and functional equation, generalized function, variational method, integral transformation integral equation.

Differential equation: functional differential equation, characteristic and spectrum theory and its inverse problem, qualitative theory, stability theory, bifurcation theory, chaos theory, singular perturbation theory, dynamic system, nonlinear elliptic (and parabolic) ordinary differential equation, partial differential equation, micro-local analysis and general partial differential operator theory, mixed and other singular equations, nonlinear evolution equation, infinite dimensional dynamic system.

In functional analysis, the work of many people, including Kasparov, extends the continuous K- theory to noncommutativity.

C*- algebraic lattice of. Continuous functions in space form commutative algebra in the sense of function product. But in its

In his case, similar discussions about noncommutative cases naturally occur, and at this time, functional analysis naturally occurs.

It has become a hotbed of these problems.

Therefore, K theory is another way to apply this simplicity to many different aspects of mathematics.

In each case, there are many fields specific to this aspect and can be connected with other parts.

K theory is not a unified tool, it is more like a unified box.

There are similarities and similarities between different parts of the framework.

Alain Connes extended many contents of this work to noncommutative differential geometry.

Very interestingly, just recently, Witten adopted his latest ideas in string theory (basic physics).

) It is found that many interesting methods are related to K theory, which seems to be the so-called "conservation".

Quantity provides a natural "home". Although homology theory was considered as the natural framework of these theories in the past, however,

Now it seems that K- 1 theory can provide a better answer.

Let's talk about geometry first: Euclid geometry, plane geometry, space geometry, straight line geometry, all of which are available.

Everything is linear, and the basis is discussed from different stages of non-Euclidean geometry to Riemann's more general geometry.

It is nonlinear in nature. In differential equations, the real study of nonlinear phenomena has been dealt with by many of us.

New phenomena invisible to classical methods. Here I only give two examples, soliton and chaos, which are two theories of differential equations.

Two completely different aspects have become extremely important and famous research topics in this century. They represent different.

Soliton represents the unpredictable and organized behavior of nonlinear differential equations, while chaos represents unpredictability.

Unorganized behavior). The material. They all appear in different fields, which is very interesting.

Important, but their basic soil is a nonlinear phenomenon. We can also compare some earlier work on solitons.

History can be traced back to the second half of the 19th century, but that is only a small part.

Of course, in physics, Maxwell's equation (the basic equation of electromagnetism) is a linear partial differential equation, which corresponds to it.

Young-Mills equation is a nonlinear equation, and the force related to the material structure should be adjusted.

These equations are nonlinear, because the Yang-Mills equation is essentially a matrix embodiment of Maxwell's equation.

The fact that the matrix is not commutative leads to the nonlinear term in the equation, so here we see a nonlinear line.

The interesting relationship between sex and non-exchangeability. Noncommutativity produces a special nonlinearity, which is really intentional.

Thinking about peace is very important.

Geometry and algebra

So far, I have talked about some general topics. Now I want to talk about a dichotomy phenomenon in mathematics. Let's talk about it

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The dichotomy between geometry and algebra, which is the two pillars of mathematics, has a long history.

His remnant snow can be traced back to ancient Greece or even earlier; Algebra originated from ancient Arabs and ancient Indians. Therefore, it

Students become the foundation of mathematics, but there is an unnatural relationship between them.

Let me start with the history of this problem. . Euc 1id geometry is the earliest example in mathematical theory until d.

Escat was purely geometric until he introduced algebraic coordinates into what we now call Cartesian plane.

Rtes is an attempt to turn geometric thinking into algebraic operation. From an algebraic point of view, this is of course.

A major breakthrough or influence on geometry, if Newton and Leibniz are compared by points.

Analyzing the work, we will find that they belong to different traditions. Newton was basically a geometer and Le 1bn.

Iz basic soil is an algebraic scientist, which has a profound reason. For Newton, geometry, or

The calculus he developed is a mathematical attempt to describe the laws of nature. What he cares about is in a broad sense.

In his view, if someone wants to understand things, he must make use of the physical world.

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Calculus is an expression form that can be as close as possible to the hidden physical connotation, so he uses geometry to demonstrate.

Because it can keep a close relationship with practical significance, on the other hand, Leibniz has a goal and an ambition.

The goal is to formalize the whole mathematics and turn it into a huge algebraic machine, which is completely different from Newton's approach.

No, they have many different marks. We know that this scene between Newton and Leibniz.

In the big debate, Leibniz's notation finally won. We still use his symbol to write the essence of partial derivative Newton.

God is still alive, but he has been buried for a long time.

/kloc-At the end of 0/9th century, that is, a hundred years ago, Poincare and Hilbert were two main figures. I'm in front.

As mentioned earlier, roughly speaking, they are descendants of Newton and Leibniz respectively.

His thoughts are more about the spirit of geometry and topology, and he regards these thoughts as his basic insight tools. . Hilbert is more.

Is a formalist, he wants to be axiomatic and formal, and give a strict and formal description. although

No great mathematician can easily fall into which category, but obviously, they belong to different categories.

Tradition.

In preparing this report, I think I should write down the characteristics that our generation can inherit these traditions.

Name of representative. It's very difficult to talk about people who are still alive-who should be on this list? And then I- ...

I thought to myself: Who would mind being put on such a famous list? So I chose two names a.

Rnold Bourbaki, the former is the successor of Poincare-Newton tradition, while the latter, I think, is Hilber.

. Arnold, T's most famous successor, unequivocally believes that his views on mechanics and physics are basically geometric, but

Derived from Newton; I think there is something in between, except something like Riemann (he is really biased against both)

Except for a few people, this is a misunderstanding. . Bourbaki tried to continue Hilbert's formal research and put mathematics

Axiomatization and formalization have advanced to a remarkable extent and achieved some success. Every point of view has its advantages.

But it is difficult to reconcile them.

Let me explain how I see the difference between geometry and algebra. Geometry is of course about space.

There is no doubt about it. If I face the audience in this room, I can see it in a second or a microsecond.

A lot, received a lot of information, of course, this is not an accident. The structure and vision of our brains are very similar.

Important relationship. I learned from some friends who are engaged in neurophysiology that vision occupies% of the cerebral cortex.

Eighty or ninety. There are about seventeen centers in the brain, each of which is responsible for different parts of visual activities.

Some parts are related to vertical direction, some parts are related to horizontal direction, and some parts are related to color and perspective.

Yes, the last part involves the specific meaning and explanation of what we see. Understand and perceive the world we see.

Boundary is a very important part of our human development and evolution. Therefore, spatial intuition or

Spatial perception is a very powerful tool, which is also occupied by geometry in mathematics.

Important position, it can be used not only for those things with obvious geometric properties, but also for those things that are not clear.

Something with geometric properties can also be used. We try to simplify them into geometric forms, because this enables us to

Use our intuition. Our intuition is our most powerful weapon, especially when explaining a kind of mathematics to students or colleagues.

You can see it clearly. When you explain a long and difficult argument, you finally make the students understand it. students

What do you say when you talk? He would say, "I see (I see)!" " "See and understand are synonyms, I.

Students can also use the word "perception" at the same time, at least this is correct in English, and compare this phenomenon with others.

Language contrast is equally interesting. I think one thing is very basic: mankind has passed this great moment of ability and vision.

Activities get a lot of information, thus developing, teaching participating in it and making it perfect.

On the other hand (some people may not think so), algebra is essentially about time.

What kind of algebra is a series of operations listed one after another, meaning "one after another"

We must have the concept of time. In a static universe, we can't imagine algebra, but the essence of geometry is static.

Stateful: I can sit here and observe, nothing has changed, but I can continue to observe. However, algebra is related to time.

Pass, this is because we have a series of operations. When I say algebra, I don't just mean modern algebra.

Any algorithm, any calculation process, gives a series of successive steps, which is made by the development of modern computers.

Everything is clear. Modern computers use a series of zeros and 1 to reflect their information, thus giving the answer to the question.

Algebra involves the operation of time, and geometry involves space. They are two perpendicular aspects of the world.

Moreover, they represent two different concepts in mathematics, so algebra and geometry were relatively important among mathematicians in the past.

Sexual arguments or conversations represent very, very basic things.

Of course, it's just to demonstrate which side lost and which side won. Not worth it. When I think about this problem.

There is an image metaphor: "Do you want to do algebra or geometry?" This question is like asking

Would you rather be deaf or blind? Same. If people's eyes are blind, they can't see space; If people's ears

If you are deaf, you can't hear. Hearing happens in time. Generally speaking, we'd rather have both.

In physics, there is a similar and roughly parallel division between physical concepts and physical experiments. physics

Learning has two parts: theory-concepts, ideas, languages, rules-and experimental tools. I think the concept is within a certain range.

Meanings are geometric because they relate to what happens in the real world. On the other hand, there are more experiments.

Just like an algebraic calculation, people always spend time doing things, measuring some numbers and substituting them into formulas. But now,

The basic concept behind the experiment is a part of the geometric tradition.

To put the above bifurcation phenomenon in a more philosophical or literary language, algebra is for geometricians.

It is the so-called "Faust's dedication". As we all know, in Goethe's story, Faust can

To get what he wants (that is, the love of a beautiful woman), the price is to sell his soul. Algebra was put forward by the devil.

For mathematicians. The devil will say, "I will give you this powerful machine, which can answer any questions you have." .

You just need to give me your soul: give up geometry and you will have this powerful machine.

Think of it as a computer! Of course, we want to have them at the same time. Maybe we can cheat the devil.

Pretend we sell our souls, but we don't really give. But the threat to our soul still exists, because when we turn to it.

In algebraic calculation, we will essentially stop thinking, stop thinking about problems with geometric concepts, and stop thinking about their meaning.

I'll talk more about algebra here, but the goal of algebra is always to establish a formula.

Put it in a machine and turn the handle to get the answer, that is, bring something meaningful.

Turn it into a formula and get the answer. In this process, people no longer need to consider this algebra.

What is the geometry corresponding to these different stages? This loses insight, which is very different at those different stages.

Important. We must never give up these opinions! Finally, we will come back to this question, which is what I said.

Faust's dedication. I know it's a little sharp.

This choice of geometry and algebra has led to the emergence of some interdisciplinary subjects and the relationship between algebra and geometry.

The difference between "Diagra" and "diagra" is not as straightforward and unpretentious as I said.

m)。 Besides geometric intuition, what can a schema be?

General technology

Now I don't want to talk too much about topics divided by content, but I want to talk about those according to the technologies and practices that have been used.

Seeing the theme of method definition, I want to describe some common methods that have been widely used in many fields. first

Yes:

Homologous theory

Homology theory developed as a branch of topology in history. It involves the following situations. There is one.

Complex topological space, from which we want to get some simple information, such as the number of holes or something like that,

Some additive linear invariants related to it are obtained, which is a construction of linear invariants under nonlinear conditions.

From the geometric point of view, closed chains can be added and subtracted, thus obtaining the so-called homology group of a space. Homologous theory

, as a basic algebraic tool to obtain some information from topological space, was discovered in the first half of this century.

Algebra that benefits a lot from geometry.

The concept of homology also appears in other aspects, and its other source can be traced back to Hilbert and his theory of polynomials.

In our research, polynomials are nonlinear functions and can be multiplied to get higher-order polynomials. It's Hilbert.

This great insight prompted him to discuss "ideal", that is, the linear combination of polynomials with common zeros. He wants to find this.

Some ideal generators. There may be many generators. He examined the relationship between them and the relationship between them. therefore

He got the hierarchical pedigree of these relationships, which is called "Hilbert conjunction". Hilbert's theory is that

A very complicated method, he tried to turn a nonlinear situation (polynomial research) into a linear situation. nature

Basically, Hilbert has built a complex linear relational system, which can integrate such nonlinear things as polynomials.

Include some information.

In topology, Hirzebruch and I copied these ideas and applied them to a pure topological paradigm.

In a sense, if Grothendieck's work is related to Hilbert's work,

Then our work is closer to Riemann-Poincare's work on homology, and we use continuous functions.

He used polynomials. K theory also plays an important role in exponential theory and linear analysis of elliptic operators.

From different angles, Milnor, Quillen and others developed the algebraic aspect of K- theory, which is shown in the following

It has great potential application in the study of number theory. The development in this direction leads to many interesting problems.