What does Langlands refer to in mathematics and geometry Langlands

1] The Langlands Program is a series of far-reaching ideas in mathematics, linking number theory, algebraic geometry and reducing group representation theory; The program was first proposed by Robert Langlands in a letter to Wey in 1967.

2] Origin: We can generalize the second law of mutual reciprocity as the starting point of the Langlands program: Given a field of numbers on q and the Galois group as commutative, the Atin's reciprocal law assigns an l function to any one-dimensional representation of the Galois group and asserts:

These L-functions are equal to some Dirichlet L-functions (an analogy of the Riemann function, expressed by the Dirichlet feature). The exact connection between these two L-functions constitutes Artin's law of mutual inversion.

Given the noncommutative Galois group and its high-dimensional representations, we can still define some naturally matched L-functions, the Atin L-function.

3] Langlands further promoted:

Replace the general linear group gln above with any connected reducing group g;

construct complex Lie group G (the so-called Langlands dual group, or L group);

Replace the self-defending representation with the L package of self-defending representation; Each L-packet is a finite set of self-defending representations, and representations belonging to the same L-packet are called L-indistinguishable.

to the self-defying cusp of each g and the finite dimension of each g, with an L-function; Representations in the same l-packet have the same l-function and -factor. Langlands also conjectured that these two l-functions satisfy a functional equation.

Number theory is one of the branches of pure mathematics that mainly studies the properties of integers. An integer can be the solution of an equation. Some analytic functions include the properties of integers and prime numbers, and some number theory problems can also be understood through these functions.

Through number theory, the relationship between real and rational numbers can also be established, and rational numbers can be used to approximate real numbers.

According to the research method, number theory can be roughly divided into elementary number theory and advanced number theory. Elementary number theory is a number theory studied by elementary methods, and its research method is essentially to use the divisible properties of integer rings, mainly including divisibility theory, congruence theory, and continuous fraction theory. Advanced number theory includes more profound mathematical research tools.

It broadly includes algebraic number theory, analytic number theory, computational number theory, and many more.

Number theory is a theory that studies the properties of integers. The basic element of integers is primes, so the essence of number theory is the study of the properties of prime numbers in Euclid's Geometric Primitives. 2,000 years ago, Euclid proved that there are infinite prime numbers.

Since there are infinite, there must be a general formula for prime numbers that represents all prime numbers, or a universal formula for prime numbers. It is a discipline with the same long history as plane geometry. Gauss hailed it as the "crown of mathematics" According to the difficulty of research methods, number theory can be roughly divided into elementary number theory (classical number theory) and advanced number theory (modern number theory).

Number theory is the study of the knowledge of number gauge, the founder of number theory is afraid to count Fermat, he proposed Fermat's theorem, there is still a lot of interest, number theory is pure mathematical knowledge, many great mathematicians have a deep insight into number theory, think Gauss, Euler, etc.

1. Come on, your virtue is still yelling at me? Height constant function and weight power function.

Standing is a nonsense, the fifth-order complete diagram is lying down is a Mei's triangle.

The number of people who are scared to death by your face every year can be ranked Fibonacci, the breadth of failure everywhere in life is comparable to that of the Longlands program, the depth is as never-ending, and the Lagrangian interpolation identity will not change you for a bright future.

A constant function image is a straight line parallel to the y-axis with a constant y-value.

The power function is an increasing function, and the growth rate is getting faster and faster (in fact, I personally think that it is not very good to use the power function, because there are many kinds of power functions, and it uses the exponential function.

It's better to describe it, the exponential function grows much faster than the power function in the later stage! ）

You can look up the fifth-order diagram to understand what kind of figure it is.

Menelaus triangle, indicating lying down on the stomach highest.

Fibonacci sequence.

It's like this 、...Liquid light touch....(nth term = n-1 term + n-2 key source, which means that the number of people in the third year is the sum of the number of people in the first and second years, and so on).

Da vinci. This sequence also keeps appearing in the code, and it is one of the most amazing sequences in the world, not only in nature, but also in human events. Here, of course, it is said that the number of people is always the sum of the previous two years, and more and more.

The Langlands Program is to establish an essential connection between some seemingly irrelevant contents. This is the equivalence transformation of our mathematical propositions (meaning that everything can be related to your failure, and the equivalent conversion is to various other failures).

Everyone understands that infinite does not cycle decimals.

Lagrangian interpolation identity refers to the interpolation method.

It is a universal method to solve the problem of discrete data modeling after experiments.

It means that no panacea will save you!

Mainly write about the main work content, achievements, and shortcomings, and finally put forward reasonable suggestions or new directions...

The work summary is to let the superior know what contribution you have made and reflect the value of your work.

So several points should be written:

1. Your understanding of the position and work 2. What exactly you have done.

3. How do you work hard and what things do you use your brain to solve? Even if it's nothing, write about some difficult questions and how you solved them with effort.

4. What abilities do you need to improve or what knowledge do you need to enrich in your future work?

5. Superiors like to take the initiative to work. You must be prepared for everything in your part, that is, the preparation work in advance is as follows for your reference:

Summarization is to conduct a comprehensive and systematic general evaluation and analysis of the situation in a period of time, and analyze achievements, shortcomings, and experience. Summarizing is a type of applied writing, which is the rational thinking of the work that has already been done.

The basic requirements for summarizing.

1 The summary must have an overview and description of the situation, some of which are relatively simple, and some of which are more detailed.

2 Achievements and shortcomings. This is the main thing to summarize. The purpose of summarizing is to affirm the achievements and find out the shortcomings. What are the achievements, how big are them, what are the aspects, and how they were achieved; The number of shortcomings, the aspects in which they manifest themselves, and how they arise should be clearly written.

3 Lessons learned. In order to facilitate future work, it is necessary to analyze, study, generalize, and form theoretical knowledge of previous work experience and lessons.

Notes on Summary:

1. We must seek truth from facts, and the achievements are basically not exaggerated, and the shortcomings are basically not reduced. This is the basis for analysis and drawing lessons.

2 Be organized. The sentences are smooth and easy to understand.

3 Be detailed. There are important and secondary, and the main points should be highlighted when writing. The questions in the summary should be prioritized and detailed.

Basic format of the summary:

1. Title. 2. Text.

Beginning: an overview of the situation, an overall evaluation; The outline is summarized and the whole text is summarized.

Subject: Analyze the shortcomings of the achievements and summarize the lessons learned.

Conclusion: Analyze the problem and clarify the direction.

3. Payment. Attribution and date.