If you have the leisure to ask a freshman in the mathematics department what it is like to study mathematics, the 99% of the face will tell you meaningfully, it is like a person who does not know English to go to the cinema to watch an English movie, but the shame is that the director team intentionally or unintentionally removed the subtitles, which is called a sour.

If you ask a student in Hua Luogeng's class what kind of experience it is to learn mathematics, he will tell you with a blank face, isn't it just addition, subtraction, multiplication and division?

Mathematics is a discipline that attaches great importance to mathematical talent, and it is also the most thorough and in-depth natural science studied by human beings at present. Mathematics is the only subject that puts knowledge from hundreds of years ago in a textbook and you will feel that it is beyond the curriculum, and that is mathematics.

Ordinary mathematics students learn the theories founded by Euler, Gauss, and even Fermat, Descartes, and others hundreds of years ago. At the same time, physics students are definitely learning about the theory of relativity, quantum mechanics and other problems that are at the forefront of physics.

But for some gifted geniuses, it's not enough to just be in a book. With their ability, the things in the book can be regarded as free reading, and the knowledge in it can be mastered at a glance, and it is not challenging at all. At this time, what they need is to further deepen their research, go deep into the frontier or quasi-frontier of various fields, and understand the dynamics of all aspects. And according to these dynamics, choose the direction you want to study. The students of Hua Luogeng's class are such a group of people.

The responsibility of Junxin and a group of other teachers is to help them fill in the knowledge they lack between the knowledge in books and the latest developments in various fields of mathematical research. This is already the scope of the curriculum for graduate students and even doctoral students.

Therefore, in the face of such a group of proud men in the mathematics world, any teacher, or even professor, will have a lot of pressure. Without exception, this kind of pressure is due to the fear that he is shallow in his studies and misleads others, and the same is true for Junxin.

However, Junxin has a little more confidence than others, after all, he has more development in this era in the past 30 years, and understands the trend of mathematics development, so many things can be applied to his own courses. In addition, he has listened to the lectures and lectures of some world-class mathematics masters many times, and has somewhat understood their views on mathematics, and has formed his own views on mathematics by learning from each other's strengths.

"I am the teacher of your algebraic geometry course for a period of time, and you will receive the knowledge I have transmitted to you about the basics of algebraic geometry in the next month, do you have any questions?" Junxin said indifferently to the students of Hua Luogeng's cram school sitting below. When he actually stood on the podium, the nervousness suddenly disappeared and was replaced by a feeling that everything was under control.

In the eyes of the students in the eighties, the teacher was an elder-level figure, so although they were surprised to see Junxin standing on the podium at such a young age, they did not say anything against it. Instead, he looked at Junxin without saying a word, waiting for his next move.

Seeing that no one was speaking, Junxin nodded and continued: "Let's first explain what algebraic geometry is from the development of mathematics. The idea of using algebraic methods to study geometry developed into another branch of geometry after the emergence of analytic geometry, which is algebraic geometry. Algebraic geometry is studied with algebraic curves of planes, algebraic curves of space, and algebraic surfaces. ”

"The study of algebraic geometry began in the first half of the 19th century with regard to the study of plane curves of three or higher order. For example, in his research on elliptic integrals, Abel discovered the biperiodicity of elliptic functions, thus laying the theoretical foundation for elliptic curves.

Riemann introduced and developed the theory of algebraic functions in 1857, which made a key breakthrough in the study of algebraic curves. Riemann defined his function on some kind of multi-layered complex plane of complex planes, thus introducing the concept of the so-called Riemann surface. Using this concept, Riemann defined one of the most important numerical invariants of an algebraic curve: the deficit. It was also the first absolute invariant in the history of algebraic geometry. For the first time, we consider the problem of a parametric cluster of double rational equivalence classes of all Riemannian surfaces with the same deficit g, and find that the dimensionality of this parametric cluster should be 3g-3, although Riemann has not been able to strictly prove its existence.

After Riemann, the German mathematician Noether and others used geometric methods to obtain many profound properties of algebraic curves. Knott also conducted research on the properties of algebraic surfaces. His work laid the foundation for the work of the later Italian school.

From the end of the 19th century, the Italian school represented by Castelnovo, Enriques and Severi and the French school represented by Poincaré, Picca and Lefschetz appeared. They did a lot of very important work on the classification of low-dimensional algebraic clusters on complex number fields, especially the theory of algebraic surface classification, which is considered to be one of the most beautiful theories in algebraic geometry. However, due to the lack of a rigorous theoretical basis in the early study of algebraic geometry, there are many loopholes and errors in these works, some of which have not been filled until now.

One of the most important advances in algebraic geometry since the 20th century has been the establishment of its theoretical foundations in the most general context. In the 30s of the 20th century, Zariski and Van der Walden were the first to introduce the method of commutative algebra in the study of algebraic geometry. On this basis, in the 40s, Weiy used the method of abstract algebra to establish the theory of algebraic geometry on the abstract field, and then in the mid-50s of the 20th century, the French mathematician Serge based the theory of algebraic clusters on the concept of layers, and established the theory of upper cohomology of condensed layers, which laid the foundation for Grothendieck's subsequent establishment of generalization theory, and his lecture notes "Foundations of Algebraic Geometry" (EGA, SGA, FGA) became the bible in this field. The establishment of generalization theory has brought the study of algebraic geometry to a new stage. The concept of generalizations is a generalization of algebraic clusters, which allow the coordinates of points to be picked in arbitrary commutative loops with unit elements, and to allow the existence of power zeros in the structure layer. ”

Speaking of this, Jun Xin paused and continued: "Not long ago, because I solved the problem of Modelle's conjecture in the field of number theory with the thinking method of algebraic geometry, I won the favor of Mr. Grotendieck, and had a discussion with him about this knowledge. ”

As soon as Junxin's words fell, the people sitting below suddenly became excited, and the eyes looking at Junxin immediately became extremely different.

Jun Xin ignored the people below and continued: "In view of the historical changes in the development of algebraic geometry, we need to learn this course a certain foundation, you need to understand abstract algebra, commutative algebra, cohomology algebra, in addition, you also need to understand and add the knowledge of algebraic topology, differential geometry, complex analysis, multiple complex variable functions, Riemannian geometry, number theory, etc. Over the next two weeks, I will pick the most important ones to explain, and in two weeks you will be faced with the real knowledge of algebraic geometry. ”