The article first appeared in Mathematics | A Pure Intellectual Adventure.
Every subject, when we don't regard it as a tool of ability and dominance, but as an adventure of knowledge that we have been striving for for for generations, is just such a harmony, which is more or less huge and rich from one period to another; In different times and centuries, it shows us subtle and subtle responses to different themes that appear in turn, as if from vanity.
The above passage is taken from the autobiography Harvest and Sowing by the famous French mathematician Grothendieck. I quote this passage because I agree with him very much-mathematics is such a subject-it is "the adventure of knowledge I have been striving for", and if nothing else, it is "such harmony".
Although everyone has been learning mathematics since primary school, it is inseparable from mathematics, but when it comes to mathematics, I think most people still lack understanding. As a student of the Department of Basic Mathematics, I want to talk about my views on the Department of Mathematics.
1, professional introduction
First of all, there are some general situations of the Department of Mathematics. The direction of mathematics department can be roughly divided into basic mathematics, applied mathematics and computational mathematics; Among them, basic mathematics tends to study the problems arising from mathematics itself or some problems from theoretical physics, theoretical computer and other related disciplines; Applied mathematics and computational mathematics tend to apply mathematical tools to computer, engineering, economics and other disciplines through modeling and other means; But generally speaking, the main purpose of the department of mathematics (especially the department of mathematics in a good school) is to cultivate mathematical research-oriented talents, so the curriculum and training plan are all around this purpose.
Take my undergraduate course as an example, most of them are basic courses of modern mathematics, specialized courses and some basic courses of physics and computer. I think the department of mathematics is lacking or even out of touch on some issues such as employment outside scientific research and teaching; Although the department of mathematics also has some courses that tend to be "applied", such as mathematical statistics and numerical analysis, when studying these courses, they are basically to analyze and solve theoretical problems of other disciplines. To apply them to practical work and transform them into productive forces, it takes time to cultivate them after employment and adapt to the thinking of "industry", so it is hard to say what advantages they can bring to future employment.
So I think if you want to study in the department of mathematics, it is very important to consider: ask yourself if you are really interested in mathematics. Otherwise, after entering the department of mathematics, faced with many difficult courses, it is very easy to feel boring and painful.
Nowadays, many people in the society say that "mathematics is the foundation, and it is easy to have an advantage after learning mathematics well", so as to encourage students to study mathematics at undergraduate level and then transfer to finance or computer field. I think this statement is irresponsible.
It is really unnecessary to choose a math department for the purpose of "laying a good foundation"-as mentioned above, first of all, you may not get the so-called "advantage" that some people imagine, and it is easy to make yourself miserable; In fact, my undergraduate students have such an example. So I think this kind of behavior is a waste of your time and is not worth encouraging. As the saying goes, good steel is used in the cutting edge, and students should choose their real interests. On the other hand, if everyone can give priority to their favorite majors and compete reasonably, I think it will be more beneficial to everyone.
So, how do you confirm whether you are interested in mathematics?
I think, first of all, we should try our best to understand mathematics, especially modern mathematics. Modern mathematics in the department of mathematics is quite different from elementary mathematics in middle schools in terms of research objects and research methods, and it is more systematic. If you are interested in mathematics learned in middle school, you may not be interested in modern mathematics; On the other hand, I don't like elementary mathematics very much, and I may not be attracted by modern mathematics.
Fortunately, the developed network now makes it relatively convenient for each of us to obtain information. You might as well take a look at the open mathematics courses (such as mathematical analysis and linear algebra) on Netease Open Class or Coursera, and try to study a course seriously-I think in this learning process, you can accumulate some new knowledge about mathematics and produce some experience yourself; Websites such as "Zhihu, Douban" also have some good questions about mathematics, and you can find many if you are interested. After you have some basic knowledge about yourself, you can also chat with the seniors of the mathematics department you know, or the high school math teacher and the college enrollment teacher to learn more information. I think, if you are still willing to study mathematics after all this, it shows that you are interested in mathematics, and mathematics major is a good choice for you.
2. Graduation way out
As for the way out after graduating from the Department of Mathematics, as far as I know (students around me, seniors and seniors), there are roughly four kinds:
1, continue to study, go abroad or go to graduate school;
2. Engaged in mathematics education;
3, engaged in computer or finance and other related fields;
4, engaged in work that has little to do with mathematics, such as sales.
(Note: To work in computer and financial related fields, you need to have a certain understanding and even professional knowledge in the corresponding fields. Compared with students who have been trained for a long time, they have some disadvantages, which need to be compensated by internship and other experiences; However, some enterprises will have a relative preference for students majoring in mathematics)
Or take my undergraduate school as an example, most students choose to go abroad for further study or postgraduate study, and continue to study mathematics or statistics, economics, computer, cryptography and other related disciplines; Among them, most students who choose to go abroad begin to prepare language scores and other materials (TOEFL, GRE, etc. ) From the sophomore year to the senior year (mainly in the United States, followed by Europe), and the graduate students are all graduate students, mostly from the junior year, contact the tutor; Individual students work directly out of choice. (However, in view of the particularity of the author's undergraduate school, the situation in other schools may be different; For students who pay more attention to employment, the author suggests that when filling in their volunteers, they can directly consult the admissions teachers of the schools they apply for, such as the employment situation in previous years and whether there are long-term cooperative scientific research institutions or enterprises. In this way, you can get the first-hand information of the school and help you make your own judgment.
3. Mathematics learning itself
Having said so much about the basics, I think we can talk about mathematics itself next. The undergraduate study of mathematics department can be roughly divided into two stages: the first is to study some basic courses; The second is to study by major.
elementary course
The basic courses are divided into three categories: analysis, algebra and geometry.
Analysis includes mathematical analysis, real analysis and complex analysis, and the main content is the establishment and popularization of calculus;
Algebra includes linear algebra and abstract algebra, which mainly studies various algebraic structures;
Geometry includes differential geometry and topology, and studies specific geometric objects (such as space, curves and surfaces). ) and their properties under some transformation.
Professional courses
Specialized courses vary according to the major direction.
If it is basic mathematics, you may continue to learn some foundations of modern mathematics, such as functional analysis, partial differential equations, algebraic topology, algebraic geometry and so on;
If you are applied mathematics or computational mathematics, you may study some subjects with application background, such as mathematical statistics, numerical methods, finite element and so on. And began to choose a tutor to start some research work.
Generally speaking, compared with other majors, life in the department of mathematics may be boring: there are not many opportunities for communication (of course, this may also be related to the school), and internship or scientific research is late. But the department of mathematics also has its own unique fun-that is mathematics itself.
Different from the scattered phenomena and formulas in middle school mathematics, modern mathematics emphasizes naturalness, universality and overall picture.
For a subject, it is necessary to understand its motivation, that is, to "follow the course";
Then there is the structure, tools, technology and results.
Then what conditions they need, what is their essence, and whether their ideas and methods can be applied to other places, that is, whether they are "widely used";
For different disciplines, we should study their relationship and find out their relative position;
We should also compare them and find out the similarities and differences, that is, how the "overall situation" is.
It is this feature that makes modern mathematics powerful and attractive.
Mathematicians use "modern" ideas and tools to solve classic problems that seemed extremely difficult before, such as Fermat's last theorem, Poincare conjecture and so on. It creates suitable tools for physicists to further uncover the secrets of nature and the universe, such as Riemann geometry to general relativity, fiber bundle theory to gauge field theory; Even in unexpected places, it has played a huge role, such as the application of group representation theory in the study of crystal structure and the application of number theory in cryptography; These achievements have proved the power of modern mathematics.
The connection and abstraction reflect the charm of modern mathematics: the so-called connection is the intersection of different fields, and the same thing is seen from different angles. Conversely, the research of the intersection of different fields has an impact on the original field. For example, there is a concept in geometry called Riemann surface, which is simply a surface that looks like a complex plane locally and meets certain conditions; Because of its local nature, complex analysis can be applied to it; As a geometric object, it can be studied by means of topology and differential geometry. Even algebraic tools can be applied to it. Moreover, the study of Riemannian surfaces in turn promotes people's understanding of analysis, geometry and algebra. Abstraction is to extract the essence from phenomena and apply it to other places.
For example, in the discipline of algebraic geometry, algebraic geometry has long been committed to studying a geometric object called "algebraic cluster"; Later, a group of mathematicians represented by French personality Rottendick greatly developed the subject of algebraic geometry. They popularized the concept of "algebraic cluster", defined a geometric object called "probability type" for any commutative ring, and applied it to number theory, which achieved great success. Even in various methods or tools, mathematicians try to find a deeper connection between them-the "philosophy" standing behind them-and abstract it to guide the development of the discipline. Typical representatives are "local integral method" and "quantitative method".
Whenever I learn such things, I will be shocked by their beauty. To quote a teacher of my undergraduate course, "In the afternoon, when you were walking on a tree-lined path or in the dead of night, when you were thinking about generate, you inadvertently thought of that theorem/problem and marveled at its ingenious conception. This is mathematics. "
Senior said:
But it is also undeniable that in the process of development, mathematics has gradually become complex and abstract, and it is increasingly unfriendly to ordinary people. In order to master some seemingly cool terms, a student often needs months or even years to learn and adapt, and constantly accumulate knowledge and examples. I spent a long time studying abstract algebra, algebraic geometry and so on to adapt to various concepts and symbols when I was an undergraduate. Therefore, as a math student, you may need to work much harder than you think, and it is also common that you have worked hard for a long time without being rewarded (or temporarily). It is particularly easy to feel lost and depressed at this time, but it is at this time that you need to work hard and persist.
In a word, it can be said that students majoring in mathematics are both happy and miserable. Happiness lies in being able to pursue beauty and truth for a period of time. According to Sir Atia, "combining art and science in a great cause, trying to understand the universe"; And have the opportunity to meet some teachers and friends who are also seeking truth. They can benefit you a lot, and a period of time with them will make you unforgettable. During my undergraduate course, I had several discussion classes with my friends. Whenever I recall the days when I had a heated argument about a definition or theorem and reached an agreement after discussion, I feel very nostalgic. The pain is that you need to pay more, you can't expect worldly rewards, and sometimes you may not get the understanding of your relatives and friends; Loneliness may often bother you.
However, as I said at the beginning, mathematics is a pure subject. If you are really interested in it and are willing to embark on a "journey of knowledge adventure", if nothing else, it is "such harmony", then I think it is all worthwhile. Even if I don't continue to walk on the road of mathematics for various reasons in the future, this experience can be a beautiful memory.
From a further point of view, studying mathematics or other majors is only a part of life; Exploring the meaning of life is a big problem in life. Finally, let me end my speech with an ancient Greek Olympic motto: "Don't ask for victory, just ask for courage."
refer to
(1) "As if from vanity-the life of Alexander Grothendieck"
(2) Sir 2)M.F.Atiyah's article:
(3) Introduction to Riemannian Geometry.