It's really a mild, soft question.
So far, I am an undergrad student, contemplating on several majors.
What will be the major difference if I become a math major and go to a graduate school to study math?
Will a graduate math major generally focus on reading/studying recent research papers? Or will the graduate math major be taught by professors on the things not covered in undergrad programs?
The reason why I am asking this question is that undergrad math programs seem to me at this point so well-covered that after studying the programs, a person will be able to do anything he wants to do, and if one wants to do research, he will be able to catch up with recent progress by reading research papers.
If this is true, why is the graduate school even needed?
By the way, I am a first-year undergraduate student 🙂
Best Answer
Here is what I tell my grad students:
The difference between undergrad mathematics and graduate mathematics is the difference between art history, or art appreciation, and learning to be an artist.
As an undergraduate you see a lot of mathematics, but you don't create new mathematics. The goal of graduate school (and here I am speaking from experience with top fifty U.S. graduate schools, so what I am saying probably applies best in that context) is to learn how to create new mathematics, and then to create that new mathematics.
One specific consequence of this (in my view) is the following: often in undergraduate mathematics classes, proofs and rigor are presented almost as moral imperatives --- as if it is a moral failing to know a statement without knowing why it is true; consequently, people often put a lot of effort into learning arguments just for the sake of having learnt them. (This is exaggerated, perhaps, but I think it reflects something real.) On the other hand, in research, one learns arguments for different reasons: to learn technique, to pick out important ideas --- there is a professional aspect to the way one looks at pieces of mathematics which is not usually present in undergraduate mathematics. One gives proofs in order to be sure that one hasn't blundered; one's interaction with the mathematics and the arguments is much more visceral than in undergraduate courses.
(I am not speaking from any experience now, but I think of the difference between learning how to interact with a block of marble, and bring a new form out of it, however rough it might be, in comparison to looking and learning about a lot of existing beautiful statues, masterpieces that they are.)