Some arrogant guys classified mathematics into noble and public level: They consider algebraic topology, algebraic geometry, non-commutative geometry…noble, and PDEs, complex variables…public.
Sorry if my English is difficult to understand. I don’t know how to type Vietnamese in this site.
I just want to say in this post that students (especially of HCM university), just feel confident about your ability to study mathematics. Don’t afraid about arrogant words of someone.
Below is one suggestions about books that a competent (under)graduate student can read if you don’t know how to spend your time:
1. Do good on your research’s topic. This help you get familiar to research’s mathematics, and most important, help you to find a fellowship to learn at a foreign university. (you understand my statement, right??:) )
2. Grasp calculus (Stokes theorem), Serge Lang’s Algebra, Gilbard and Trudinger’s PDEs, Rudin’s Real and Complex Analysis, Brezis’s Functional Analysis, Gameline’s Complex Variable
3. After Step 2, you can read do Carmo’s Riemann Geometry, and Algebraic Topology. I don’t suggest Hatcher’s book, although it is a great book. In my opinion that book is very thick and detailed. There are many references, I don’t know what is the best sources.
4. After Step 3, you can read Hatshorne’s Algebraic Geometry and Griffith and Harris’s Principle of Algebraic Geometry.
5. Some more stuff: K -Theory and Several complex variables.
After Step 5, I think you have enough fundamental knowledge to do research yourself. (Of course you need to read more stuffs on narrow fields of mathematics, but I think you can understand the materials not very difficult.)