I'm a first year graduate student of mathematics and I have an important question. I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks and moreover I try to understand the concepts rather than worry about the exams. Despite this, months after such an intense study, I forget inexorably most things that I have learned. For example if I study algebraic geometry, commutative algebra or differential geometry, In my minds remain only the main ideas at the end. Viceversa when I deal with arguments such as linear algebra, real analysis, abstract algebra or topology, so more simple subjects that I studied at first or at the second year I'm comfortable. So my question is: what should remain in the mind of a student after a one semester course? What is to learn and understand many demostrations if then one forgets them all?
I'm sorry for my poor english.