What remains in a student's mind 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.
 A: I am in the same boat at the moment, I am partway through my first year as a postgrad and I feel I am forgetting many results that I used to know through lack of use/practice or full understanding.
However, I think at this level maths is more about understanding the situation rather than studying the results. For example, I find myself thinking "so this is what person X is trying to do" and "so this is what the concepts capture" rather than "Wow, here is yet another nice theorem to remember".
Of course studying results is important too but the bigger picture is more important I feel. You will always have access to enough material to jog your memory on any given topic. It is always easier to pick something up for the second time.
A: As a student who is suffering from the very same problem, I want to share my less professional solution. With this method, I feel like my studying became much more efficient.
When I read, I tend to be generous. I used to pick out every single detail and I gave an author or a lecturer a criticism about not writing "for all" or writing $x$ instead of $\bar{x}$ for an element of $\mathbb{R}[x]/(x^{2} + 1)$. However, I realized that dropping this habit of being extremely nit-picky is necessary in order to understand more difficult ideas. I usually nod my head if I read what I expect or arguments similar to my intuitive ideas. For example, I saw an author saying that $R/(p)$ is a field when $p \in R$ is a prime element and $R$ is a PID. Of course, it is technically wrong if you take $R = \mathbb{Z}$ and $p = 0$, but in the proof that contained the "mistake," it was more intuitive to think that $R/(p)$ was a field, and later I found that the case $p = 0$ gives a trivial proof of the theorem where $R/(p)$ was used. It is still very difficult for me to proceed the readings without going through details like this, but I try my best to give the author/lecturer trust and underline the details that I do not understand at the moment so that I can review it later (and most of the time, I do understand them when I calm myself down and look at them again).
On the other hand, I try to be as rigorous as possible when I write my own proof. I had bad experiences from letting my intuition conquer all of my mathematical activities. The problem of being too intuitive is that everything seems "obvious," which is not always the best scenario when it comes to writing mathematics.
Long story short, I read more loosely and write more rigorously. I found that this method only works under certain circumstance. It only works when I have great determination and patience that makes me be willing to endure slow-pace studying or getting stuck. It is also important to balance yourself to find some interesting (but not-too-difficult) problems related to the reading/lecture that give you more motivations for learning.
A: After many years of semi-non-use, some useful things remain: process, existance, etc.  
You might know for example, that calculus exists, and there are handsome volumes of pre-made calculus, to modify to the current end.
You might know how to read the runes in mathematical papers, and generally follow the arguments, even if you might not do it that way yourself.  
You might be able do interesting things with matricies, such as work with oblique vectors, or a generalised product of two vectors.  
Generalising the specific is also a handy effect.  I often calculate specific polynomials with $x=100$ or $x=1000$, to save the drudgery of algebra.  You just have to know to watch to see no carries happen.
It's something like visiting a town from before.  You might not be able to say, but years from now, you can still run the maze.  
A: Let's say you can't remember what Theorem 42 says, from a course you took nine semesters ago.  Then 20 years later someone mentions a term that sounds vaguely familiar to you---you're not sure why.  It's in a discussion of a topic you're curious about.  You look in book indexes for terms sounding vaguely related, and google those terms, and after pursuing vague memories, it turns out to lead you to Theorem 42, which you'd forgotten, and that's just what you need to answer the question that was on your mind.  So all was not lost.
However, it is also often useful to actually remember things.  Two things that help are (1) you find surprising connections among seemingly disparate things, and it impresses you, especially if one of them was something you were interested in; and (2) You teach a course that includes the statement and proof of Theorem 42.  You teach that course six or seven times, and grade students' answers to exercises in which they must use Theorem 42, or prove it by another method that is roughly sketched for them, or prove another result by the same method, and you answer lots of students' questions about all this, and help them through difficulties they have with it.
When I took topology as an undergraduate I remember the instructor putting on the blackboard a huge list of various spaces and said this one has this property and this property but not this property, and we were supposed to learn to identify an example given those properties.  They were quite different from each other: manifolds pasted together, spaces of sequences of real numbers, things put together out of transfinite ordinals, and I wondered: How can one remember all of this?  But then some years later I found much of it fresh in my mind whenever a question arises that such examples answer.  So another part of the answer is: just keep going.
A: Usually what remains in your mind is the general idea: a vague outline of the terminology and theorems. It may sound bad, but it's fine. I can honestly say that I hardly remember anything from most courses I took, except the things I had to teach, or directly relate to my work. This includes things from a course I took the previous semester and even things from a course I am currently taking whose topic is in fact close to my own research.
It is not that bad. What important is to learn how to store this data so the next time you run into it you will immediately recognize it - or at least recognize that you should recognize it.
I can give an example from my own experience, after learning and forgetting most of the basic course in Galois theory, I was still able to identify a similar picture when looking at the covering spaces of a topological space and the $\pi_1(X,x_0)$ structure between them. I went to ask the professor who taught me Galois if there is any connection and indeed there was.
Remember that a good mathematician should be able to see analogies between theorems, and even theories. So identifying similar ideas and similar patterns is more or less essential to this point. 
To the actual details of all the theorems, I wish I could tell you that your memory can fight the good war of remembering and win. It is not usually the case, and even friends that in our undergrad courses remembered everything in details have now - not too long into grad school - forgotten a lot of the material.
Let me sum up with what I think you should take from courses. You should be able at least locally (during the course and up to one semester later) be able to remember most of the proofs, or at least the theorems. Later on you need to remember the idea, and the methods used in the proofs. The methods can take you an extra mile later on when you approach proving things on your own because if you see similarities you can often use similar methods.
A: When a newborn baby sees his mothers face
The first of all objects, the eyes first light
It may forget
It may pass on
But it will not forget all
Something remains, a hidden sight.
Seen again, again, again,
Her face becomes familiar
Then sought for
Then loved.

You may not remember the theorems,
You may not remember the proofs,
You may forget and pass on into your future studies,
But seen again, again, again,
Something will stir within you
Something will change inside you
It will become familiar,
Then natural,
Then loved.

Children take time,
So do students. Good luck!
A: I am a late-comer to this question but I hope I have a good answer to this question.
I am a HS math teacher, inevitably after some times into a semester, an overwhelmed student would come up to me and asks if he has to remember all of these formulas when going to college. Generally, this is my answer: No, you don't, you need to remember only the most basic concepts, but what I expect to remain in a your mind is the mysterious thing called  math maturity.
Math maturity is the chemistry that clicks in a student's mind when he is tackling a tough question in SAT, it is also the lubricant that makes his brain runs faster when in college.
Perhaps we can extrapolate this to learning higher math during college, as asked by the OP? Thank you.
