"Advice to young mathematicians"
My advice would be:
$\bullet $ Do many calculations
$\bullet \bullet$ Ask yourself concrete questions whose answer is a number.
$\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!)
$\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in your subject.
I have answered many questions tagged algebraic geometry on this site and I was struck by the contrast between the excellent quality of the beginners in that field and the nature of their questions: they would know and really understand abstract results (like, say, the equivalence between the category of commutative rings and that of affine schemes) but would have difficulties answering more down-to-earth questions like: "how many lines cut four skew lines in three-dimensional projective space ?" or "give an example of a curve of genus $17$".
In summary the point of view of some quantum physicists toward the philosophy of their subject
Shut up and calculate ! contains more than a grain of truth for mathematicians too (although it could be formulated more gently...)
Nota Bene
The above exhortation is probably due to David Mermin, although it is generally misattributed to Richard Feynman.
Edit
Since @Mark Fantini asks for more advice in his comment below, here are some more (maybe too personal!) thoughts:
$\bigstar$ Learn mathematics pen in hand but after that go for a stroll and think about what you have just learned. This helps classifying new material in the brain, just as sleep is well known to do.
$\bigstar \bigstar$ Go to a tea-room with a mathematician friend and scribble mathematics for a few hours in a relaxed atmosphere.
I am very lucky to have had such a friend since he and I were beginners and we have been working together in public places ( also in our shared office, of course) ever since.
$\bigstar \bigstar \bigstar$ If you don't understand something, teach it!
I had wanted to learn scheme theory for quite a time but I backed down because I feared the subject.
One semester I agreed to teach it to graduate students and since I had burned my vessels I really had to learn the subject in detail and invent simple examples to see what was going on.
My students did not realize that I was only one or two courses ahead of them and my teaching was maybe better in that the material taught was as new and difficult for me as it was for them.
$\bigstar \bigstar \bigstar \bigstar$ Last not least: use this site!
Not everybody has a teaching position, but all of us can answer here.
I find using this site and MathOverflow the most efficient way of learning or reviewing mathematics . The problems posed are often quite ingenious, incredibly varied and the best source for questions necessitating explicit calculations (see points $\bullet$ and $\bullet \bullet$ above).
New Edit (December 9th)
Here are a few questions posted in the last 12 days which I find are in the spirit of what I recommend in my post: a), b), c), d), e), f), g), h).
Newer Edit(December 17th)
Here is a fantastic question, brilliantly illustrating how to aggressively tackle mathematics, asked a few hours ago by Clara: very concrete, low-tech and naïve but quite disconcerting.
This question also seems to me absolutely original : I challenge everybody to find it in any book or any on-line document !
The best advise I can share was given to me by my mother, (she was a researcher in medicine) when I was a first-year student (of mathematics): find a good adviser and follow his/her advice.
As a beginner, you usually cannot judge yourself about research areas of mathematics, and what to do and what to learn. In all this you should rely on a good adviser, who must be a mathematician with well-established reputation, and a person you feel comfortable working with. So investigate carefully all potential advisers around and choose the best one. Once you make your choice, follow his/her advises in everything.
I don't know how many of these advices are already present in the pdfs, but I found these really valuable pieces of advice.
- Choose a subject, an area of mathematics, which is "your favourite one". Live there as it was your home.
- Relentlessly go back to the very basic fundamentals of that subject. Re-study everything from scratch once a year, re-do things you know using all you've learned in the last months. Do what professional basketball players do: fundamentals, all the time.
- Don't wait for others to learn what you want to learn. The question "Hi, I took only a course in algebra, but I want to have an idea of what the hell is Galois theory." is perfectly legitimate, and it's your teacher's fault if they can't give you a simple, well posed and enlightening elementary example.
- Recall yourself that old mathematics done in a deeper and more elegant way is new mathematics. This might be a very opinion-based piece of advice, and yet.
- Don't fear to travel outside your preferred field. Your home will look the same, but totally different after each trip.
- Don't indulge in the thought that you don't want to check if an idea is a good idea because it might be wrong and spoil your last month's work. We already are ignorant about almost everything in mathematics, there is no need need to be also coward.