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I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the Career Advice section of Terence Tao's blog, and I am amazed by the intelligence of the pieces of advice given in these pages.

Now, I ask to the many accomplished mathematicians who are active on this website if they would mind adding some of their own contributions to these already rich set of advice to novice mathematicians.

I realize that this question may be seen as extremely opinion-based. However, I hope that it will be well-received (and well-answered) because, as Timothy Gowers put it,

"The most important thing that a young mathematician needs to learn is of course mathematics. However, it can also be very valuable to learn from the experiences of other mathematicians. The five contributors to this article were asked to draw on their experiences of mathematical life and research, and to offer advice that they might have liked to receive when they were just setting out on their careers."

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    $\begingroup$ I would really like to hear (detailed, not vague) advice on the problem solving process, both strategies, techniques, and dealing with the emotional aspect of it. $\endgroup$ – abnry Dec 7 '14 at 18:58
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    $\begingroup$ If the question is to be understood literally, the answer is "No, I wouldn't mind." If you prefer me to answer the implied question rather than the formal one, can you, please, be more specific? @nayrb I spent 3 hours today in the classroom (preparing people for qualifiers) showing how a problem solving process is akin to the bargaining on the market (we did 4 reasonably hard problems in measure theory), but do you really expect me to post a transcript of my crazy spiel here? The most I can do (if I have a long evening with nothing else to do) is to choose one problem and to go over it... $\endgroup$ – fedja Dec 21 '14 at 1:13
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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 !

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    $\begingroup$ +1! I certainly fell prey to this a bit in my wild and reckless youth. $\endgroup$ – Qiaochu Yuan Nov 29 '14 at 19:11
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    $\begingroup$ Dear @Qiaochu, I couldn't help smiling (a bit wistfully) when I saw you referring, even in jest, to your youth in the past tense :-) $\endgroup$ – Georges Elencwajg Nov 29 '14 at 19:32
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    $\begingroup$ @GeorgesElencwajg +1. An excellent answer even though you appear to be algebraic geometer. I would like to add just couple more comments. - Work on concrete problems first, i.e., problems you can get your hand and head around. "Premature and unnecessary abstraction is the root cause of all evil". - Always try to construct counterexamples by violating some assumption of the problem/theorem. $\endgroup$ – Adhvaitha Dec 2 '14 at 22:22
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    $\begingroup$ Dear @Adhvaitha, your outspokenness is quite refreshing: I found your implicit evaluation of algebraic geometers quite amusing! And thanks for what is, after all, a flattering comment... $\endgroup$ – Georges Elencwajg Dec 3 '14 at 17:56
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    $\begingroup$ Thank you very much for the additional thoughts: as always, your comments are rich of intelligent insights and precious experiences. I wish I could upvote twice! $\endgroup$ – Dal Dec 3 '14 at 21:57
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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 advises.

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.

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    $\begingroup$ Find a good adviser is IMO the most difficult part of a science carrer. $\endgroup$ – reuns Dec 9 '16 at 12:50
  • $\begingroup$ But occasionally ignoring your advisor (and evaluating in hindsight whether this was a good idea) is also important! $\endgroup$ – Remy Aug 22 at 2:49
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$ \clubsuit$ Try to view every thing to one dimension or two dimension where we can see geometrically..

$\quad$ It is quite often helps to realize the things and to generalize it.

$\clubsuit$ Aim to guess the answer first. If we realize that the answer is correct then proving

$\quad$ is not that difficult.

$\clubsuit$ If you are finding something wrong. Try to give counter examples or prove that

$\quad$ why this is wrong.

$\clubsuit $ Teaching and explaining is the best way to understand the mathematics...

$\clubsuit$ It is better to view the old things in new ways.

$\quad$ For, example nowadays we are thinking widely through all subjects.

$\quad$ If we wish to study about groups try to give examples in analysis like $\mathcal C[a,b]$ etc.

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While there are many excellent suggestions, I would like to add that it is crucial to go abroad/other cities to get acquainted with many people, from whom you can learn a lot in many ways. This is something that's not good to delay, as later you might have no chance for family reasons!

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  • $\begingroup$ +1 I cannot overstate how much talking with other mathematicians has allowed me to move forward after being stuck. $\endgroup$ – Ben Blum-Smith Dec 29 '17 at 15:32
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I don't know how many of these advices are already present in the pdfs, but I found these really valuable pieces of advice.

  1. Choose a subject, an area of mathematics, which is "your favourite one". Live there as it was your home.
  2. 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.
  3. 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.
  4. 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.
  5. Don't fear to travel outside your preferred field. Your home will look the same, but totally different after each trip.
  6. 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.
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    $\begingroup$ I disagree with the advice to entrench oneself fully in one area of mathematics. It's all connected, and sometimes unexpected connections (that lead to amazing theorems) can pop up between seemingly unrelated fields of mathematics. $\endgroup$ – Franklin Pezzuti Dyer Nov 17 '18 at 21:42
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Here is a late answer (several years late it seems):

1) One minor piece of advice is to have respect for others. (Well, perhaps that is a major piece of advice.)

Here is a minor example: People on stackexchange that answer a question two hours after a major hint has been given, using the basic idea of the hint, but who provide no mention of the hint or the person who gave it, do not demonstrate respect for others. To me, this behavior suggests the person is rude, unsafe, and may have other work that is heavily borrowed from others (without sufficient referencing).

An example on the flipside: If this kind of behavior does happen, give the offender the benefit of the doubt: Maybe that person accidentally left a reference off, or was somehow unaware.

2) I resonate with not being afraid to "ask dumb questions" from your link here: https://terrytao.wordpress.com/career-advice/

If you strive to always be right, and to always ask "smart questions," you likely will not get very many new results. To make progress you must also make mistakes.

3) Do not put arbitrary constraints on yourself.

I give some examples on that last point in these slides I made on "thinking outside the box." A simple example that I repeatedly observed when tutoring middle school and high school students is when they set up their math homework by first boxing in the amount of space needed for each problem. [These slides are from a powerpoint talk I gave several years ago (as a second part of a larger seminar at the University of Southern California). These slides are supposed to change when clicked, though some PDF viewers only seem to allow scrolling.]

http://ee.usc.edu/assets/008/61688.pdf

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$\bullet$ Patience!

$\bullet\bullet$ Persistence.

$\bullet\bullet$ Work hard.

$\bullet\bullet\bullet$ Learn things very well. (in detail)

$\bullet\bullet\bullet\bullet$ Ask yourself lots of questions, even stupid ones! (when does this lemma work? when it doesn't? is there a generalization of it? is there a similar lemma about ...)

$\bullet\bullet\bullet\bullet\bullet$ Don’t base career decisions on glamour or fame.

$\bullet\bullet\bullet\bullet\bullet\bullet$ Think about the “big picture”.

$\bullet\bullet\bullet\bullet\bullet\bullet\bullet$ Professional mathematics is not a sport (in sharp contrast to mathematics competitions).

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    $\begingroup$ What does "Professional mathematics is not a sport (in sharp contrast to mathematics competitions)" mean? That the point is not to have fun? That the point is not to find out how good you are compared to others? That the point is not to try your best and see how far it gets you? $\endgroup$ – JiK Feb 24 '15 at 22:17
  • $\begingroup$ What is the mean of big picture ? Sorry, my English is poor. $\endgroup$ – lanse7pty Dec 12 '15 at 9:44
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If I could give a younger me advice, I would tell myself to lower my pride and accept as much help as I could in the times it was available to me, and shed my fear of humiliation and inferiority that kept me from even considering talking to others entirely let alone other mathematicians.

While you should strive to be more receptive to constructive criticism every day, I also think it's important to disregard criticism when it's source is from an individual that considers their authority in the field in question to be a conclusive presumption that must be held by yourself and all others.

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