"Advice to young mathematicians"

Question

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."

Answer 1

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 !

Answer 2

$ \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.

Answer 3

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!

Answer 4

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.

Answer 5

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

Answer 6

$\bullet$ 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).

Answer 7

I don't know how many of these advices are already present in the pdfs, but I found these really valuable pieces of advice.

0. Choose a subject, an area of mathematics, which is "your favourite one". Live there as it was your home.
1. 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.
2. 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.
3. 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.
4. Don't fear to travel outside your preferred field. Your home will look the same, but totally different after each trip.
5. 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 are already ignorant about almost everything in mathematics, there is no need to be also coward.

Answer 8

Here is my answer in short

1. Use YouTube videos .Some of them are good and explain basic content.

2. The Math Sorcerer channel is an excellent for math advice and
   inspiration

3. George Polya’s Mathematics and Plausible Reasoning vol 1 &2 These are great books to build mathematical skills

4. Circle around a particular problem to try solving it,instead of a attacking it head on.

5. If you can’t solve a problem, go to another one and when you go to bed, think about it and forget it. Let “Deep mind”, ie,subconscious mind ,find a way to solve it. You might be surprised it finds hints for your conscious mind to solve it

6. Use this site,it is wealth of info.