I would really like to hear from any professional mathematicians who didn't just sail through their university education. If one looks at the pages of many of today's mathematicians, one finds that they usually aced their university exams (often coming top of the year at prestigious universities etc). This can be disheartening (at times) for someone who hasn't followed that path. Is there actually a non-negligible hope for such people? Thank you.
I didn't always ace my exams in school. Being a mathematician has nothing to do with test scores.
My advice is to make mathematics personal. It's not about anybody else but you. In my opinion, it's best to compete with yourself. Try and have your own relationship with mathematics, understand it in a way that is your own, and, as always, engage mathematics because it gives you great joy.
My advice has two parts:
1) Don't worry about other people. Chances are, the successful people you refer to got that way from thinking about their work and not by focusing on the performance or others. I often found myself falling into this trap as an undergraduate. I'd say about half of what should have been math time was spent worrying about my relative abilities. Not coincidentally, my absolute abilities improved dramatically when I focused on work alone. It's not easy, but it's worth it. (Also, for every unimpeded genius you find, there's another mathematician, just as successful, with a rockier path.)
2) Don't be afraid to do mathematical work outside of pure mathematics. You may feel comfortable in a computational field outside of pure mathematics. I studied pure math for my B.A. and M.A. but then switched to an applied field for my PhD. I sense less competition in the applied fields because of the large, messy and probably insoluble problems applied scientists all face. I still get to do lots of rewarding and advanced mathematics, without what could be called the sanctimonious attitudes of some people in the pure domain. Since my switch, I've felt much better about my abilities. By feedback, my abilities have improved as a result. Maybe this would work for you too.
I wanted to post the excerpt below somewhere because I think many will find it interesting and, as far as I can tell, it seems to be nowhere on the internet (see my comments here). After several minutes of searching, this question wound up being the closest fit that I was able to find.
What follows are the italicized (except for 9 words) introductory comments to Part 3. Mathematics in Isaac Asimov’s 1969 book Opus 100 (amazon.com page and Wikipedia page)—pages 89−91 in my 1969 Dell paperback version and, I think, pages 87−90 in the 1969 Houghton Mifflin hardback version. Asimov has three more personal commentaries in Part 3 (pages 94−95, 102−104, 113−115 in my paperback version), but those other commentaries veer away from the main topic of what follows.
WHEN I WAS IN GRADE SCHOOL, I had an occasional feeling that I might be a mathematician when I grew up. I loved the math classes because they seemed so easy. As soon as I got my math book at the beginning of a new school term, I raced through it from beginning to end, found it all beautifully clear and simple, and then breezed through the course without trouble.
It is, in fact, the beauty of mathematics, as opposed to almost any other branch of knowledge, that it contains so little unrelated and miscellaneous factual material one must memorize. Oh, there are a few definitions and axioms, some terminology—but everything else is deduction. And, if you have a feel for it, the deduction is all obvious, or becomes obvious as soon as it is once pointed out.
As long as this holds true, mathematics is not only a breeze, it is an exciting intellectual adventure that has few peers. But then, sooner or later (except for a few transcendent geniuses), there comes a point when the breeze turns into a cold and needle-spray storm blast. For some it comes quite early in the game: long division, fractions, proportions, something shows up which turns out to be no longer obvious no matter how carefully it is explained. You may get to understand it but only by constant concentration; it never becomes obvious.
And at that point mathematics ceases to be fun.
When there is a prolonged delay in meeting that barrier, you feel lucky, but are you? The longer the delay, the greater the trauma when you do meet the barrier and smash into it.
I went right through high school, for instance, without finding the barrier. Math was always easy, always fun, always an “A-subject” that required no studying.
To be sure, I might have had a hint there was something wrong. My high school was Boys High School of Brooklyn and in the days when I attended (1932 to 1935) it was renowned throughout the city for the skill and valor of its math team. Yet I was not a member of the math team.
I had a dim idea that the boys on the math team could do mathematics I had never heard of, and that the problems they faced and solved were far beyond me. I took care of that little bit of unpleasantness, however, by studiously refraining from giving it any thought, on the theory (very widespread among people generally) that a difficulty ignored is a difficulty resolved.
At Columbia I took up analytical geometry and differential calculus and, while I recognized a certain unaccustomed intellectual friction heating up my mind somewhat, I still managed to get my A’s.
It was when I went on to integral calculus that the dam broke. To my horror, I found that I had to study; that I had to go over a point several times and that even then it remained unclear; that I had to sweat away over the homework problems and sometimes either had to leave them unsolved or, worse still, worked them out incorrectly. And in the end, in the second semester of the year course, I got (oh shame!) a B.
I had, in short, reached my own particular impassable barrier, and I met that situation with a most vigorous and effective course of procedure—I never took another math course.
Oh, I’ve picked up some additional facets of mathematics on my own since then, but the old glow was gone. It was never the shining gold of “Of course” anymore, only the dubiously polished pewter of “I think I see it.”
Fortunately, a barrier at integral calculus is quite a high one. There is plenty of room beneath it within which to run and jump, and I have therefore been able to write books on mathematics. I merely had to remember to keep this side of integral calculus.
In June, 1958, Austin Olney of Houghton Mifflin (whose acquaintance I had first made the year before and whose suggestion is responsible for this book you are holding) asked me to write a book on mathematics for youngsters. I presume he thought I was an accomplished mathematician and I, for my part, did not see my way clear to disabusing him. (I suppose he is disabused now, though.)
I agreed readily (with one reservation which I shall come to in due course) and proceeded to write a book called Realm of Numbers which was as far to the safe side of integral calculus as possible.
In fact, it was about elementary arithmetic, to begin with, and it was not until the second chapter that I as much as got to Arabic numerals, and not until the fourth chapter that I got to fractions.
However, by the end of the book I was talking about imaginary numbers, hyperimaginary numbers, and transfinite numbers—and that was the real purpose of the book. In going from counting to transfiinites [sic], I followed such a careful and gradual plan that it never stopped seeming easy.
Anyway, here’s part of a chapter from the book, rather early on, while I am still reveling in the simplest matters, but trying to get across the rather subtle point of the importance of zero.
Rob Kirby may be the example you are looking for (https://www.simonsfoundation.org/science_lives_video/robion-kirby/):
In 1963, rising mathematical star John Milnor set forth a list of what he considered the seven hardest and most important problems in the nascent field of geometric topology. Just five years later, no fewer than four of those problems had been laid to rest, largely through the efforts of a young mathematics professor whose entry into mathematics research had seemed anything but auspicious. Described by colleagues and students as “slow,” “non-threatening,” and “deliberate,” Robion Kirby had followed a desultory path through higher education, marked by failed exams, lost fellowships and recommendations that he go study somewhere else. Yet just three years out of graduate school, he pulled off a mathematical coup, one that helped define the future of his field.
“I sometimes felt like the Virgin Mary,” Kirby says. “How could this happen to me?”
Another example (coincidentally also in topology) is Stephen Smale: https://en.wikipedia.org/wiki/Stephen_Smale
Smale entered the University of Michigan in 1948. Initially, he was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly in his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrandt, threatened to kick Smale out that he began to work hard. Smale finally earned his Ph.D. in 1957, under Raoul Bott.