Three astounding feats accomplished quickly:

1. As related in Mark Kac‘s memoir “Enigmas of Chance”. I recommend reading this. Two quotes to give a flavour of Kac:

* There are two kinds of geniuses: the ‘ordinary’ and the ‘magicians’. An ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better: there is no mystery as to how his mind works – once we understand what they’ve done, we feel certain that we, too, could have done it. It is different with the magicians … [Richard] Feynman is a magician of the highest caliber.

* I then reached for a time honored tactic used by mathematicians: if you can’t solve the real problem, change it into one you can solve.

In Enigmas of Chance, Kac relates that in the 1940s an important problem in Topology (generalised geometry) was what is the dimension of the set of rational points in Hilbert Space (a set now called “Erdos Space”). Don’t worry about what that means.

[Caveat: the following is from my memory: some details may be incorrect, but the main point is what I read in Enigmas of Chance.]

Some mathematicians were discussing this problem, for which at the time the experts thought that the dimension of this set was either zero or infinity, but nothing in between, which shows how tricky the problem was. Paul Erdos wandered up and asked them what they were talking about, so they told him. Demonstrating his (lack of!) knowledge of the subject, he asked them what they meant by dimension. (For clarity, Erdos would have been quite well aware of the normal definition of dimension: what he wanted to know was how was dimension defined for this problem. There is more than one way to define dimension.) So they told him – basically an inductive definition, starting with a point has zero dimension, from which you inductively get the dimension of a line as 1, of a plane as 2, and so on – and then said now please go away Paul and stop bothering us. Which he did, only to return about an hour later with the (correct) answer that the dimension of this set was 1.

That’s going from a start of knowing absolutely nothing about a subject to solving an important problem in the subject which the experts hadn’t yet solved, and doing it in about an hour. If there is anything more impressive, I haven’t yet heard it.

A similar feat by Erdos, in the words of someone who saw it is here.

2. From the book “Love and Math” by Edward Frenkel.

The Russian mathematician Gelfand had Vladimir Kazakov, a physicist, present a series of talks about Kazakov’s work on applying quantum physics methods to mathematics. During this series John Harer and Don Zagier found a beautiful solution to a difficult combinatorial problem. Frenkel comments that Zagier was known for solving intractable problems and for also being quick: reputedly this took him six months. Gelfand suspected that Kazakov’s methods could be useful for these combinatorial problems, and during a presentation asked Kazakov to solve the the Harer-Zagier problem. Kazakov did not know about Harer and Zagier’s work, and on his feet thought for about two minutes and then wrote down a formula that would solve the problem using his methods.

This amazed everyone present, except for Gelfand, who mischievously asked Kazakov how long he had been working on his theory: Kazakov replied perhaps six years. Frenkel (who was present) reports Gelfand as then saying: “So it took you six years plus two minutes, and it took Don Zagier six months. Hmmmm… You see how much better he is?”

3. In a way, back to Paul Erdos, to be precise to someone who later collaborated with him.

Fan Chung” has published 13 joint papers with Paul Erdos, which strongly suggests her mathematics ability is not so dusty. The following story comes from her biography at a St Andrews University website:

Fan Chung’s father was an engineer. She attended high school in Kaoshiung, Taiwan and was encouraged to take up mathematics by her father who told her:

“… in math all you need is pencil and paper.”

She entered the National Taiwan University to read for a B.S. in mathematics. In [1] she described how she was encouraged to think in terms of a career in mathematics by interaction with her fellow students:

“As an undergraduate in Taiwan, I was surrounded by good friends and many women mathematicians. We enjoyed talking about mathematics and helping each other. A large part of education is learning from your peers, not just the professors. Seeing other women perform well is a great confidence builder, too!”

It was during her years as an undergraduate in Taiwan that she was first attracted to combinatorics, the area in which she was soon to begin research:

“… many problems from combinatorics were easily explained, you could get into them quickly, but getting out was often very hard … Later on I discovered that there were all sorts of connections to other branches of mathematics as well as to many applications.”

Chung graduated with a B.S. in mathematics in 1970 and then went to the United States for her graduate studies. She entered the University of Pennsylvania but at first Herbert Wilf, Professor of Mathematics at the University of Pennsylvania, hardly noticed her. Wilf writes (see [1]):-

“I never paid any attention to the graduate students until they got past their qualifying exams. My policy then was to go after the best student and try to get him to go into combinatorics. The year she took the exam, 1971, she had the highest score by far-there was a huge gap between her and the next best student. So I immediately sought her out – I had never spoken to her before – and asked her if she knew anything about combinatorics. She said she knew a little from her days at Taiwan National University but not too much. I pulled out one of my magnetic subjects, Ramsey theory, that is guaranteed to get graduate students hooked on combinatorics because it is very pretty stuff. I gave her a book and told her to read the chapter on Ramsey theory. We set up an appointment in a week to talk about it. When she came to the appointment, I asked her how she liked the chapter. She smiled and said it was fine. Then she flipped the book open to a key theorem and said, gently, “I think I can do a little better with the proof.” My eyes were bulging. I was very excited. I asked her to go to the blackboard and show me. What she wrote was incredible! In just one week, from a cold start, she had a major result in Ramsey theory. I told her she had just done two-thirds of a doctoral dissertation. “Really?” she said softly. In fact, the result did become a major part of her dissertation.”

…

[1] Fan Chung’s home page

***** Now that’s impressive!

***** Tip – go to her home page (making sure you have javascript enabled), wait for it to fully load, and then move your computer screen cursor around the webpage! (Update 17.May.2017: alas it seems that the entertaining graphic may no longer be working.)

***** Researching this post (in other words wwww searching) I found an article “On Buckets and Fires” written by Herbert Wilf in 2010 on the subject of mathematics education in schools. The title comes from a saying, possibly by Plutarch: “A mind is a fire to be kindled, not a bucket to be filled.” I recommend reading this because he makes a lot of sense on education, not just mathematics education, and not just in schools. An extract:

… I have seen in my own office many undergraduates who enter their freshman years propelled like rockets toward particular careers—medicine, the law, computer science, whatever. My advice to them has always been the same: “Cool it. You are a student in one of the great universities of the world [*]. You can study medicine later. For now, try archaeology, Sanskrit, art appreciation, mathematics, etc., all of which are available in abundance here. Make your choices based on who will be teaching the various courses. Sign up for the ones that have stellar teachers. Don’t worry about what they’re teaching. Just listen, think, and enjoy.” …

[*] University of Pennsylvania

… One of Penn’s most well known academic qualities is its emphasis on interdisciplinary education, which it promotes through numerous joint degree programs, research centers and professorships, a unified campus, and the ability for students to take classes from any of Penn’s schools (the “One University Policy”) …

[That reminds me of the University of Warwick Mathematics Department freedom *and* encouragement to take courses from other departments, at least when I was there many years ago.]

***** In the opposite direction from astounding feats accomplished quickly, an entertaining and instructive anecdote about someone not doing something very easy. The story (from Manifold, a Warwick University mathematics magazine for undergraduates) tells of Stephen Smale https://en.wikipedia.org/wiki/Stephen_Smale (an indisputably great mathematician, who at one point in his student years at university had mediocre grades) giving a seminar at Warwick on (I think) some aspect of topology, and in the course of his talk he had occasion to put on the blackboard a pair of simple simultaneous equations not dissimilar to (but not exactly like) (a) x + 5y = 7, (b) 13x + 7y = 42, and looked at these for some (I assume not many) minutes before somone else at the seminar told him how to solve them. (Caveat: assuming the story is true, I think it unlikely that Smale couldn’t usually solve such a simple pair of simultaneous equations, but I am prepared to believe that in the context of giving a seminar on a complicated topic he became temporarily unable to deal with a very simple problem.)

***** On the subject of mathematics and its relation to other sciences and arts (is mathematics a science or an art? maybe), an xkcd cartoon

(Aside on 5.May.2015 as a defence of psychology: in a 2013 email to a psychology researcher – who is also a concert pianist – I wrote that if I had a time machine I’d go back to university and combine studying mathematics with studying psychology – mostly human, but with some non-human animal. Further addition 17.May.2017: expanding on that, I suspect that it might be more difficult to do psychology (or sociology, etc) really well than to do physics really well, and that to deal with a lot of humanity’s current and future problems, really good psychology and sociology may be more important than really good physics.)

Note the XKCD Warning: this comic occasionally contains strong language (which may be unsuitable for children), unusual humor (which may be unsuitable for adults), and advanced mathematics (which may be unsuitable for liberal-arts majors).

Other XKCD cartoons I like include: Science Montage (movie compared to reality), stories or a large version; surreal; sweet & touching physics.