Question

E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what about the last one hundred years? Is it still possible for an amateur to make a significant contribution to mathematics? Can anyone cite examples of important works done by amateur mathematicians in the last one hundred years?

For a definition of amateur:

I think that to make the term "amateur" meaningful, it should mean someone who has had no formal instruction in mathematics past undergraduate school and does not maintain any sort of professional connection with mathematicians in the research world. – Harry Gindi

Answer 1

About ten years ago Ahcène Lamari and Nicholas Buchdahl independently proved that all compact complex surfaces with even first Betti number are Kahler. This was known since 1983, but earlier proofs made use of the classification of surfaces to reduce to hard case-by-case verification.

At the time, Lamari was a teacher at a high school in Paris. Apparently he announced his result by crashing a conference in Paris and going up to Siu (who had proved the last case in the earlier proof in 1983) with a copy of his proof. Lamari's proof was published in the Annales de l'Institut Fourier in 1999 (link: http://tinyurl.com/2e6cbj5).

Answer 2

After Martin Gardner published what one mathematician claimed to be a complete list of convex pentagons that could tile the plane, amateurs (Richard James III, a computer scientist, and Marjorie Rice, who had no mathematical training beyond high school) discovered several more classes of pentagons that could tile.

Answer 3

Greg Egan. He's a very renowned science fiction writer who holds a bachelor degree in mathematics. He wrote, as a coauthor, 2 articles which were published in peer-reviewed journals, one of them is with John Baez. The first one was written when he was approximately 40 years old.

There's also more eccentric example of Andrew Beal, which is much more known in the world of poker. He made however one minor conjecture in number theory for whose proof or disproof he offers $100,000.

And there's also a list on wikipedia which might be worth going through.

Answer 4

There are many interesting discoveries made by mathematical distributed computing projects.

• The Great Internet Mersenne Prime Search: http://www.mersenne.org/
• ABC@Home: http://en.wikipedia.org/wiki/ABC@Home
• PrimeGrid: http://en.wikipedia.org/wiki/PrimeGrid
• Ramsey@Home: http://boinc.berkeley.edu/wiki/Ramsey@Home

Their discoveries don't have an impact in the same way that theorems do, but from time to time resolving a theorem boils down to computation, and most of the participants are probably interested amateurs.

Answer 5

Kenneth A. Perko Jr. is a lawyer and an amateur topologist. In 1974 he found -while manipulating loops of rope on his living floor- that two knots that were listed as separate knots in C. N. Little's "On knots, with a census for order 10" (1885) and similar tables, were actual identical.

Mathoverflow-user Daniel Moskovich recounted earlier on this site:

Little (with Tait and Kirkman) compiled his tables combinatorially. He drew all possible 4-valent graphs with some number of vertices (in this case 10), and resolved 4-valent vertices into crossings in all possible ways. He ended up with 2¹⁰ knots. Then he worked BY HAND to eliminate doubles, by making physical models with string. He failed to bring these two knots to the same position, and concluded that they must be different. It took almost 100 years to find the ambient isotopy which shows that there are the same knot.

The book "Knots and Links" by Dale Rolfsen, published two years after Perko's publication, still lists the knots as different, they are knots [; 10_{161} ;] and [; 10_{162} ;] in Appendix C.

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Argh, I don't have enough reputation to add image links. Someone please edit this post and add the following image tag:

Answer 6

I think Escher qualifies. See Doris Schattschneider, The mathematical side of M. C. Escher, Notices of the American Mathematical Society 57 (2010) 706-718, http://www.ams.org/notices/201006/rtx100600706p.pdf

Answer 7

Robert Ammann had some extremelly important contributions to the study of aperiodic tilings, and to Quasi-crystals.

Answer 8

$K_n$ is not planar for $n \geq 5.$ One may ask: what is the minimum Euler characteristic $\gamma(K_n)$ among all compact orientable surfaces into which $K_n$ may be embedded? It is a nice exercise to embed $K_5,$ $K_6,$ and $K_7$ into the torus. The final result was that $\gamma(K_n) = 2 \lfloor \frac{n (7 - n)}{12} \rfloor.$ In 1968 this theorem had been proven for "all cases except $n = 18,20,$ and $23.$ The proof was completed, at the end of the sixties, by Jean Mayer, a professor of French literature (!), when he found embeddings for these three values." (Surface topology, Firby and Gardiner, p. 111).

Answer 9

Bill Gates co-authored the following paper in the 1970s with Christos Papadimitriou:

"Bounds for sorting by prefix reversal," Discrete Mathematics 27 (1979), no. 1, 47–57, MR0534952.

Not sure if Gates counts as an amateur, but he is at least a college dropout. :)

The only reason I know this is because once I ran across a book or article that discusses the results in this paper and then says something like, "Yes, this is THE Bill Gates." I was almost certain the book or article was by Knuth, but now I can't find the reference in any of my Knuth books. If someone else knows the reference I'm talking about, I would be grateful if they would post it as a comment to my answer. (It now bothers me that I can't find that reference. :) )

Answer 10

While this is on the front page again, I wanted to make mention of Joan Taylor, who discovered an aperiodic single tile, which she published with Joshua Socolar of Duke University in 2010. This is her bio blurb as it appears on their article in The Mathematical Intelligencer:

JOAN M. TAYLOR took up mathematics in 1991 at age 34 after being inspired by a magazine article on quasicrystals featuring Penrose’s rhombus tiling. She began but did not complete a degree, preferring to conduct her own research. Since then she has pursued tiling and related topics in abstract algebra and number theory including original work on constructible polygons. She likes to unwind with knitting and reading.

Answer 11

The important artists Anthony Hill and John Ernest proposed an upper bound for the crossing number of complete graphs (published by Richard Guy in 1960). Hill made other contributions to graph theory and was elected to the London Mathematical Society in 1979.

Answer 12

I don't know if this really qualifies, but I would say that Scott Draves can be viewed as an amateur mathematician for inventing/discovering the fractals known as Flame fractals. His work is more towards art, but there is a decent amount of math behind to optimize the aesthetically properties of the fractals. (The "nicest" fractal dimension is 1.52, for example).

Answer 13

The American Institute of Mathematics, a nonprofit organization, was founded in 1994 by Silicon Valley businessmen John Fry and Steve Sorenson, longtime supporters of mathematical research.

That's Fry, as in Fry's Electronics, a retail chain in California.