Bruce Bartlett Wins Levi Conant Prize
Posted by John Baez
Hurrah! Bruce Bartlett has won the 2027 Levi L. Conant Prize for the best expository paper in either the Notices or Bulletin of the American Mathematical Society. He won it for his paper The quintic, the icosahedron and elliptic curves, and I recommend that you stop reading this right now and read that instead.
No? You’re still reading this? Okay, here’s what the AMS said about Bruce’s prize-winning paper:
The article describes a deep and beautiful connection between the roots of a quintic polynomial, the icosahedron, and elliptic curves. As Bartlett explains, the original discovery of this connection was made by Felix Klein in 1878. The goal of the article is to recount Klein’s story and update it with new ideas and perspectives that have developed over the last 150 years. It succeeds in doing this in a a clear, engaging, and exciting fashion.
More specifically, the article draws a direct correspondence between (a) the “evenly ordered” roots for a specific family of polynomials, the Brioschi quintics, (b) points on the icosahedron, and (c) elliptic curves equipped with a primitive basis for their 5-torsion. The article then uses this correspondence to give an elegant algorithm for computing the roots of an arbitrary quintic polynomial.
Bartlett invites the reader to join “on a mathematical carpet ride through the mathematics of the last four centuries. Along the way we will marvel at Kepler’s Platonic model of the solar system from 1597, witness Gauss’ excitement in his diary entry from 1799, and experience the atmosphere in Trinity College Hall during the wonderful moment Ramanujan burst onto the scene in 1913.” Again, the article lives up to this billing.
On the way to describing the root-finding algorithm, the article wends its way through a wide variety of rich mathematics. It touches upon classical invariant theory, elliptic curves and their moduli space, covering space theory, the structure of the symmetric group, and the group theory of the modular group and its subgroups.
The reader also learns about Vladimir Arnold’s topological proof of the unsolvability of the quintic by radicals (originally presented to high school students in Moscow), the complex arithmetic-geometric mean of Cremona and Thongjunthug, and the continued fraction of Rogers and Ramanujan. Amazingly, along with all the fantastic mathematical ideas, the article manages to give salient insights into the people behind those ideas. There are cameos from Hermite, Hardy, and Russell, among many others.
The article is beautifully illustrated. It ends with a step-by-step description of the root-finding algorithm, as well as a link to an implementation of the algorithm in Mathematica, a fitting exclamation point to this broad, engaging, and hands-on tour through the wonders of the quintic polynomials.
Bruce replied, in part:
I am greatly honoured and chuffed to receive the 2027 Levi L. Conant prize for my April 2024 paper in the Notices on “The Quintic, the Icosahedron and Elliptic Curves.” I had long been fascinated by this topic, but I could never quite understand it, until I got hold of Sander Bessels’ wonderful Master’s thesis from 2006, which explained that the “five things” which are being permuted when we rotate the icosahedron are the five inscribed octahedra. I gradually became aware that there are a number of simple but remarkable geometric insights like this, each individually known to a group of experts, but apparently no-one had made a coherent expository account of it all. (Another example is how the six vertex axes of the icosahedron can be placed in an intriguing correspondence with the six points in the projective line over the field with five elements, giving an explanation for the group isomorphism A5 = PSL(2,5)… this had to wait for my follow-up article though!) Once Ramanujan entered the story, I knew I could write up a nice article.
Nice use of the word “chuffed”! I’m glad that Bruce likes many of the things I like, and has managed to fit a bunch of them together more clearly than I ever could.