Wednesday, August 13, 2008
Wednesday Math, Vol. 34: Bernhard Riemann
What are the criteria for measuring greatness in an art form? The standards are not the same across the different disciplines.
The most rigorous standard would be to produce a vast body of work with several acknowledged masterpieces interspersed among these works. By that standard, the greatest artists would be people like Shakespeare, Bach, Mozart, Dickens, Rembrandt and others. In literature, there are masters given the highest rank who were not particularly prolific, or only produced one or two masterworks. Leo Tolstoy stands at the top of the pantheon of world literature based on two large and sprawling novels; the rest of his work is much less well known. Herman Melville is given a lofty perch as well, though perhaps I think more of him than he deserves because we are both Americans. He died in complete obscurity, and after Moby Dick, his second best known work today, Billy Budd, was published after his death because he could not find a publisher when he was alive. That novel's success is greatly enhanced by Benjamin Britten turning it into an opera and Peter Ustinov turning it into a play and film.
In music, the top of the pantheon is largely reserved for those who meet the most difficult criteria, the massive body of work with many masterpieces. Georges Bizet, for example, is not often mentioned among the greatest of the opera composers. After Carmen, clearly an important and popular opera, his second best known work is... The Pearl Fishers? La Jolie Fille de Perth? Sadly, Bizet died of a heart attack at the age of 36 only a few months after composing Carmen. For all that work's success, the artist himself does not get placed in the pantheon beside Verdi, Puccini and Wagner.
In math, the criteria are much the same. Some mathematicians produce one truly great result, but that cannot put them at the top of the heap. Both Neils Abel and Evariste Galois are given credit for independently proving the impossibility of solving the quintic equation algebraically, which was the most important open problem of the early 19th Century. Sadly, both died young, Abel of tuberculosis and Galois in a duel, and no one would put their work at the same level of someone like Archimedes, Newton or Gauss.
If I were to make a list of the Top Ten mathematicians of all time, and I limited the list to the deceased, my list would obviously include the Big Three mentioned above, and I would be joined by the vast majority of mathematicians when I include David Hilbert, Leonhard Euler and last week's star John Von Neumann. Other names like Poincaré and LaGrange and Kolmogorov might also be included, all of whom meet the standard of prolific work and many important results.
But one man, and as far as I can think one man alone, would make many mathematicians' lists of the greatest of all time without meeting the criterion of prolific work. Bernhard Riemann did some of the most important mathematical work of the 19th Century, working with many of the top mathematicians of his day, but he only produced a handful of papers. Each one of those papers is a treasure trove of mathematical ideas, and the work that was done to extend those ideas is some of the most important work in math and physics ever.
There are many things in mathematics named for Riemann, though like a good mathematician he did not name them after himself. One of his best works he called the Dirichlet Principle, naming it after his favorite professor and advisor. Today, there is the Riemann integral, its improvement the Riemann-Stieltjes integral, the Riemann Hypothesis, the Riemann zeta function and most importantly, the Riemannian manifold.
The Riemannian manifold is a very important extension of integral calculus, the idea of being able to take an integral over a complex and possibly curved surface, instead of just taking a measure over a simple flat surface like a line or a plane. The pictures here are a model of the real projective plane, a two dimensional surface that cannot be truly built in three dimensions because it must pass through itself without having a hole in the surface. The idea of the manifold is that this odd looking shape, and many other odd looking shapes, can be defined by covering the surface with a patchwork quilt of flat or nearly flat patches, and if two patches overlap, there is an "easy" method of matching points on one patch to the corresponding points on the other. Of all of Riemann's ideas, the manifold may be the most useful, because without it, Einstein would not have had a mathematical model for the idea of curved space.
Like way too many great artists of the 19th Century, Bernhard Riemann died young, and again like too many, he died of tuberculosis. He was 39 when died, but his ideas and his name live on, and if he is not one of the top ten mathematicians of all time, he is most certainly in the top twenty.