
OBSERVER
Jilted by Sweden, Feted by Norway, Mathematics Finally Gets Its Due
By DANIEL ROCKMORE
Like clockwork each fall since 1901, the announcements have come forth from Sweden heralding the new class of science heroes, that year's Nobel laureates. Almost always the honored discoveries in physics, chemistry, medicine, and economics are written in the language of mathematics. Nevertheless, to the public's surprise and often to the irritation of the discipline's practitioners, there is no Nobel Prize in mathematics.
Last year, however, a math award equal in stature arrived on the circuit. Announced each spring, the Abel Prize, currently worth about $900,000, recognizes outstanding achievement in mathematics.
The prize is named for Niels Abel (180229), a Norwegian mathematician whose tragically short life was marked by heights of intellectual achievement and depths of personal misfortune. Over the course of a mere 10 years he managed to lay the groundwork for a wide range of mathematical fields. His inventions and discoveries are still crucial to any mathematician, physicist, chemist, or engineer. Even so, he received no real benefit in his own life.
The Abel Prize was created by the Norwegian government to remedy the Nobel neglect of mathematics. A variety of stories purport to explain the missing Nobel in math. The best is that its omission was the sweet revenge of Alfred Nobel for his cuckolding at the hands of the Swedish mathematician Magnus Gösta MittagLeffler. But there is little evidence for that tabloidready tale. By now, the accepted dogma is simply that Nobel didn't have any real interest in mathematics as a standalone discipline. Rather, his appreciation was for the application of mathematics in the form of engineering. Born into a family of engineers, over a tremendously productive lifetime Nobel obtained more than 350 patents, among which are inventions related to synthetic forms of rubber, silk, and leather. The primary source of his fame and fortune, of course, was his contribution to the invention of dynamite. Nobel is credited with helping to stabilize the explosive in order to increase its safety, not only in military applications but also as a tool for rock blasting in construction.
To salve his pacifist conscience and spur applied scientific achievement, Nobel created the eponymous prizes, decreeing that they be presented "to those who, during the preceding year, shall have conferred the greatest benefit on mankind." It is likely that pure mathematics did not pass that test in the engineer's teleological view.
Nobel's prizes recognize discoveries that shake the scientific firmament. Winning theories blow away old beliefs, revealing new scientific bedrock. Aristotle's earth, water, air, and fire fall to the elements of Lavoisier, which give way to the atoms of Dalton, which step aside for Bohr's electronnucleon model, which surrenders to the charmquark model of GellMann.
That subsuming of the old by the new is not the aesthetic of mathematics. The mathematician's results are not of a time, but for all time; that is the main difference between theories and theorems. In mathematics, the old informs the new, not just as straw man or benchmark, but as inspiration, and even source material. It is not uncommon for a mathematician to need a 100yearold paper to work on current problems.
Until last year, mathematicians would point to the Fields Medal as the discipline's Nobel equivalent. The puregold medal is named in honor of the Canadian mathematician J.C. Fields, who endowed the prize in 1932.
One way in which the Fields Medal differs from its cousins in the physical sciences is that the latter generally arrive late in a scientist's career, marking a fundamental discovery that has stood the test of time, whereas the Fields, by convention, is given to as many as four of the best and brightest mathematicians younger than 40. After all, the nature of mathematics guarantees the work's irrefutability.
Now it is the Abel that stands as the highest recognition. The first Abel Prize went to JeanPierre Serre, an emeritus professor at the Collège de France, in Paris. He was honored for his landmark work in the fields of algebraic geometry and algebraic topology. The former makes algebra of Euclidean geometry, turning the literary descriptions of lines and curves into equations of x's and y's. The latter knits the seemingly disparate fields of algebra and topology. Topology takes the perfect shapes of Euclid and considers their general properties. To a topologist, a circle is a triangle is a square, and algebraic topology finds formulas to quantify the geometric essence of topologically similar shapes. Serre's greatness is in the groundbreaking ways in which he applied the formal manipulations of algebra to the squishy world of topology.
This year's Abel, which went last month to Sir Michael Francis Atiyah, of the University of Edinburgh, and Isadore M. Singer, of the Massachusetts Institute of Technology, once more recognizes crossover achievement. In particular, their shared prize rewards the work represented by a series of papers written in the early 1960s, which culminated in the proof of the AtiyahSinger Index Theorem  in the words of the academy, "one of the great landmarks of 20thcentury mathematics." The theorem, at the nexus of analysis, differential geometry, and algebraic topology, over the past generation has provided a mathematical foothold in the search for physics' Holy Grail  a Grand Unified Theory, or socalled "theory of everything," which might knit all the laws of nature into one mathematical haiku of symbology.
We've already touched upon algebraic topology. Differential geometry is a second branch of mathematics that finds its roots in Euclid's geometric world, in which the implications of unvarying straight lines and perfect circles are chased to their logical conclusions. Differential geometry uses Euclidean inspiration to allow us to quantify a world that is both crinkled seashore and powderpuff cloud. Analysis, the third ingredient of the Index Theorem, is an offshoot of calculus, that bag of mathematical tricks invented for physics by Newton and Leibniz as a means of quantifying change as it occurs on both macroscopic and microscopic scales.
Those three subjects meet in the AtiyahSinger Index Theorem to produce a single number (that's the "index") that helps measure the complexity of the possibilities for physical laws in a given geometric setting. Written in the language of mathematics, physical laws become differential equations, symbolic sentences that relate the rates of change of various quantities of interest. Perhaps the most famous example is the wave equation, which relates, say, the upanddown displacement of a point on a guitar string to the rate of that displacement. Any wave, represented by the solution of a wave equation, can be considered an amalgamation of certain fundamental waves, much like any musical sound can be thought of as the simultaneous sounding of a collection of variouslength strings, each played at a particular volume or plucked at a particular strength. The Index Theorem computes the measure of complexity (i.e., the number of strings needed to make the music) from the geometric and topological descriptors of the ambient space.
Atiyah and Singer pursued their mathematical quarry only with the goal of going where no mathematician had gone before. But in mathematics, work done purely in the pursuit of pushing boundaries turns out to be just the thing needed to help explain the world around us. That mysterious ability of mathematics to provide a means of quantifying and predicting natural phenomena is what the physics Nobel laureate Eugene Wigner called "the unreasonable effectiveness of mathematics." The Index Theorem happened to provide physicists in the 1970s with a means for understanding the unity between electromagnetic and weak forces. More recently, it has helped them to understand, in the superstring model of the universe, the jungle of elementary particles that may be the primary constituents of matter.
The AtiyahSinger Index Theorem is mathematics at its best  beautiful in and of itself, but also providing the connective tissue for understanding phenomena beyond itself. Physicists can point to the stars, chemists to the elements, biologists to the genome, economists to the markets, and in each of those cases there is something tangible, empirical, and experiential. In distinction, mathematics is in the fabric of those disciplines, the ghost in the machine. While recognized by some as the Queen of the Sciences, it is perhaps better seen as science's Cinderella, possessing the understated nobility of the fairytale stepchild who does so much of the heavy lifting out of the limelight. In that sense, it is better represented by the humble origins of Abel than the illustrious background of Nobel. Thanks to the generosity of a fairygodmother Norwegian government, mathematics is finally assuming its rightful place of public honor.
Daniel Rockmore is a professor of mathematics and computer science at Dartmouth College.
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Section: The Chronicle Review
Volume 50, Issue 31, Page B5
