A Strange Wilderness

by Amir D. Aczel

  Ramanujan was awarded a doctorate for his work at Cambridge, and he eventually became a Fellow of Trinity College, a Fellow of the Cambridge Philosophical Society, and in 1918 was elected Fellow of the Royal Society. But his health suffered. He had endured a variety of ailments throughout his life, and in England he felt worse. After five years in Cambridge, he required frequent hospitalizations. Although he was believed to have been suffering from tuberculosis, later findings suggested a parasitic infection affecting his liver.

  Ramanujan was said to be a friend of every integer, and no incident demonstrates his love for numbers better than an exchange he had with Hardy not long before his death. While he was lying ill, Hardy came to visit him. In an attempt to lift the ailing mathematician’s spirits, Hardy led off with a comment about a number, numbers being Ramanujan’s favorite topic: “I came here in a taxi with a very boring number: 1729.” To his surprise Ramanujan gathered whatever strength he still possessed, jumped up in bed, and cried, “No, Hardy, no, Hardy—it’s a very interesting number! It’s the smallest number expressible as sums of two cubes in two different ways!” (The number 1,729 = 103 + 93 = 13 + 123.) From this event, the mathematical study of taxicab numbers—the smallest numbers that can be expressed as sums of two (positive) cubes in n distinct ways—emerged. To date, only the first six taxicab numbers have been found.

  Shortly after Hardy’s visit, Ramanujan’s condition worsened. In despair, he decided to return home to India, thinking that the warm weather there would help him improve. In February 1919 he left for India, arriving in Madras the following month. Unfortunately, he was already too sick to recover. On April 26, 1920, in Kumbakonam, Ramanujan died. He was only thirty-two years old. His widow, Amal, continued to live in Madras until her death seventy-four years later, in 1994.

  FELIX KLEIN AND THE

  ERLANGEN PROGRAM

  One of the most prominent mathematicians in Germany in the late 1800s was Felix Klein (1849–1925), who taught at the University of Erlangen, in Bavaria. After attending the gymnasium in his hometown of Düsseldorf, he matriculated from the University of Bonn, where he had plans to earn a degree in physics.

  Under the direction of Julius Plücker, the chair of mathematics and experimental physics, Klein studied many types of geometry that were emerging at the time. He also studied the connections between geometry and group theory—the area discovered by Galois. In 1868, he received his doctorate. When Plücker died in the middle of writing a book on geometry, Klein took up where his former professor had left off and completed it. The resulting work, New Geometry, included ideas on such emergent geometries as the Riemannian geometry of the brilliant German mathematician Georg Friedrich Bernhard Riemann (1826–66), which is very important in both mathematics and theoretical physics, as well as the non-Euclidean geometries developed by Gauss, János Bolyai (1802–60), and Nikolai Lobachevsky (1792–1856). Klein visited Paris and was greatly impressed by the methods of group theory developed at that time. Groups can be connected with geometries. Once a group associated with a geometrical object can be identified, this gives the mathematician information about the geometry of an object that might otherwise not be discoverable.

  At the University of Göttingen, Germany, Felix Klein supervised the first Ph.D. thesis ever written by a woman in the mathematics department. The degree was awarded to Grace Chisholm Young in 1895.

  While lecturing at Göttingen, Klein met Sophus Lie, with whom he collaborated in research on groups and their properties. When he became a professor of mathematics in 1872, at the age of twenty-three, Klein gave an address that has become well known around the world. In this lecture, he inaugurated the Erlangen Program, through which he hoped to classify geometries by their associated groups of symmetries. Three years after he became professor at Erlangen, Klein married Anna Hegel, the granddaughter of the famous German philosopher Georg Wilhelm Friedrich Hegel.

  In 1882 Klein discovered a nonorientable surface now called the Klein bottle. It is an extension to three dimensions of the previously known Möbius strip, a nonorientable surface named after the German mathematician August Ferdinand Möbius, who discovered it in 1859. Both surfaces are shown below.

  The two-dimensional Möbius strip (left), and the three-dimensional shape known as the Klein bottle (right), are nonorientable surfaces.

  An ant walking on the top face of a Möbius strip would eventually find itself walking on the bottom face of the same strip, and then again on the top face, and so on as it continued. The Klein bottle, by comparison, is a tube looped back through itself to join its other opening, so it has no well-defined “inside” or “outside.” (It doesn’t really exist in Euclidean space.)

  The Georg-August-University of Göttingen was founded in 1737 by Elector George Augustus II of Hanover—a.k.a. King George II of Great Britain. This late-nineteenth-century lithograph shows the school’s Auditorium Maximum, built in 1826.

  In 1886 Klein moved to the more renowned University of Göttingen, where leading research in mathematics was taking place. But the shameful story of how Weierstrass in Berlin had to resort to teaching the brilliant Sofia Kovalevskaya on the side because women were not permitted in classes weighed on enlightened German academics at Göttingen. Klein, in particular, was pained by the German higher education system’s treatment of women.

  In 1893 he managed to convince the University of Göttingen to admit women, which was a great step forward. Under the newly passed law, Klein had the opportunity to teach a young Englishwoman named Grace Chisholm Young (1868–1944), who had been a student of the inventor of the matrix, Arthur Cayley, at Cambridge. At Göttingen, Young earned her Ph.D. working under Klein. Sexism was still rampant in Europe, however, and her first works were published under her husband’s name.

  In 1895, through the influence of Klein, Göttingen hired David Hilbert (1862–1943), who, along with Klein, made Göttingen even more prominent in mathematical research. As mentioned previously, David Hilbert gave the keynote address at the 1900 Congress of Mathematicians, in which he listed the ten problems he hoped mathematicians would solve in the twentieth century.

  Klein and Hilbert were both interested in group theory and mathematical physics. Then, in 1900, Max Planck (1858–1947), a prominent German physicist working at the Berlin Academy, discovered the quantum principle. The new discovery virtually shouted for help from mathematicians and increased interest in mathematical physics. Coincidentally, the mathematical “space” that best describes the quantum world happens to be Hilbert space, a concept developed by David Hilbert within the context of pure mathematics. This was the milieu of pure and applied mathematics in Germany when the brilliant Emmy Noether appeared on the scene.

  EMMY NOETHER

  We now turn to the life of one of the most interesting and important female mathematicians in history. Writing in The New York Times after her death, Albert Einstein described her as the most important woman in mathematical history. She lived and worked in Germany during the same period that Ramanujan worked in England and India, but whereas Ramanujan’s contributions were in analysis and number theory, hers were in abstract algebra and in the application of algebra to theoretical physics. The two celebrated Noether’s theorems laid a strong mathematical foundation for conservation laws in physics, the first theorem being essential in quantum field theory and the second in general relativity.

  The mathematician Emmy Noether, whose theorems deal with the conservation of energy in physics, appears in this undated photograph.

  Emmy Noether (1882–1935) was born to a Jewish family in Erlangen. Her father, Max Noether, came from a wealthy merchant family and taught himself mathematics. As professor of mathematics at the University of Erlangen, he was involved in the emerging field of algebraic geometry, which would later be vastly expanded and extended by Alexander Grothendieck, who is discussed in the next chapter.

  Emmy grew up in a happy home in Bavaria; she loved to dance and enjoyed music. At the public girls’ school she atten
ded, she performed very well but showed no interest in the area that would become her life’s work. She excelled in language, studying English and French, and prepared for a career as a language teacher in German state schools. She even took and passed the teacher’s qualifying examination.

  But somewhere along the way, the young woman changed her mind and decided not to teach. Instead, she received special permission—as was required of all women who attempted to study at a German university—to sit in on courses. She audited mathematics courses at the University of Erlangen, and then moved to Göttingen and audited mathematics courses taught by some of the most famous mathematicians, including Hilbert and Klein. She also sat in on courses given by the chair of the mathematics department, Hermann Minkowski (1846–1909), a Russian-born German mathematician who had previously been a professor at the Swiss Federal Institute of Technology in Zurich (ETH). There, in the 1890s, Albert Einstein had been a student of his, and Minkowski—who had done key work on number theory, studying sums of squares and the geometry of quadratic forms—acquired a strong interest in mathematical physics. Minkowski is the mathematician to whom we owe the geometrical understanding of Einstein’s four-dimensional space-time.

  Hilbert, who was developing his own interest in mathematical physics, had lured Minkowski away from Zurich to the mathematics department at Göttingen just a year before Emmy Noether arrived at the university in 1903. When she returned to Erlangen, the university gave her a degree based on all the courses she had taken unofficially at Erlangen, Göttingen, and Nuremberg, where she studied for a while. She then started her doctoral work at Erlangen and was awarded a Ph.D. in 1907, after presenting her dissertation on invariants. (Roughly speaking, invariants are things that remain mathematically the same after some change is made to the parameters of the problem.)

  Having obtained her doctorate, Noether was well qualified for a position at a university, but the persistent sexist atmosphere in Germany prevented the brilliant young woman from being able to even apply for a job. This depressed her greatly, but she began helping her father—by then in poor health—with his research. She also started publishing papers on her own, which were so well received that she was invited to join a number of European mathematical societies, including the German Mathematical Association, which had been founded by Georg Cantor.

  EINSTEIN’S REVOLUTION

  In 1905, the world of science was shocked when a twenty-six-year-old patent clerk in Bern, Switzerland, named Albert Einstein showed that space and time are interlinked as something called space-time. Likewise, Klein, Hilbert, and Minkowski were shocked by the news that time slows down for a fast-moving object and that the speed of light is the only “invariant” in Einstein’s theory, the special theory of relativity. In an effort to explain Einstein’s revolutionary new findings within mathematics rather than within the language of theoretical physics, Minkowski—Einstein’s former professor from Zurich—wrote his paper on four-dimensional space-time.

  The three great mathematicians at Göttingen also realized that Einstein’s new physics was “invariant” under the action of a group. Thus Galois and Lie both entered the story of relativity. The Lorenz group, named after the Dutch physicist Hendrik Lorentz (1853–1928), was identified as the group of all transformations that leave the Minkowski four-dimensional space-time invariant, in that it allows us to correctly change the coordinate system to reconcile what different observers see. A larger group of transformations comprise the Poincaré group, named after the French mathematician Henri Poincaré (1854–1912). Incidentally, Poincaré has been called “the last universalist” because he was a master of so many different areas of mathematics.3 (Interestingly, a celebrated conjecture Poincaré had made in 1904 about the geometry of three-dimensional surfaces was proved almost a century later by the reclusive Russian mathematician Grigori Perelman, who then refused both the prestigious Fields Medal and the $1 million Millennium Prize for his breakthrough.)

  The need for new mathematical tools in physics became more acute ten years after special relativity was proposed. In 1915 Einstein shocked the world again by presenting his general theory of relativity, which included gravity and thus changed our understanding of Newton’s work, taking us far beyond its realm to extremes of speed and gravitational force in which Newtonian mechanics does not work. Ever since finishing his special theory of relativity in 1905, Einstein had been working hard to try to apply the relativity principle to Newton’s theory of gravity. In 1907 he tried an approach that didn’t work and then realized that he needed new mathematics. His friend Marcel Grossmann lent Einstein his notebooks from classes they had both taken at the ETH in Zurich (Einstein, apparently, was bad at taking notes). Through them Einstein was led to non-Euclidean geometry, the work of Riemann, and finally to obscure results by Italian mathematician Gregorio Ricci-Curbastro (1853–1925), who had studied with Felix Klein, and his collaborator Tullio Levi-Civita (1873–1941). The two had developed the “absolute differential calculus”—a calculus method that used tensors, which are generalizations of the matrices invented by Cayley.

  In Switzerland and Prague, Einstein labored on his tensors, trying to make them yield a mathematical system that would explain Newtonian mechanics in a relativistic context. Then he was invited to work at the Berlin Academy and moved back to his native Germany. By this time, he had realized that gravitation curves space-time, and in 1914 he sent the Berlin astronomer Erwin Freundlich to the Crimea in order to prove the validity of his emerging theory of relativity by making observations of stars during a total solar eclipse. Einstein had predicted that starlight passing by the sun would curve around it, distorting the star’s position. An eclipse in the Crimea offered the opportunity to prove such curvature. But World War I intervened, and Freundlich was arrested by the Russians. By the time of his release, the eclipse had already occurred.

  In 1915 Einstein accepted an invitation to visit Göttingen and agreed to give a talk to the mathematics department about his ongoing research on a tensor equation, in which the elements are arrays of variables, for general relativity. David Hilbert was in the audience, and he took copious notes on what Einstein was writing on the board. Then he went to his office and did perhaps the most unkind thing he had ever done: starting from Einstein’s work, he tried to derive the right equation. When he thought he had it, he sent his equation of general relativity, which used tensors of dimension 14 (meaning fourteen rows of variables), to a journal for publication.

  At the same time, Einstein made the breakthrough he had been looking for and sent his general relativity equation, with ten-dimensional tensors, to a journal. Research by Jürgen Renn of the Max Planck Institute in Berlin has established the exact time line of both paper submissions in November 1915 and the fact that Hilbert indeed tried to beat Einstein to the finish line. But the mathematician got his equation wrong, while the physicist got it right!

  This photograph of Einstein was taken during a lecture he gave in Vienna in 1921.

  As it turned out, Hilbert’s cumbersome fourteen-dimensional equation lacked an essential property: invariance. Relativity requires a certain kind of invariance called general covariance, which means that the physics an equation describes should not depend on the frame of reference from which it is observed. Whether you are looking at a rocket launch from the north or from the south, you should see the same physical process. Einstein’s equation was also far more elegant and compact than Hilbert’s. The Pythagoreans, who worshipped the number 10—their tetractys—would have been proud of it.

  Somehow, the mathematician who had proven several theorems about invariance failed to imbue his equation with this key property. Hilbert clearly needed someone who understood invariance even better than he did if he wanted to understand the mathematics of Einstein’s general theory of relativity. It was then that he and Klein remembered the exceptional young woman who had sat in on their courses a decade earlier and who was now publishing papers without an academic home.

  BY 19
15 NOETHER WAS a famous mathematician in her own right, and her papers were read with interest throughout the world. They dealt primarily with algebra. Like Hardy, who invited Ramanujan to come and work with him in Cambridge, Klein and Hilbert invited Emmy Noether to work with them at Göttingen.

  Noether arrived at Göttingen and began her work on invariance in mathematical physics. Meanwhile, Klein rallied to get her appointed a professor at Göttingen, but he had to struggle with the administration until 1919, when his request was finally granted. While she was at Göttingen, and during visits to Moscow State University in Russia in the 1920s, Emmy Noether accomplished incredible achievements in mathematics. In 1915, she presented the two Noether’s theorems, which show a paramount connection between the concept of symmetry in mathematics and the very important conservation laws in physics. More specifically, the theorems show that when the Lagrangian (the mathematical expression invented by Lagrange that captures the elements of a physical situation) enjoys a certain kind of symmetry—i.e., when it is invariant under the action of a Lie group, such as the group of all rotations of a circle—then the physical system modeled by the Lagrangian implies a conservation law. Thus Noether’s theorems explained the conservation of energy, momentum, and electric charge in the language of group theory.