Abram Besicovitch and Hermann Weyl
Let me now introduce you to some notable frogs and birds that I knew personally. I came to Cambridge University as a student in 1941 and had the tremendous luck to be given the Russian mathematician Abram Samoilovich Besicovitch as my supervisor. Since this was in the middle of World War Two, there were very few students in Cambridge, and almost no graduate students. Although I was only seventeen years old and Besicovitch was already a famous professor, he gave me a great deal of his time and attention, and we became life-long friends. He set the style in which I began to work and think about mathematics. He gave wonderful lectures on measure-theory and integration, smiling amiably when we laughed at his glorious abuse of the English language. I remember only one occasion when he was annoyed by our laughter. He remained silent for a while and then said, “Gentlemen. Fifty million English speak English you speak. Hundred and fifty million Russians speak English I speak.”
Besicovitch was a frog, and he became famous when he was young by solving a problem in elementary plane geometry known as the Kakeya problem. The Kakeya problem was the following. A line segment of length one is allowed to move freely in a plane while rotating through an angle of 360 degrees. What is the smallest area of the plane that it can cover during its rotation? The problem was posed by the Japanese mathematician Kakeya in 1917 and remained a famous unsolved problem for ten years. George Birkhoff, the leading American mathematician at that time, publicly proclaimed that the Kakeya problem and the fourcolor problem were the outstanding unsolved problems of the day. It was widely believed that the minimum area was π /8, which is the area of a three-cusped hypocycloid. The three-cusped hypocycloid is a beautiful three-pointed curve. It is the curve traced out by a point on the circumference of a circle with radius one-quarter, when the circle rolls around the inside of a fixed circle with radius three-quarters. The line segment of length one can turn while always remaining tangent to the hypocycloid with its two ends also on the hypocycloid. This picture of the line turning while touching the inside of the hypocycloid at three points was so elegant that most people believed it must give the minimum area. Then Besicovitch surprised everyone by proving that the area covered by the line as it turns can be less than for any positive .
Besicovitch had actually solved the problem in 1920 before it became famous, not even knowing that Kakeya had proposed it. In 1920 he published the solution in Russian in the Journal of the Perm Physics and Mathematics Society, a journal that was not widely read. The university of Perm, a city 1,100 kilometers east of Moscow, was briefly a refuge for many distinguished mathematicians after the Russian revolution. They published two volumes of their journal before it died amid the chaos of revolution and civil war. Outside Russia the journal was not only unknown but unobtainable. Besicovitch left Russia in 1925 and arrived at Copenhagen, where he learned about the famous Kakeya problem that he had solved five years earlier. He published the solution again, this time in English in the Mathematische Zeitschrift. The Kakeya problem as Kakeya proposed it was a typical frog problem, a concrete problem without much connection with the rest of mathematics. Besicovitch gave it an elegant and deep solution, which revealed a connection with general theorems about the structure of sets of points in a plane.
The Besicovitch style is seen at its finest in his three classic papers with the title, “On the fundamental geometric properties of linearly measurable plane sets of points”, published in Mathematische Annalen in the years 1928, 1938, and 1939. In these papers he proved that every linearly measurable set in the plane is divisible into a regular and an irregular component, that the regular component has a tangent almost everywhere, and the irregular component has a projection of measure zero onto almost all directions. Roughly speaking, the regular component looks like a collection of continuous curves, while the irregular component looks nothing like a continuous curve. The existence and the properties of the irregular component are connected with the Besicovitch solution of the Kakeya problem. One of the problems that he gave me to work on was the division of measurable sets into regular and irregular components in spaces of higher dimensions. I got nowhere with the problem, but became permanently imprinted with the Besicovitch style. The Besicovitch style is architectural. He builds out of simple elements a delicate and complicated architectural structure, usually with a hierarchical plan, and then, when the building is finished, the completed structure leads by simple arguments to an unexpected conclusion. Every Besicovitch proof is a work of art, as carefully constructed as a Bach fugue.
A few years after my apprenticeship with Besicovitch, I came to Princeton and got to know Hermann Weyl. Weyl was a prototypical bird, just as Besicovitch was a prototypical frog. I was lucky to overlap with Weyl for one year at the Princeton Institute for Advanced Study before he retired from the Institute and moved back to his old home in Zürich. He liked me because during that year I published papers in the Annals of Mathematics about number theory and in the Physical Review about the quantum theory of radiation. He was one of the few people alive who was at home in both subjects. He welcomed me to the Institute, in the hope that I would be a bird like himself. He was disappointed. I remained obstinately a frog. Although I poked around in a variety of mud-holes, I always looked at them one at a time and did not look for connections between them. For me, number theory and quantum theory were separate worlds with separate beauties. I did not look at them as Weyl did, hoping to find clues to a grand design.
Weyl’s great contribution to the quantum theory of radiation was his invention of gauge fields. The idea of gauge fields had a curious history. Weyl invented them in 1918 as classical fields in his unified theory of general relativity and electromagnetism, . He called them “gauge fields” because they were concerned with the non-integrability of measurements of length. His unified theory was promptly and publicly rejected by Einstein. After this thunderbolt from on high, Weyl did not abandon his theory but moved on to other things. The theory had no experimental consequences that could be tested. Then in 1929, after quantum mechanics had been invented by others, Weyl realized that his gauge fields fitted far better into the quantum world than they did into the classical world, . All that he needed to do, to change a classical gauge into a quantum gauge, was to change real numbers into complex numbers. In quantum mechanics, every quantum of electric charge carries with it a complex wave function with a phase, and the gauge field is concerned with the non-integrability of measurements of phase. The gauge field could then be precisely identified with the electromagnetic potential, and the law of conservation of charge became a consequence of the local phase invariance of the theory.
Weyl died four years after he returned from Princeton to Zürich, and I wrote his obituary for the journal Nature, . “Among all the mathematicians who began their working lives in the twentieth century,” I wrote, “Hermann Weyl was the one who made major contributions in the greatest number of different fields. He alone could stand comparison with the last great universal mathematicians of the nineteenth century, Hilbert and Poincaré. So long as he was alive, he embodied a living contact between the main lines of advance in pure mathematics and in theoretical physics. Now he is dead, the contact is broken, and our hopes of comprehending the physical universe by a direct use of creative mathematical imagination are for the time being ended.” I mourned his passing, but I had no desire to pursue his dream. I was happy to see pure mathematics and physics marching ahead in opposite directions.
The obituary ended with a sketch of Weyl as a human being: “Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half joking, ‘My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful’. This remark sums up his personality perfectly. It shows his profound faith in an ultimate harmony of Nature, in which the laws should inevitably express themselves in a mathematically beautiful form. It shows also his recognition of human frailty, and his humor, which always stopped him short of being pompous. His friends in Princeton will remember him as he was when I last saw him, at the Spring Dance of the Institute for Advanced Study last April: a big jovial man, enjoying himself splendidly, his cheerful frame and his light step giving no hint of his sixty-nine years.”
The fifty years after Weyl’s death were a golden age of experimental physics and observational astronomy, a golden age for Baconian travelers picking up facts, for frogs exploring small patches of the swamp in which we live. During these fifty years, the frogs accumulated a detailed knowledge of a large variety of cosmic structures and a large variety of particles and interactions. As the exploration of new territories continued, the universe became more complicated. Instead of a grand design displaying the simplicity and beauty of Weyl’s mathematics, the explorers found weird objects such as quarks and gamma-ray bursts, weird concepts such as supersymmetry and multiple universes. Meanwhile, mathematics was also becoming more complicated, as exploration continued into the phenomena of chaos and many other new areas opened by electronic computers. The mathematicians discovered the central mystery of computability, the conjecture represented by the statement P is not equal to NP. The conjecture asserts that there exist mathematical problems which can be quickly solved in individual cases but cannot be solved by a quick algorithm applicable to all cases. The most famous example of such a problem is the traveling salesman problem, which is to find the shortest route for a salesman visiting a set of cities, knowing the distance between each pair. All the experts believe that the conjecture is true, and that the traveling salesman problem is an example of a problem that is P but not NP. But nobody has even a glimmer of an idea how to prove it. This is a mystery that could not even have been formulated within the nineteenth-century mathematical universe of Hermann Weyl.
Frank Yang and Yuri Manin
The last fifty years have been a hard time for birds. Even in hard times, there is work for birds to do, and birds have appeared with the courage to tackle it. Soon after Weyl left Princeton, Frank Yang arrived from Chicago and moved into Weyl’s old house. Yang took Weyl’s place as the leading bird among my generation of physicists. While Weyl was still alive, Yang and his student Robert Mills discovered the Yang-Mills theory of non-Abelian gauge fields, a marvelously elegant extension of Weyl’s idea of a gauge field, . Weyl’s gauge field was a classical quantity, satisfying the commutative law of multiplication. The Yang-Mills theory had a triplet of gauge fields which did not commute. They satisfied the commutation rules of the three components of a quantum mechanical spin, which are generators of the simplest non-Abelian Lie algebra A2 . The theory was later generalized so that the gauge fields could be generators of any finite-dimensional Lie algebra. With this generalization, the Yang-Mills gauge field theory provided the framework for a model of all the known particles and interactions, a model that is now known as the Standard Model of particle physics. Yang put the finishing touch to it by showing that Einstein’s theory of gravitation fits into the same framework, with the Christoffel three-index symbol taking the role of gauge field, .
In an appendix to his 1918 paper, added in 1955 for the volume of selected papers published to celebrate his seventieth birthday, Weyl expressed his final thoughts about gauge field theories (my translation), : “The strongest argument for my theory seemed to be this, that gauge invariance was related to conservation of electric charge in the same way as coordinate invariance was related to conservation of energy and momentum.” Thirty years later Yang was in Zürich for the celebration of Weyl’s hundredth birthday. In his speech, , Yang quoted this remark as evidence of Weyl’s devotion to the idea of gauge invariance as a unifying principle for physics. Yang then went on, “Symmetry, Lie groups, and gauge invariance are now recognized, through theoretical and experimental developments, to play essential roles in determining the basic forces of the physical universe. I have called this the principle that symmetry dictates interaction.” This idea, that symmetry dictates interaction, is Yang’s generalization of Weyl’s remark. Weyl observed that gauge invariance is intimately connected with physical conservation laws. Weyl could not go further than this, because he knew only the gauge invariance of commuting Abelian fields. Yang made the connection much stronger by introducing non-Abelian gauge fields. With non-Abelian gauge fields generating nontrivial Lie algebras, the possible forms of interaction between fields become unique, so that symmetry dictates interaction. This idea is Yang’s greatest contribution to physics. It is the contribution of a bird, flying high over the rain forest of little problems in which most of us spend our lives.
Another bird for whom I have a deep respect is the Russian mathematician Yuri Manin, who recently published a delightful book of essays with the title Mathematics as Metaphor . The book was published in Moscow in Russian, and by the American Mathematical Society in English. I wrote a preface for the English version, and I give you here a short quote from my preface. “Mathematics as Metaphor is a good slogan for birds. It means that the deepest concepts in mathematics are those which link one world of ideas with another. In the seventeenth century Descartes linked the disparate worlds of algebra and geometry with his concept of coordinates, and Newton linked the worlds of geometry and dynamics with his concept of fluxions, nowadays called calculus. In the nineteenth century Boole linked the worlds of logic and algebra with his concept of symbolic logic, and Riemann linked the worlds of geometry and analysis with his concept of Riemann surfaces. Coordinates, fluxions, symbolic logic, and Riemann surfaces are all metaphors, extending the meanings of words from familiar to unfamiliar contexts. Manin sees the future of mathematics as an exploration of metaphors that are already visible but not yet understood. The deepest such metaphor is the similarity in structure between number theory and physics. In both fields he sees tantalizing glimpses of parallel concepts, symmetries linking the continuous with the discrete. He looks forward to a unification which he calls the quantization of mathematics.
“Manin disagrees with the Baconian story, that Hilbert set the agenda for the mathematics of the twentieth century when he presented his famous list of twenty-three unsolved problems to the International Congress of Mathematicians in Paris in 1900. According to Manin, Hilbert’s problems were a distraction from the central themes of mathematics. Manin sees the important advances in mathematics coming from programs, not from problems. Problems are usually solved by applying old ideas in new ways. Programs of research are the nurseries where new ideas are born. He sees the Bourbaki program, rewriting the whole of mathematics in a more abstract language, as the source of many of the new ideas of the twentieth century. He sees the Langlands program, unifying number theory with geometry, as a promising source of new ideas for the twenty-first. People who solve famous unsolved problems may win big prizes, but people who start new programs are the real pioneers.”
The Russian version of Mathematics as Metaphor contains ten chapters that were omitted from the English version. The American Mathematical Society decided that these chapters would not be of interest to English language readers. The omissions are doubly unfortunate. First, readers of the English version see only a truncated view of Manin, who is perhaps unique among mathematicians in his broad range of interests extending far beyond mathematics. Second, we see a truncated view of Russian culture, which is less compartmentalized than English language culture, and brings mathematicians into closer contact with historians and artists and poets.
Adapted from NOTICES OF AMS 2/2009