Birds and Frogs Freeman Dyson
ome mathematicians are birds, others skill as a mathematician. In his later years he hired are frogs. Birds fly high in the air and younger colleagues with the title of assistants to survey broad vistas of mathematics out do mathematical calculations for him. His way of to the far horizon. They delight in con- thinking was physical rather than mathematical. cepts that unify our thinking and bring He was supreme among physicists as a bird who togetherS diverse problems from different parts of saw further than others. I will not talk about Ein- the landscape. Frogs live in the mud below and see stein since I have nothing new to say. only the flowers that grow nearby. They delight in the details of particular objects, and they solve Francis Bacon and René Descartes problems one at a time. I happen to be a frog, but At the beginning of the seventeenth century, two many of my best friends are birds. The main theme great philosophers, Francis Bacon in England and of my talk tonight is this. Mathematics needs both René Descartes in France, proclaimed the birth of birds and frogs. Mathematics is rich and beautiful modern science. Descartes was a bird, and Bacon because birds give it broad visions and frogs give it was a frog. Each of them described his vision of intricate details. Mathematics is both great art and the future. Their visions were very different. Bacon important science, because it combines generality said, “All depends on keeping the eye steadily fixed of concepts with depth of structures. It is stupid on the facts of nature.” Descartes said, “I think, to claim that birds are better than frogs because therefore I am.” According to Bacon, scientists they see farther, or that frogs are better than birds should travel over the earth collecting facts, until because they see deeper. The world of mathemat- the accumulated facts reveal how Nature works. ics is both broad and deep, and we need birds and The scientists will then induce from the facts the frogs working together to explore it. laws that Nature obeys. According to Descartes, This talk is called the Einstein lecture, and I am scientists should stay at home and deduce the grateful to the American Mathematical Society laws of Nature by pure thought. In order to deduce for inviting me to do honor to Albert Einstein. the laws correctly, the scientists will need only Einstein was not a mathematician, but a physicist the rules of logic and knowledge of the existence who had mixed feelings about mathematics. On of God. For four hundred years since Bacon and the one hand, he had enormous respect for the Descartes led the way, science has raced ahead power of mathematics to describe the workings by following both paths simultaneously. Neither of nature, and he had an instinct for mathematical Baconian empiricism nor Cartesian dogmatism beauty which led him onto the right track to find has the power to elucidate Nature’s secrets by nature’s laws. On the other hand, he had no inter- itself, but both together have been amazingly suc- est in pure mathematics, and he had no technical cessful. For four hundred years English scientists have tended to be Baconian and French scientists Freeman Dyson is an emeritus professor in the School of Cartesian. Faraday and Darwin and Rutherford Natural Sciences, Institute for Advanced Study, Princeton, were Baconians; Pascal and Laplace and Poincaré NJ. His email address is [email protected]. were Cartesians. Science was greatly enriched by This article is a written version of his AMS Einstein Lecture, the cross-fertilization of the two contrasting cul- which was to have been given in October 2008 but which tures. Both cultures were always at work in both unfortunately had to be canceled. countries. Newton was at heart a Cartesian, using
212 NNoticesotices ofof thethe AMAMSS Volumeolume 56, Numberumber 2 pure thought as Descartes intended, and using wave mechanics in 1926. Schrödinger was a bird it to demolish the Cartesian dogma of vortices. who started from the idea of unifying mechanics Marie Curie was at heart a Baconian, boiling tons with optics. A hundred years earlier, Hamilton had of crude uranium ore to demolish the dogma of unified classical mechanics with ray optics, using the indestructibility of atoms. the same mathematics to describe optical rays In the history of twentieth century mathematics, and classical particle trajectories. Schrödinger’s there were two decisive events, one belonging to idea was to extend this unification to wave optics the Baconian tradition and the other to the Carte- and wave mechanics. Wave optics already existed, sian tradition. The first was the International Con- but wave mechanics did not. Schrödinger had to gress of Mathematicians in Paris in 1900, at which invent wave mechanics to complete the unification. Hilbert gave the keynote address, Starting from wave optics as a model, charting the course of mathematics he wrote down a differential equa- for the coming century by propound- tion for a mechanical particle, but the ing his famous list of twenty-three equation made no sense. The equation outstanding unsolved problems. Hil- looked like the equation of conduction bert himself was a bird, flying high of heat in a continuous medium. Heat over the whole territory of mathemat- conduction has no visible relevance to ics, but he addressed his problems to particle mechanics. Schrödinger’s idea the frogs who would solve them one seemed to be going nowhere. But then at a time. The second decisive event came the surprise. Schrödinger put was the formation of the Bourbaki the square root of minus one into the group of mathematical birds in France equation, and suddenly it made sense. in the 1930s, dedicated to publish- Suddenly it became a wave equation ing a series of textbooks that would instead of a heat conduction equation. establish a unifying framework for Francis Bacon And Schrödinger found to his delight all of mathematics. The Hilbert prob- that the equation has solutions cor- lems were enormously successful in responding to the quantized orbits in guiding mathematical research into the Bohr model of the atom. fruitful directions. Some of them were It turns out that the Schrödinger solved and some remain unsolved, equation describes correctly every- but almost all of them stimulated the thing we know about the behavior of growth of new ideas and new fields atoms. It is the basis of all of chem- of mathematics. The Bourbaki project istry and most of physics. And that was equally influential. It changed the square root of minus one means that style of mathematics for the next fifty nature works with complex numbers years, imposing a logical coherence and not with real numbers. This dis- that did not exist before, and moving covery came as a complete surprise, the emphasis from concrete examples to Schrödinger as well as to every- to abstract generalities. In the Bour- body else. According to Schrödinger, René Descartes baki scheme of things, mathematics is his fourteen-year-old girl friend Itha the abstract structure included in the Junger said to him at the time, “Hey, Bourbaki textbooks. What is not in the textbooks you never even thought when you began that so is not mathematics. Concrete examples, since they much sensible stuff would come out of it.” All do not appear in the textbooks, are not math- through the nineteenth century, mathematicians ematics. The Bourbaki program was the extreme from Abel to Riemann and Weierstrass had been expression of the Cartesian style. It narrowed the creating a magnificent theory of functions of scope of mathematics by excluding the beautiful complex variables. They had discovered that the flowers that Baconian travelers might collect by theory of functions became far deeper and more the wayside. powerful when it was extended from real to com- plex numbers. But they always thought of complex Jokes of Nature numbers as an artificial construction, invented by For me, as a Baconian, the main thing missing in human mathematicians as a useful and elegant the Bourbaki program is the element of surprise. abstraction from real life. It never entered their The Bourbaki program tried to make mathematics heads that this artificial number system that they logical. When I look at the history of mathematics, had invented was in fact the ground on which I see a succession of illogical jumps, improbable atoms move. They never imagined that nature had coincidences, jokes of nature. One of the most got there first. profound jokes of nature is the square root of Another joke of nature is the precise linearity minus one that the physicist Erwin Schrödinger of quantum mechanics, the fact that the possible put into his wave equation when he invented states of any physical object form a linear space.
February 2009 Notices of the AMS 213 Before quantum mechanics was invented, classical second commandment says: “Let your acts be di- physics was always nonlinear, and linear models rected towards a worthy goal, but do not ask if they were only approximately valid. After quantum can reach it: they are to be models and examples, mechanics, nature itself suddenly became linear. not means to an end.” Szilard practiced what he This had profound consequences for mathemat- preached. He was the first physicist to imagine ics. During the nineteenth century Sophus Lie nuclear weapons and the first to campaign ac- developed his elaborate theory of continuous tively against their use. His second commandment groups, intended to clarify the behavior of classical certainly applies here. The proof of the Riemann dynamical systems. Lie groups were then of little Hypothesis is a worthy goal, and it is not for us to interest either to mathematicians or to physicists. ask whether we can reach it. I will give you some The nonlinear theory of Lie groups was too compli- hints describing how it might be achieved. Here I cated for the mathematicians and too obscure for will be giving voice to the mathematician that I was the physicists. Lie died a disappointed man. And fifty years ago before I became a physicist. I will then, fifty years later, it turned out that nature was talk first about the Riemann Hypothesis and then precisely linear, and the theory of linear represen- about quasi-crystals. tations of Lie algebras was the natural language of There were until recently two supreme unsolved particle physics. Lie groups and Lie algebras were problems in the world of pure mathematics, the reborn as one of the central themes of twentieth proof of Fermat’s Last Theorem and the proof of century mathematics. the Riemann Hypothesis. Twelve years ago, my A third joke of nature is the existence of quasi- Princeton colleague Andrew Wiles polished off crystals. In the nineteenth century the study of Fermat’s Last Theorem, and only the Riemann Hy- crystals led to a complete enumeration of possible pothesis remains. Wiles’ proof of the Fermat Theo- discrete symmetry groups in Euclidean space. rem was not just a technical stunt. It required the Theorems were proved, establishing the fact that discovery and exploration of a new field of math- in three-dimensional space discrete symmetry ematical ideas, far wider and more consequential groups could contain only rotations of order three, than the Fermat Theorem itself. It is likely that four, or six. Then in 1984 quasi-crystals were dis- any proof of the Riemann Hypothesis will likewise covered, real solid objects growing out of liquid lead to a deeper understanding of many diverse metal alloys, showing the symmetry of the icosa- areas of mathematics and perhaps of physics too. hedral group, which includes five-fold rotations. Riemann’s zeta-function, and other zeta-func- Meanwhile, the mathematician Roger Penrose tions similar to it, appear ubiquitously in number discovered the Penrose tilings of the plane. These theory, in the theory of dynamical systems, in are arrangements of parallelograms that cover a geometry, in function theory, and in physics. The plane with pentagonal long-range order. The alloy zeta-function stands at a junction where paths lead quasi-crystals are three-dimensional analogs of in many directions. A proof of the hypothesis will the two-dimensional Penrose tilings. After these illuminate all the connections. Like every serious discoveries, mathematicians had to enlarge the student of pure mathematics, when I was young I theory of crystallographic groups to include quasi- had dreams of proving the Riemann Hypothesis. crystals. That is a major program of research which I had some vague ideas that I thought might lead is still in progress. to a proof. In recent years, after the discovery of A fourth joke of nature is a similarity in be- quasi-crystals, my ideas became a little less vague. havior between quasi-crystals and the zeros of I offer them here for the consideration of any the Riemann Zeta function. The zeros of the zeta- young mathematician who has ambitions to win function are exciting to mathematicians because a Fields Medal. they are found to lie on a straight line and nobody Quasi-crystals can exist in spaces of one, two, understands why. The statement that with trivial or three dimensions. From the point of view of exceptions they all lie on a straight line is the physics, the three-dimensional quasi-crystals are famous Riemann Hypothesis. To prove the Rie- the most interesting, since they inhabit our three- mann Hypothesis has been the dream of young dimensional world and can be studied experi- mathematicians for more than a hundred years. mentally. From the point of view of a mathemati- I am now making the outrageous suggestion that cian, one-dimensional quasi-crystals are much we might use quasi-crystals to prove the Riemann more interesting than two-dimensional or three- Hypothesis. Those of you who are mathematicians dimensional quasi-crystals because they exist in may consider the suggestion frivolous. Those who far greater variety. The mathematical definition are not mathematicians may consider it uninterest- of a quasi-crystal is as follows. A quasi-crystal ing. Nevertheless I am putting it forward for your is a distribution of discrete point masses whose serious consideration. When the physicist Leo Fourier transform is a distribution of discrete Szilard was young, he became dissatisfied with the point frequencies. Or to say it more briefly, a ten commandments of Moses and wrote a new set quasi-crystal is a pure point distribution that has of ten commandments to replace them. Szilard’s a pure point spectrum. This definition includes
214 Notices of the AMS Volume 56, Number 2 Photograph of A. Besicovitch from the AMS archives. Photo of Hermann Weyl Hermann of Photo archives. AMS the from Besicovitch A. of Photograph
as a special case the ordinary crystals, objects is a quintessentially Baconian which are periodic distributions with activity. It is an appropriate activity periodic spectra. for mathematical frogs. We shall then Excluding the ordinary crystals, find the well-known quasi-crystals quasi-crystals in three dimensions associated with PV numbers, and come in very limited variety, all of also a whole universe of other quasi- them associated with the icosahedral crystals, known and unknown. Among group. The two-dimensional quasi- the multitude of other quasi-crystals crystals are more numerous, roughly we search for one corresponding to one distinct type associated with each the Riemann zeta-function and one regular polygon in a plane. The two- corresponding to each of the other dimensional quasi-crystal with pentag- zeta-functions that resemble the Rie- onal symmetry is the famous Penrose mann zeta-function. Suppose that tiling of the plane. Finally, the one- Abram Besicovitch we find one of the quasi-crystals in dimensional quasi-crystals have a far our enumeration with properties richer structure since they are not tied that identify it with the zeros of the to any rotational symmetries. So far as Riemann zeta-function. Then we have I know, no complete enumeration of proved the Riemann Hypothesis and one-dimensional quasi-crystals exists. we can wait for the telephone call It is known that a unique quasi-crystal announcing the award of the Fields exists corresponding to every Pisot- Medal. Vijayaraghavan number or PV number. These are of course idle dreams. A PV number is a real algebraic inte- The problem of classifying one- ger, a root of a polynomial equation dimensional quasi-crystals is horren- with integer coefficients, such that all dously difficult, probably at least as the other roots have absolute value difficult as the problems that Andrew less than one, [1]. The set of all PV Wiles took seven years to explore. But numbers is infinite and has a remark- Hermann Weyl if we take a Baconian point of view, able topological structure. The set the history of mathematics is a his- of all one-dimensional quasi-crystals tory of horrendously difficult prob- has a structure at least as rich as the lems being solved by young people too ignorantpermisssion. with used Roquette, Peter of archives the tofrom Weyl H. of Photo archives. AMS from Besicovitch A. of Photograph set of all PV numbers and probably much richer. know that they were impossible. The classification We do not know for sure, but it is likely that a of quasi-crystals is a worthy goal, and might even huge universe of one-dimensional quasi-crystals turn out to be achievable. Problems of that degree not associated with PV numbers is waiting to be of difficulty will not be solved by old men like me. discovered. I leave this problem as an exercise for the young Here comes the connection of the one- frogs in the audience. dimensional quasi-crystals with the Riemann hypothesis. If the Riemann hypothesis is true, Abram Besicovitch and Hermann Weyl then the zeros of the zeta-function form a one- Let me now introduce you to some notable frogs dimensional quasi-crystal according to the defini- and birds that I knew personally. I came to Cam- tion. They constitute a distribution of point masses bridge University as a student in 1941 and had on a straight line, and their Fourier transform is the tremendous luck to be given the Russian likewise a distribution of point masses, one at each mathematician Abram Samoilovich Besicovitch of the logarithms of ordinary prime numbers and as my supervisor. Since this was in the middle prime-power numbers. My friend Andrew Odlyzko of World War Two, there were very few students has published a beautiful computer calculation of in Cambridge, and almost no graduate students. the Fourier transform of the zeta-function zeros, Although I was only seventeen years old and Besi- [6]. The calculation shows precisely the expected covitch was already a famous professor, he gave structure of the Fourier transform, with a sharp me a great deal of his time and attention, and we discontinuity at every logarithm of a prime or became life-long friends. He set the style in which prime-power number and nowhere else. I began to work and think about mathematics. He My suggestion is the following. Let us pretend gave wonderful lectures on measure-theory and that we do not know that the Riemann Hypothesis integration, smiling amiably when we laughed at is true. Let us tackle the problem from the other his glorious abuse of the English language. I re- end. Let us try to obtain a complete enumera- member only one occasion when he was annoyed tion and classification of one-dimensional quasi- by our laughter. He remained silent for a while and crystals. That is to say, we enumerate and classify then said, “Gentlemen. Fifty million English speak all point distributions that have a discrete point English you speak. Hundred and fifty million Rus- spectrum. Collecting and classifying new species of sians speak English I speak.”
February 2009 Notices of the AMS 215 Besicovitch was a frog, and he became famous into a regular and an irregular component, that when he was young by solving a problem in el- the regular component has a tangent almost ementary plane geometry known as the Kakeya everywhere, and the irregular component has a problem. The Kakeya problem was the following. projection of measure zero onto almost all direc- A line segment of length one is allowed to move tions. Roughly speaking, the regular component freely in a plane while rotating through an angle looks like a collection of continuous curves, while of 360 degrees. What is the smallest area of the the irregular component looks nothing like a con- plane that it can cover during its rotation? The tinuous curve. The existence and the properties of problem was posed by the Japanese mathematician the irregular component are connected with the Kakeya in 1917 and remained a famous unsolved Besicovitch solution of the Kakeya problem. One problem for ten years. George Birkhoff, the lead- of the problems that he gave me to work on was ing American mathematician at that time, publicly the division of measurable sets into regular and proclaimed that the Kakeya problem and the four- irregular components in spaces of higher dimen- color problem were the outstanding unsolved sions. I got nowhere with the problem, but became problems of the day. It was widely believed that permanently imprinted with the Besicovitch style. the minimum area was π/8, which is the area of a The Besicovitch style is architectural. He builds three-cusped hypocycloid. The three-cusped hypo- out of simple elements a delicate and complicated cycloid is a beautiful three-pointed curve. It is the architectural structure, usually with a hierarchical curve traced out by a point on the circumference plan, and then, when the building is finished, the of a circle with radius one-quarter, when the circle completed structure leads by simple arguments rolls around the inside of a fixed circle with radius to an unexpected conclusion. Every Besicovitch three-quarters. The line segment of length one can proof is a work of art, as carefully constructed as turn while always remaining tangent to the hypo- a Bach fugue. cycloid with its two ends also on the hypocycloid. A few years after my apprenticeship with Be- This picture of the line turning while touching the sicovitch, I came to Princeton and got to know inside of the hypocycloid at three points was so Hermann Weyl. Weyl was a prototypical bird, just elegant that most people believed it must give the as Besicovitch was a prototypical frog. I was lucky minimum area. Then Besicovitch surprised every- to overlap with Weyl for one year at the Princeton one by proving that the area covered by the line as Institute for Advanced Study before he retired it turns can be less than for any positive . from the Institute and moved back to his old home Besicovitch had actually solved the problem in in Zürich. He liked me because during that year I 1920 before it became famous, not even knowing published papers in the Annals of Mathematics that Kakeya had proposed it. In 1920 he published about number theory and in the Physical Review the solution in Russian in the Journal of the Perm about the quantum theory of radiation. He was one Physics and Mathematics Society, a journal that of the few people alive who was at home in both was not widely read. The university of Perm, a subjects. He welcomed me to the Institute, in the city 1,100 kilometers east of Moscow, was briefly hope that I would be a bird like himself. He was dis- a refuge for many distinguished mathematicians appointed. I remained obstinately a frog. Although after the Russian revolution. They published two I poked around in a variety of mud-holes, I always volumes of their journal before it died amid the looked at them one at a time and did not look for chaos of revolution and civil war. Outside Russia connections between them. For me, number theory the journal was not only unknown but unobtain- and quantum theory were separate worlds with able. Besicovitch left Russia in 1925 and arrived at separate beauties. I did not look at them as Weyl Copenhagen, where he learned about the famous did, hoping to find clues to a grand design. Kakeya problem that he had solved five years ear- Weyl’s great contribution to the quantum theory lier. He published the solution again, this time in of radiation was his invention of gauge fields. The English in the Mathematische Zeitschrift. The Ka- idea of gauge fields had a curious history. Weyl keya problem as Kakeya proposed it was a typical invented them in 1918 as classical fields in his frog problem, a concrete problem without much unified theory of general relativity and electromag- connection with the rest of mathematics. Besico- netism, [7]. He called them “gauge fields” because vitch gave it an elegant and deep solution, which they were concerned with the non-integrability revealed a connection with general theorems about of measurements of length. His unified theory the structure of sets of points in a plane. was promptly and publicly rejected by Einstein. The Besicovitch style is seen at its finest in After this thunderbolt from on high, Weyl did his three classic papers with the title, “On the not abandon his theory but moved on to other fundamental geometric properties of linearly things. The theory had no experimental conse- measurable plane sets of points”, published in quences that could be tested. Then in 1929, after Mathematische Annalen in the years 1928, 1938, quantum mechanics had been invented by others, and 1939. In these papers he proved that every Weyl realized that his gauge fields fitted far bet- linearly measurable set in the plane is divisible ter into the quantum world than they did into the
216 Notices of the AMS Volume 56, Number 2 classical world, [8]. All astronomy, a golden that he needed to do, to age for Baconian travel- change a classical gauge ers picking up facts, for into a quantum gauge, frogs exploring small was to change real patches of the swamp numbers into complex in which we live. Dur- numbers. In quantum ing these fifty years, the mechanics, every quan- frogs accumulated a de- tum of electric charge tailed knowledge of a carries with it a com- large variety of cosmic plex wave function with structures and a large a phase, and the gauge variety of particles and field is concerned with interactions. As the the non-integrability of Chen Ning (Frank) Yuri Manin exploration of new ter- measurements of phase. Yang ritories continued, the The gauge field could universe became more then be precisely identified with the electromag- complicated. Instead of a grand design displaying netic potential, and the law of conservation of the simplicity and beauty of Weyl’s mathematics, charge became a consequence of the local phase the explorers found weird objects such as quarks invariance of the theory. and gamma-ray bursts, weird concepts such as su- Weyl died four years after he returned from persymmetry and multiple universes. Meanwhile, Princeton to Zürich, and I wrote his obituary for the mathematics was also becoming more compli- journal Nature, [3]. “Among all the mathematicians cated, as exploration continued into the phenom- who began their working lives in the twentieth ena of chaos and many other new areas opened century,” I wrote, “Hermann Weyl was the one who by electronic computers. The mathematicians made major contributions in the greatest number discovered the central mystery of computability,University. Northwestern of courtesy Manin Y. of Photo Brook. Stony SUNY of courtesy Yang F. of Photo of different fields. He alone could stand compari- the conjecture represented by the statement P is son with the last great universal mathematicians not equal to NP. The conjecture asserts that there of the nineteenth century, Hilbert and Poincaré. exist mathematical problems which can be quickly So long as he was alive, he embodied a living con- solved in individual cases but cannot be solved tact between the main lines of advance in pure by a quick algorithm applicable to all cases. The mathematics and in theoretical physics. Now he most famous example of such a problem is the is dead, the contact is broken, and our hopes of traveling salesman problem, which is to find the comprehending the physical universe by a direct shortest route for a salesman visiting a set of cit- use of creative mathematical imagination are for ies, knowing the distance between each pair. All the time being ended.” I mourned his passing, but the experts believe that the conjecture is true, and I had no desire to pursue his dream. I was happy that the traveling salesman problem is an example to see pure mathematics and physics marching of a problem that is P but not NP. But nobody has ahead in opposite directions. even a glimmer of an idea how to prove it. This is The obituary ended with a sketch of Weyl as a mystery that could not even have been formu- a human being: “Characteristic of Weyl was an lated within the nineteenth-century mathematical aesthetic sense which dominated his thinking on universe of Hermann Weyl. all subjects. He once said to me, half joking, ‘My work always tried to unite the true with the beau- Frank Yang and Yuri Manin tiful; but when I had to choose one or the other, The last fifty years have been a hard time for I usually chose the beautiful’. This remark sums birds. Even in hard times, there is work for birds up his personality perfectly. It shows his profound to do, and birds have appeared with the courage to faith in an ultimate harmony of Nature, in which tackle it. Soon after Weyl left Princeton, Frank Yang the laws should inevitably express themselves in arrived from Chicago and moved into Weyl’s old a mathematically beautiful form. It shows also house. Yang took Weyl’s place as the leading bird his recognition of human frailty, and his humor, among my generation of physicists. While Weyl which always stopped him short of being pomp- was still alive, Yang and his student Robert Mills ous. His friends in Princeton will remember him discovered the Yang-Mills theory of non-Abelian as he was when I last saw him, at the Spring Dance gauge fields, a marvelously elegant extension of of the Institute for Advanced Study last April: a Weyl’s idea of a gauge field, [11]. Weyl’s gauge field big jovial man, enjoying himself splendidly, his was a classical quantity, satisfying the commuta- cheerful frame and his light step giving no hint of tive law of multiplication. The Yang-Mills theory his sixty-nine years.” had a triplet of gauge fields which did not com- The fifty years after Weyl’s death were a golden mute. They satisfied the commutation rules of the age of experimental physics and observational three components of a quantum mechanical spin,
February 2009 Notices of the AMS 217 which are generators of the simplest non-Abelian the worlds of geometry and dynamics with his Lie algebra A2. The theory was later generalized so concept of fluxions, nowadays called calculus. In that the gauge fields could be generators of any the nineteenth century Boole linked the worlds finite-dimensional Lie algebra. With this general- of logic and algebra with his concept of symbolic ization, the Yang-Mills gauge field theory provided logic, and Riemann linked the worlds of geometry the framework for a model of all the known par- and analysis with his concept of Riemann sur- ticles and interactions, a model that is now known faces. Coordinates, fluxions, symbolic logic, and as the Standard Model of particle physics. Yang put Riemann surfaces are all metaphors, extending the finishing touch to it by showing that Einstein’s the meanings of words from familiar to unfamiliar theory of gravitation fits into the same framework, contexts. Manin sees the future of mathematics with the Christoffel three-index symbol taking the as an exploration of metaphors that are already role of gauge field, [10]. visible but not yet understood. The deepest such In an appendix to his 1918 paper, added in 1955 metaphor is the similarity in structure between for the volume of selected papers published to number theory and physics. In both fields he sees celebrate his seventieth birthday, Weyl expressed tantalizing glimpses of parallel concepts, symme- his final thoughts about gauge field theories (my tries linking the continuous with the discrete. He translation), [12]: “The strongest argument for my looks forward to a unification which he calls the theory seemed to be this, that gauge invariance quantization of mathematics. was related to conservation of electric charge in “Manin disagrees with the Baconian story, that the same way as coordinate invariance was related Hilbert set the agenda for the mathematics of the to conservation of energy and momentum.” Thirty twentieth century when he presented his famous years later Yang was in Zürich for the celebration list of twenty-three unsolved problems to the In- of Weyl’s hundredth birthday. In his speech, [12], ternational Congress of Mathematicians in Paris Yang quoted this remark as evidence of Weyl’s de- in 1900. According to Manin, Hilbert’s problems votion to the idea of gauge invariance as a unifying were a distraction from the central themes of principle for physics. Yang then went on, “Sym- mathematics. Manin sees the important advances metry, Lie groups, and gauge invariance are now in mathematics coming from programs, not from recognized, through theoretical and experimental problems. Problems are usually solved by apply- developments, to play essential roles in determin- ing old ideas in new ways. Programs of research ing the basic forces of the physical universe. I have are the nurseries where new ideas are born. He called this the principle that symmetry dictates in- sees the Bourbaki program, rewriting the whole of teraction.” This idea, that symmetry dictates inter- mathematics in a more abstract language, as the action, is Yang’s generalization of Weyl’s remark. source of many of the new ideas of the twentieth Weyl observed that gauge invariance is intimately century. He sees the Langlands program, unifying connected with physical conservation laws. Weyl number theory with geometry, as a promising could not go further than this, because he knew source of new ideas for the twenty-first. People only the gauge invariance of commuting Abelian who solve famous unsolved problems may win big fields. Yang made the connection much stronger prizes, but people who start new programs are the by introducing non-Abelian gauge fields. With real pioneers.” non-Abelian gauge fields generating nontrivial Lie The Russian version of Mathematics as Meta- algebras, the possible forms of interaction between phor contains ten chapters that were omitted from fields become unique, so that symmetry dictates the English version. The American Mathematical interaction. This idea is Yang’s greatest contribu- Society decided that these chapters would not be tion to physics. It is the contribution of a bird, of interest to English language readers. The omis- flying high over the rain forest of little problems sions are doubly unfortunate. First, readers of the in which most of us spend our lives. English version see only a truncated view of Manin, Another bird for whom I have a deep respect who is perhaps unique among mathematicians in is the Russian mathematician Yuri Manin, who his broad range of interests extending far beyond recently published a delightful book of essays with mathematics. Second, we see a truncated view of the title Mathematics as Metaphor [5]. The book Russian culture, which is less compartmentalized was published in Moscow in Russian, and by the than English language culture, and brings math- American Mathematical Society in English. I wrote ematicians into closer contact with historians and a preface for the English version, and I give you artists and poets. here a short quote from my preface. “Mathematics as Metaphor is a good slogan for birds. It means John von Neumann that the deepest concepts in mathematics are Another important figure in twentieth century those which link one world of ideas with another. mathematics was John von Neumann. Von Neu- In the seventeenth century Descartes linked the mann was a frog, applying his prodigious tech- disparate worlds of algebra and geometry with nical skill to solve problems in many branches his concept of coordinates, and Newton linked of mathematics and physics. He began with the
218 Notices of the AMS Volume 56, Number 2 foundations of mathematics. He found the first divide the atmosphere at any moment into stable satisfactory set of axioms for set-theory, avoiding regions and unstable regions. Stable regions we the logical paradoxes that Cantor had encountered can predict. Unstable regions we can control.” Von in his attempts to deal with infinite sets and Neumann believed that any unstable region could infinite numbers. Von Neumann’s axioms were be pushed by a judiciously applied small perturba- used by his bird friend Kurt Gödel a few years later tion so that it would move in any desired direction. to prove the existence of undecidable propositions The small perturbation would be applied by a fleet in mathematics. Gödel’s theorems gave birds a new of airplanes carrying smoke generators, to absorb vision of mathematics. After Gödel, mathematics sunlight and raise or lower temperatures at places was no longer a single structure tied where the perturbation would be most together with a unique concept of effective. In particular, we could stop truth, but an archipelago of structures an incipient hurricane by identifying with diverse sets of axioms and di- the position of an instability early verse notions of truth. Gödel showed enough, and then cooling that patch that mathematics is inexhaustible. No of air before it started to rise and form matter which set of axioms is chosen a vortex. Von Neumann, speaking in as the foundation, birds can always 1950, said it would take only ten years find questions that those axioms can- to build computers powerful enough not answer. to diagnose accurately the stable and Von Neumann went on from the unstable regions of the atmosphere. foundations of mathematics to the Then, once we had accurate diagno- foundations of quantum mechanics. sis, it would take only a short time To give quantum mechanics a firm for us to have control. He expected mathematical foundation, he created John von Neumann that practical control of the weather a magnificent theory of rings of op- would be a routine operation within erators. Every observable quantity is the decade of the 1960s. represented by a linear operator, and Von Neumann, of course, was the peculiarities of quantum behav- wrong. He was wrong because he ior are faithfully represented by the did not know about chaos. We now algebra of operators. Just as Newton know that when the motion of the invented calculus to describe classi- atmosphere is locally unstable, it is cal dynamics, von Neumann invented very often chaotic. The word “chaotic” rings of operators to describe quan- means that motions that start close tum dynamics. together diverge exponentially from Von Neumann made fundamental each other as time goes on. When the contributions to several other fields, motion is chaotic, it is unpredictable, especially to game theory and to the and a small perturbation does not design of digital computers. For the Mary Cartwright move it into a stable motion that canCambridge. College, Girton Fellows, and Mistress The of courtesy Cartright Mary of Photograph last ten years of his life, he was deeply be predicted. A small perturbation involved with computers. He was so will usually move it into another cha- strongly interested in computers that he decided otic motion that is equally unpredictable. So von not only to study their design but to build one with Neumann’s strategy for controlling the weather real hardware and software and use it for doing fails. He was, after all, a great mathematician but science. I have vivid memories of the early days of a mediocre meteorologist. von Neumann’s computer project at the Institute Edward Lorenz discovered in 1963 that the so- for Advanced Study in Princeton. At that time he lutions of the equations of meteorology are often had two main scientific interests, hydrogen bombs chaotic. That was six years after von Neumann and meteorology. He used his computer during the died. Lorenz was a meteorologist and is generally night for doing hydrogen bomb calculations and regarded as the discoverer of chaos. He discovered during the day for meteorology. Most of the people the phenomena of chaos in the meteorological con- hanging around the computer building in daytime text and gave them their modern names. But in fact were meteorologists. Their leader was Jule Char- I had heard the mathematician Mary Cartwright, ney. Charney was a real meteorologist, properly who died in 1998 at the age of 97, describe the humble in dealing with the inscrutable mysteries same phenomena in a lecture in Cambridge in 1943, of the weather, and skeptical of the ability of the twenty years before Lorenz discovered them. She computer to solve the mysteries. John von Neu- called the phenomena by different names, but they mann was less humble and less skeptical. I heard were the same phenomena. She discovered them in von Neumann give a lecture about the aims of his the solutions of the van der Pol equation which de- project. He spoke, as he always did, with great con- scribe the oscillations of a nonlinear amplifier, [2]. fidence. He said, “The computer will enable us to The van der Pol equation was important in World
February 2009 Notices of the AMS 219 War II because nonlinear amplifiers fed power from the 1930s out of a drawer and dusted it off. to the transmitters in early radar systems. The The lecture was about rings of operators, a subject transmitters behaved erratically, and the Air Force that was new and fashionable in the 1930s. Noth- blamed the manufacturers for making defective ing about unsolved problems. Nothing about the amplifiers. Mary Cartwright was asked to look into future. Nothing about computers, the subject that the problem. She showed that the manufacturers we knew was dearest to von Neumann’s heart. were not to blame. She showed that the van der Pol He might at least have had something new and equation was to blame. The solutions of the van der exciting to say about computers. The audience in Pol equation have precisely the chaotic behavior the concert hall became restless. Somebody said that the Air Force was complaining about. I heard in a voice loud enough to be heard all over the all about chaos from Mary Cartwright seven years hall, “Aufgewärmte Suppe”, which is German for before I heard von Neumann talk about weather “warmed-up soup”. In 1954 the great majority of control, but I was not far-sighted enough to make mathematicians knew enough German to under- the connection. It never entered my head that the stand the joke. Von Neumann, deeply embarrassed, erratic behavior of the van der Pol equation might brought his lecture to a quick end and left the hall have something to do with meteorology. If I had without waiting for questions. been a bird rather than a frog, I would probably have seen the connection, and I might have saved Weak Chaos von Neumann a lot of trouble. If he had known If von Neumann had known about chaos when he about chaos in 1950, he would probably have spoke in Amsterdam, one of the unsolved prob- thought about it deeply, and he would have had lems that he might have talked about was weak something important to say about it in 1954. chaos. The problem of weak chaos is still unsolved Von Neumann got into trouble at the end of fifty years later. The problem is to understand his life because he was really a frog but everyone why chaotic motions often remain bounded and expected him to fly like a bird. In 1954 there was do not cause any violent instability. A good ex- an International Congress of Mathematicians in ample of weak chaos is the orbital motions of the Amsterdam. These congresses happen only once planets and satellites in the solar system. It was in four years and it is a great honor to be invited to discovered only recently that these motions are speak at the opening session. The organizers of the chaotic. This was a surprising discovery, upsetting Amsterdam congress invited von Neumann to give the traditional picture of the solar system as the the keynote speech, expecting him to repeat the act prime example of orderly stable motion. The math- that Hilbert had performed in Paris in 1900. Just as ematician Laplace two hundred years ago thought Hilbert had provided a list of unsolved problems he had proved that the solar system is stable. It to guide the development of mathematics for the now turns out that Laplace was wrong. Accurate first half of the twentieth century, von Neumann numerical integrations of the orbits show clearly was invited to do the same for the second half of that neighboring orbits diverge exponentially. It the century. The title of von Neumann’s talk was seems that chaos is almost universal in the world announced in the program of the congress. It was of classical dynamics. “Unsolved Problems in Mathematics: Address by Chaotic behavior was never suspected in the Invitation of the Organizing Committee”. After the solar system before accurate long-term integra- congress was over, the complete proceedings were tions were done, because the chaos is weak. Weak published, with the texts of all the lectures except chaos means that neighboring trajectories diverge this one. In the proceedings there is a blank page exponentially but never diverge far. The divergence with von Neumann’s name and the title of his talk. begins with exponential growth but afterwards Underneath, it says, “No manuscript of this lecture remains bounded. Because the chaos of the plan- was available.” etary motions is weak, the solar system can survive What happened? I know what happened, be- for four billion years. Although the motions are cause I was there in the audience, at 3:00 p.m. chaotic, the planets never wander far from their on Thursday, September 2, 1954, in the Concert- customary places, and the system as a whole does gebouw concert hall. The hall was packed with not fly apart. In spite of the prevalence of chaos, mathematicians, all expecting to hear a brilliant the Laplacian view of the solar system as a perfect lecture worthy of such a historic occasion. The piece of clockwork is not far from the truth. lecture was a huge disappointment. Von Neumann We see the same phenomena of weak chaos in had probably agreed several years earlier to give the domain of meteorology. Although the weather a lecture about unsolved problems and had then in New Jersey is painfully chaotic, the chaos has forgotten about it. Being busy with many other firm limits. Summers and winters are unpredict- things, he had neglected to prepare the lecture. ably mild or severe, but we can reliably predict Then, at the last moment, when he remembered that the temperature will never rise to 45 degrees that he had to travel to Amsterdam and say some- Celsius or fall to minus 30, extremes that are thing about mathematics, he pulled an old lecture often exceeded in India or in Minnesota. There
220 Notices of the AMS Volume 56, Number 2 is no conservation law of physics that forbids solve old problems that were previously unsolvable. temperatures from rising as high in New Jersey Second, the string theorists think of themselves as in India, or from falling as low in New Jersey as physicists rather than mathematicians. They as in Minnesota. The weakness of chaos has been believe that their theory describes something real essential to the long-term survival of life on this in the physical world. And third, there is not yet planet. Weak chaos gives us a challenging variety any proof that the theory is relevant to physics. of weather while protecting us from fluctuations The theory is not yet testable by experiment. The so severe as to endanger our existence. Chaos theory remains in a world of its own, detached remains mercifully weak for reasons that we do from the rest of physics. String theorists make not understand. That is another unsolved problem strenuous efforts to deduce consequences of the for young frogs in the audience to take home. I theory that might be testable in the real world, so challenge you to understand the reasons why the far without success. chaos observed in a great diversity of dynamical My colleagues Ed Witten and Juan Maldacena systems is generally weak. and others who created string theory are birds, The subject of chaos is characterized by an flying high and seeing grand visions of distant abundance of quantitative data, an unending sup- ranges of mountains. The thousands of hum- ply of beautiful pictures, and a shortage of rigor- bler practitioners of string theory in universities ous theorems. Rigorous theorems are the best way around the world are frogs, exploring fine details to give a subject intellectual depth and precision. of the mathematical structures that birds first Until you can prove rigorous theorems, you do not saw on the horizon. My anxieties about string fully understand the meaning of your concepts. theory are sociological rather than scientific. It is In the field of chaos I know only one rigorous a glorious thing to be one of the first thousand theorem, proved by Tien-Yien Li and Jim Yorke in string theorists, discovering new connections and 1975 and published in a short paper with the title, pioneering new methods. It is not so glorious to “Period Three Implies Chaos”, [4]. The Li-Yorke be one of the second thousand or one of the tenth paper is one of the immortal gems in the literature thousand. There are now about ten thousand of mathematics. Their theorem concerns nonlinear string theorists scattered around the world. This maps of an interval onto itself. The successive posi- is a dangerous situation for the tenth thousand tions of a point when the mapping is repeated can and perhaps also for the second thousand. It may be considered as the orbit of a classical particle. happen unpredictably that the fashion changes An orbit has period N if the point returns to its and string theory becomes unfashionable. Then it original position after N mappings. An orbit is could happen that nine thousand string theorists defined to be chaotic, in this context, if it diverges lose their jobs. They have been trained in a narrow from all periodic orbits. The theorem says that if a specialty, and they may be unemployable in other single orbit with period three exists, then chaotic fields of science. orbits also exist. The proof is simple and short. To Why are so many young people attracted to my mind, this theorem and its proof throw more string theory? The attraction is partly intellectual. light than a thousand beautiful pictures on the String theory is daring and mathematically elegant. basic nature of chaos. The theorem explains why But the attraction is also sociological. String theory chaos is prevalent in the world. It does not explain is attractive because it offers jobs. And why are why chaos is so often weak. That remains a task so many jobs offered in string theory? Because for the future. I believe that weak chaos will not string theory is cheap. If you are the chairperson be understood in a fundamental way until we can of a physics department in a remote place without prove rigorous theorems about it. much money, you cannot afford to build a modern laboratory to do experimental physics, but you can String Theorists afford to hire a couple of string theorists. So you I would like to say a few words about string theory. offer a couple of jobs in string theory, and you Few words, because I know very little about string have a modern physics department. The tempta- theory. I never took the trouble to learn the subject tions are strong for the chairperson to offer such or to work on it myself. But when I am at home at the jobs and for the young people to accept them. Institute for Advanced Study in Princeton, I am sur- This is a hazardous situation for the young people rounded by string theorists, and I sometimes listen and also for the future of science. I am not say- to their conversations. Occasionally I understand a ing that we should discourage young people from little of what they are saying. Three things are clear. working in string theory if they find it exciting. I First, what they are doing is first-rate mathemat- am saying that we should offer them alternatives, ics. The leading pure mathematicians, people like so that they are not pushed into string theory by Michael Atiyah and Isadore Singer, love it. It has economic necessity. opened up a whole new branch of mathematics, Finally, I give you my own guess for the future with new ideas and new problems. Most remark- of string theory. My guess is probably wrong. I ably, it gave the mathematicians new methods to have no illusion that I can predict the future. I tell
February 2009 Notices of the AMS 221 you my guess, just to give you something to think Jung, is a mental image rooted in a collective un- about. I consider it unlikely that string theory will conscious that we all share. The intense emotions turn out to be either totally successful or totally that archetypes carry with them are relics of lost useless. By totally successful I mean that it is a memories of collective joy and suffering. Manin is complete theory of physics, explaining all the de- saying that we do not need to accept Jung’s theory tails of particles and their interactions. By totally as true in order to find it illuminating. useless I mean that it remains a beautiful piece of More than thirty years ago, the singer Monique pure mathematics. My guess is that string theory Morelli made a recording of songs with words by will end somewhere between complete success Pierre MacOrlan. One of the songs is La Ville Morte, and failure. I guess that it will be like the theory the dead city, with a haunting melody tuned to of Lie groups, which Sophus Lie created in the Morelli’s deep contralto, with an accordion singing nineteenth century as a mathematical framework counterpoint to the voice, and with verbal images for classical physics. So long as physics remained of extraordinary intensity. Printed on the page, the classical, Lie groups remained a failure. They were words are nothing special: a solution looking for a problem. But then, fifty “En pénétrant dans la ville morte, years later, the quantum revolution transformed Je tenait Margot par le main… physics, and Lie algebras found their proper place. Nous marchions de la nécropole, They became the key to understanding the central Les pieds brisés et sans parole, role of symmetries in the quantum world. I expect Devant ces portes sans cadole, that fifty or a hundred years from now another Devant ces trous indéfinis, revolution in physics will happen, introducing new Devant ces portes sans parole concepts of which we now have no inkling, and the Et ces poubelles pleines de cris”. new concepts will give string theory a new mean- “As we entered the dead city, I held Margot by ing. After that, string theory will suddenly find the hand…We walked from the graveyard on our its proper place in the universe, making testable bruised feet, without a word, passing by these statements about the real world. I warn you that doors without locks, these vaguely glimpsed holes, this guess about the future is probably wrong. It these doors without a word, these garbage cans has the virtue of being falsifiable, which accord- full of screams.” ing to Karl Popper is the hallmark of a scientific I can never listen to that song without a dispro- statement. It may be demolished tomorrow by portionate intensity of feeling. I often ask myself some discovery coming out of the Large Hadron why the simple words of the song seem to resonate Collider in Geneva. with some deep level of unconscious memory, as Manin Again if the souls of the departed are speaking through Morelli’s music. And now unexpectedly in Manin’s To end this talk, I come back to Yuri Manin and book I find an answer to my question. In his chap- his book Mathematics as Metaphor. The book ter, “The Empty City Archetype”, Manin describes is mainly about mathematics. It may come as a how the archetype of the dead city appears again surprise to Western readers that he writes with and again in the creations of architecture, litera- equal eloquence about other subjects such as the ture, art and film, from ancient to modern times, collective unconscious, the origin of human lan- ever since human beings began to congregate in guage, the psychology of autism, and the role of cities, ever since other human beings began to the trickster in the mythology of many cultures. congregate in armies to ravage and destroy them. To his compatriots in Russia, such many-sided The character who speaks to us in MacOrlan’s song interests and expertise would come as no surprise. is an old soldier who has long ago been part of an Russian intellectuals maintain the proud tradition army of occupation. After he has walked with his of the old Russian intelligentsia, with scientists wife through the dust and ashes of the dead city, and poets and artists and musicians belonging to he hears once more: a single community. They are still today, as we see them in the plays of Chekhov, a group of idealists “Chansons de charme d’un clairon bound together by their alienation from a super- Qui fleurissait une heure lointaine stitious society and a capricious government. In Dans un rêve de garnison”. Russia, mathematicians and composers and film- “The magic calls of a bugle that came to life for producers talk to one another, walk together in the an hour in an old soldier’s dream”. snow on winter nights, sit together over a bottle of The words of MacOrlan and the voice of Mo- wine, and share each others’ thoughts. relli seem to be bringing to life a dream from our Manin is a bird whose vision extends far be- collective unconscious, a dream of an old soldier yond the territory of mathematics into the wider wandering through a dead city. The concept of the landscape of human culture. One of his hobbies collective unconscious may be as mythical as the is the theory of archetypes invented by the Swiss concept of the dead city. Manin’s chapter describes psychologist Carl Jung. An archetype, according to the subtle light that these two possibly mythical
222 Notices of the AMS Volume 56, Number 2 concepts throw upon each other. He describes the collective unconscious as an irrational force that powerfully pulls us toward death and destruction. The archetype of the dead city is a distillation of the agonies of hundreds of real cities that have been destroyed since cities and marauding armies 0ROFESSOR OF -ATHEMATICS were invented. Our only way of escape from the /PERATIONS 2ESEARCH insanity of the collective unconscious is a collec- 4HE $EPARTMENT OF -ATHEMATICS WWWMATHETHZCH AT %4( :~RICH tive consciousness of sanity, based upon hope INVITES APPLICATIONS FOR A FACULTY POSITION IN MATHEMATICS OF OPERATIONS and reason. The great task that faces our contem- RESEARCH 4HE DUTIES OF THE FUTURE PROFESSOR WHO WILL BE A MEMBER OF THE $EPARTMENT OF -ATHEMATICS INCLUDE TEACHING AND RESEARCH IN THE porary civilization is to create such a collective MATHEMATICS OF OPERATIONS RESEARCH AND RELATED AREAS 4HE PERSPEC consciousness. TIVE OF THE POSITION INCLUDES THE HEADING OF THE )NSTITUTE FOR /PERATIONS 2ESEARCH AT %4( :URICH References 7E ARE SEEKING CANDIDATES WITH AN INTERNATIONALLY RECOGNIZED RESEARCH RECORD IN MATHEMATICS RELATED TO OPERATIONS RESEARCH AND WITH PROVEN [1] M. J. Bertin et al., Pisot and Salem Numbers, ABILITY TO DIRECT RESEARCH OF HIGH QUALITY %XPERTISE AND A STRONG BACK Birkhäuser Verlag, Basel, 1992. GROUND IN OPTIMIZATION ANDOR COMPUTATION WILL BE ESPECIALLY APPRECI [2] M. L. Cartwright and J. E. Littlewood, On non- ATED 7ILLINGNESS TO TEACH AT ALL UNIVERSITY LEVELS AND TO COLLABORATE WITH COLLEAGUES FROM ENGINEERING AND INDUSTRY IS EXPECTED linear differential equations of the second order, I, Jour. London Math. Soc. 20 (1945), 180–189. 4OGETHER WITH THE COLLEAGUES FROM THE DEPARTMENT THE NEW PROFES SOR WILL BE RESPONSIBLE FOR UNDERGRADUATE COURSES IN MATHE MATICS [3] Freeman Dyson, Prof. Hermann Weyl, For.Mem.R.S., AT %4( :URICH FOR STUDENTS OF MATHEMATICS ENGINEERING AND NATURAL Nature 177 (1956), 457–458. SCIENCES AND FOR GRADUATE COURSES IN THE PROGRAMMES -3C IN !PPLIED [4] Tien-Yien Li and James A. Yorke, Period three implies -ATHEMATICS -3C IN #OMPUTATIONAL 3CIENCE AND %NGINEERING -3C IN chaos, Amer. Math. Monthly 82 (1975), 985–992. 3TATISTICS AND -3C IN 1UANTITATIVE &INANCE JOINT DEGREE WITH THE 5NI VERSITY OF :~RICH 4HE SUCCESSFUL CANDIDATE WILL BE EXPECTED TO TEACH [5] Yuri I. Manin, Mathematics as Metaphor: Selected UNDERGRADUATE LEVEL COURSES 'ERMAN OR %NGLISH AND GRADUATE LEVEL Essays, American Mathematical Society, Providence, COURSES %NGLISH Rhode Island, 2007. [The Russian version is: Manin, 0LEASE SUBMIT YOUR APPLICATION WITH A CURRICULUM VITAE AND LIST OF PUB Yu. I., Matematika kak Metafora, Moskva, Izdatyelstvo LICATIONS TO THE 0RESIDENT OF %4( :URICH 0ROF $R 2ALPH %ICHLER 2AE MTsNMO, 2008.] MISTRASSE :URICH 3WITZERLAND NO LATER THAN -ARCH 7ITH A VIEW TOWARD INCREASING THE NUMBER OF FEMALE PROFESSORS %4( [6] Andrew M. Odlyzko, Primes, quantum chaos and :URICH SPECIFICALLY ENCOURAGES FEMALE CANDIDATES TO APPLY computers, in Number Theory, Proceedings of a Sym- posium, National Research Council, Washington DC, 1990, pp. 35–46. [7] Hermann Weyl, Gravitation und elektrizität, Sitz. König. Preuss. Akad. Wiss. 26 (1918), 465–480. [8] ——— , Elektron und gravitation, Zeits. Phys. 56 (1929), 350–352. New Directions Short Course [9] ——— , Selecta, Birkhäuser Verlag, Basel, 1956, Applied Algebraic Topology p. 192. June 15-26, 2009 [10] Chen Ning Yang, Integral formalism for gauge fields, Phys. Rev. Letters 33 (1974), 445–447. Instructors: [11] Chen Ning Yang and Robert L. Mills, Conservation Gunnar Carlsson (Stanford University) of isotopic spin and isotopic gauge invariance, Phys. Robert Ghrist (University of Pennsylvania) Rev. 96 (1954), 191–195. From June 15-26, 2009 the IMA will host an intensive [12] ——— , Hermann Weyl’s contribution to physics, in Hermann Weyl, 1885–1985, (K. Chandrasekharan, ed.), short course designed to efficiently provide researchers Springer-Verlag, Berlin, 1986, p. 19. in the mathematical sciences and related disciplines the basic knowledge prerequisite to undertake research in applied algebraic topology. The course will be taught by Gunnar Carlsson (Department of Mathematics, Stanford University) and Robert Ghrist (Department of Electrical and Systems Engineering, Department of Mathematics, University of Pennsylvania). The primary audience for the course is mathematics faculty. No prior background in applied algebraic topology is expected. Participants will receive full travel and lodging support during the workshop. For more information and to apply: www.ima.umn.edu/2008-2009/ND6.15-26.09 Application deadline: April 1, 2009
The IMA is an NSF funded institute
February 2009 Notices of the AMS 223