School of Mathematics

After his teatime conversation with Hugh Montgomery, Freeman Dyson wrote this letter to Atle Selberg with references showing that the pair-correlation of the zeros of the zeta function is identical to that of the eigenvalues of a random matrix.

In early April 1972, Hugh Montgomery, who had been a Member in the School of Mathematics the previous year, stopped by the Institute to share a new result with Atle Selberg, a Professor in the School. The discussion between Montgomery and Selberg involved Montgomery’s work on the zeros of the Riemann zeta function, which is connected to the pattern of the prime numbers in number theory. Generations of mathematicians at the Institute and elsewhere have tried to prove the Riemann Hypothesis, which conjectures that the non-trivial zeros (those that are not easy to find) of the Riemann zeta function lie on the critical line with real part equal to 1⁄2.

Montgomery had found that the statistical distribution of the zeros on the critical line of the Riemann zeta function has a certain property, now called Montgomery’s pair correlation conjecture. He explained that the zeros tend to repel between neighboring levels. At teatime, Montgomery mentioned his result to Freeman Dyson, Professor in the School of Natural Sciences.

In the 1960s, Dyson had worked on random matrix theory, which was proposed by physicist Eugene Wigner in 1951 to describe nuclear physics. The quantum mechanics of a heavy nucleus is complex and poorly understood. Wigner made a bold conjecture that the statistics of the energy levels could be captured by random matrices. Because of Dyson’s work on random matrices, the distribution or the statistical behavior of the eigenvalues of these matrices has been understood since the 1960s.

By John Wheeler 

The paper by Niels Bohr and John Wheeler on the mechanism of nuclear fission appeared in the Physical Review of September 1, 1939, the same day the war began.

The following excerpt is from remarks given by John Archibald Wheeler on March 27, 2000, in connection with the play Copenhagen by Michael Frayn. Wheeler was a Professor of Physics at Princeton University from 1938 until his retirement in 1976 and a Member of the Institute’s School of Mathematics (prior to the founding of the School of Natural Sciences) in the spring of 1937, when it was still temporarily housed in Fine Hall (now Jones Hall) at Princeton University. Niels Bohr, who had a twenty-year association with the Institute, first visited in the academic year 1938–39, when the Institute completed Fuld Hall. For more about Bohr and his relationship with Albert Einstein, one of the Institute’s first Professors, see the Spring 2009 Institute Letter.

If two such great thinkers as Bohr and Einstein, who had such a high regard for each other, could be brought together for a prolonged period, would not something emerge of great value to all of us? This thought and this hope animated the guiding spirits of the Princeton Institute for Advanced Study to invite Niels Bohr to come as a guest of the Institute for the entire spring semester of 1939. However, four days before Bohr boarded his America-bound ship, he learned from Otto Robert Frisch that Frisch and his aunt Lisa Meitner had solid evidence that a neutron splits the nucleus of uranium. As he crossed the Atlantic, Bohr’s vision turned more and more from the problem of quantum mechanics to the problems of nuclear physics. So January and February, March and April of 1939 saw him working, discussing, calculating, and writing, day after day, not with Einstein on quantum physics as intended, but with me on the nuclear physics of fission. Yes, of course, there were meetings Bohr had with Einstein but they were occasional and did not lead to the big push it takes to formulate a solid well-argued position. No. Fission, and what it meant and how it differed from one nucleus to another, and what those differences offered in the way of using the nucleus for a chain reaction stood at the center of our attention. . . .

By Mina Teicher 

Leonardo da Vinci’s Vitruvian Man, ca. 1487

It is known that mathematicians see beauty in mathematics. Many mathematicians are motivated to find the most beautiful proof, and often they refer to mathematics as a form of art. They are apt to say “What a beautiful theorem,” “Such an elegant proof.” In this article, I will not elaborate on the beauty of mathematics, but rather the mathematics of beauty, i.e., the mathematics behind beauty, and how mathematical notions can be used to express beauty—the beauty of manmade creations, as well as the beauty of nature.

I will give four examples of beautiful objects and will discuss the mathematics behind them. Can the beautiful object be created as a solution of a mathematical formula or question? Moreover, I shall explore the general question of whether visual experience and beauty can be formulated with mathematical notions.

I will start with a classical example from architecture dating back to the Renaissance, move to mosaic art, then to crystals in nature, then to an example from my line of research on braids, and conclude with the essence of visual experience.

The shape of a perfect room was defined by the architects of the Renaissance to be a rectangular-shaped room that has a certain ratio among its walls—they called it the “golden section.” A rectangular room with the golden-section ratio also has the property that the ratio between the sum of the lengths of its two walls (the longer one and the shorter one) to the length of its longer wall is also the golden section, 1 plus the square root of 5 over 2. Architects today still believe that the most harmonious rooms have a golden-section ratio. This number appears in many mathematical phenomena and constructions (e.g., the limit of the Fibonacci sequence). Leonardo da Vinci observed the golden section in well-proportioned human bodies and faces—
in Western culture and in some other civilizations the golden-section ratio of a well-proportioned human body resides between the upper part (above the navel) and the lower part (below the navel).

By Richard Taylor 

In modular arithmetic, one thinks of the whole numbers arranged around a circle, like the hours on a clock, instead of along an infinite straight line. Here we have seven “hours” on our clock—arithmetic modulo 7. To add 3 and 5 modulo 7, you start at 0, count 3 clockwise, and then a further 5 clockwise, this time ending on 1. To multiply 3 by 5 modulo 7, you start at 0 and count 3 clockwise 5 times, again ending up at 1.

Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In this article, I will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs.

For almost all its history, the study of modular arithmetic has been driven purely by its inherent beauty and by human curiosity. But in one of those strange pieces of serendipity which often characterize the advance of human knowledge, in the last half century modular arithmetic has found important applications in the “real world.” Today, the theory of modular arithmetic (e.g., Reed-Solomon error correcting codes) is the basis for the way DVDs store or satellites transmit large amounts of data without corrupting it. Moreover, the cryptographic codes which keep, for example, our banking transactions secure are also closely connected with the theory of modular arithmetic. You can visualize the usual arithmetic as operating on points strung out along the “number line.”

By Ankur Moitra 

If two customers have a common interest in cooking, Amazon can use information about which items one of them has bought to make good recommendations to the other and vice-versa. Ankur Moitra is trying to develop rigorous theoretical foundations for widely used algorithms whose behavior we cannot explain.

How do we navigate the vast amount of data at our disposal? How do we choose a movie to watch, out of the 75,000 movies available on Netflix? Or a new book to read, among the 800,000 listed on Amazon? Or which news articles to read, out of the thousands written everyday? Increasingly, these tasks are being delegated to computers—​recommendation systems analyze a large amount of data on user behavior, and use what they learn to make personalized recommendations for each one of us.

In fact, you probably encounter recommendation systems on an everyday basis: from Netflix to Amazon to Google News, better recommendation systems translate to a better user experience. There are some basic questions we should ask: How good are these recommendations? In fact, a more basic question: What does “good” mean? And how do they do it? As we will see, there are a number of interesting mathematical questions at the heart of these issues—most importantly, there are many widely used algorithms (in practice) whose behavior we cannot explain. Why do these algorithms work so well? Obviously, we would like to put these algorithms on a rigorous theoretical foundation and understand the computational complexity of the problems they are trying to solve.

By Helmut Hofer and Derek Bermel 

This image (produced with a Java applet by Alec Jacobson at http://alecjacobåson.com/programs/three-body-chaos) shows color­ful trackings of the paths of satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits.

I can’t understand why people are frightened of new ideas. I’m frightened of the old ones.—John Cage

Helmut Hofer, Professor in the School of Mathematics, writes:

Last September, the School of Mathematics launched its yearlong program with my Member seminar talk “First Steps in Symplectic Dynamics.” About two years earlier, it had become clear that certain important problems in dynamical systems could be solved with ideas coming from a different field, the field of symplectic geometry. The goal was then to bring researchers from the fields of dynamical systems and symplectic geometry together in a program aimed at the development of a common core and ideally leading to a new field—symplectic dynamics.

Not long before, in my 2010 inaugural public lecture at IAS, “From Celestial Mechanics to a Geometry Based on the Concept of Area,” I had described the historical background and some of the interesting mathematical problems belonging to this anticipated field of symplectic dynamics. The lecture began with a computer program showing chaos in the restricted three-body problem. This problem describes the movement of a satellite under the gravity of two big bodies, say the earth and the moon, in a rotating coordinates system in which the earth and the moon stay at fixed positions. The chaos in the system is illustrated by putting about ten satellites initially at almost the same position with almost the same velocity.

When the system starts evolving, the program shows colorful trackings of the paths of the satellites as they evolve from a simple single orbit to a complex multicolored tangle of orbits, once the orbits of the different satellites start separating.

In his conjectures, now collectively known as the Langlands program, Robert Langlands drew on the work of Hermann Weyl (above), André Weil, and Harish-Chandra, among others with extensive ties to the Institute.

It has been said that the goals of modern mathematics are recon­struction and development.¹ The unifying conjectures between number theory and representation theory that Robert Langlands, Professor Emeritus in the School of Mathematics, articulated in a letter to André Weil in 1967, continue a tradition at the Institute of advancing mathematical knowledge through the identification of problems central to the understanding of active areas or likely to become central in the future.

“Two striking qualities of mathematical concepts regarded as central are that they are simultaneously pregnant with possibilities for their own development and, so far as we can judge from a history of two and a half millennia, of permanent validity,” says Langlands. “In comparison with biology, above all with the theory of evolution, a fusion of biology and history, or with physics and its two enigmas, quantum theory and relativity theory, mathematics contributes only modestly to the intellectual architecture of mankind, but its central contributions have been lasting, one does not supersede another, it enlarges it.”²

In his conjectures, now collectively known as the Langlands program, Langlands drew on the work of Harish-Chandra, Atle Selberg, Goro Shimura, André Weil, and Hermann Weyl, among others with extensive ties to the Institute. 

Weyl, whose appointment to the Institute’s Faculty in 1933 followed those of Albert Einstein and Oswald Veblen, was a strong believer in the overall unity of mathematics, across disciplines and generations. Weyl had a major impact on the progress of the entire field of mathematics, as well as physics, where he was equally comfortable. His work spanned topology, differential geometry, Lie groups, representation theory, harmonic analysis, and analytic number theory, and extended into physics, including relativity, electromagnetism, and quantum mechanics. “For [Weyl] the best of the past was not forgotten,” notes Michael Atiyah, a former Institute Professor and Member, “but was subsumed and refined by the mathematics of the present.”³

The simplest case of the fundamental lemma counts points with alternating signs at various distances from the center of a certain tree-like structure. As depicted in the above image by former Member Bill Casselman, it counts 1, 1–3=–2, 1–3+6=4, 1–3+6–12=–8, etc. But this case is deceptively simple, and Bao Châu Ngô’s final proof required a huge range of sophisticated mathematical tools.

The proof of the fundamental lemma by Bao Châu Ngô that was confirmed last fall is based on the work of many mathematicians associated with the Institute for Advanced Study over the past thirty years. The fundamental lemma, a technical device that links automorphic representations of different groups, was formulated by Robert Langlands, Professor Emeritus in the School of Mathematics, and came out of a set of overarching and interconnected conjectures that link number theory and representation theory, collectively known as the Langlands program. The proof of the fundamental lemma, which resisted all attempts for nearly three decades, firmly establishes many theorems that had assumed it and paves the way for progress in understanding underlying mathematical structures and possible connections to physics.

The simplest case of the fundamental lemma counts points with alternating signs at various distances from the center of a certain tree-like structure. As depicted in the above image by former Member Bill Casselman, it counts 1, 1–3=–2, 1–3+6=4, 1–3+6–12=–8, etc. But this case is deceptively simple, and Ngô’s final proof required a huge range of sophisticated mathematical tools.

The story of the fundamental lemma, its proof, and the deep insights it provides into diverse fields from number theory and algebraic geometry to theoretical physics is a striking example of how mathematicians work at the Institute and demonstrates a belief in the unity of mathematics that extends back to Hermann Weyl, one of the first Professors at the Institute. This interdisciplinary tradition has changed the course of the subject, leading to profound discoveries in many different mathematical fields, and forms the basis of the School’s interaction with the School of Natural Sciences, which has led to the use of ideas from physics, such as gauge fields and strings, in solving problems in geometry and topology and the use of ideas from algebraic and differential geometry in theoretical physics.

By George Dyson 

In this 1953 diagnostic photograph from the maintenance logs of the IAS Electronic Computer Project (ECP), a 32-by-32 array of charged spots––serving as working memory, not display––is visible on the face of a Williams cathode-ray memory tube. Starting in late 1945, John von Neumann, Professor in the School of Mathematics, and a group of engineers worked at the Institute to design, build, and program an electronic digital computer.

I am thinking about something much more important than bombs. I am thinking about computers.––John von Neumann, 1946 

By Cécile DeWitt-Morette 

Cécile DeWitt-Morette with (from left to right) Isadore Singer, Freeman Dyson, and Raoul Bott at the Institute in the 1950s

In Brief 

It all began with a cable from Oppenheimer that I received on March 10, 1948, in Trondheim, Norway: ON THE RECOMMENDATION OF BOHR AND HEITLER I AM GLAD TO OFFER YOU MEMBERSHIP SCHOOL OF MATHEMATICS FOR THE ACADEMIC YEAR 1948 – 1949 WITH STIPEND OF $3500. ROBERT OPPENHEIMER.

I did not know that this was a great offer. I did not even know where Princeton was, but as a general rule, I would rather say “yes” than “no.” I was then on leave from the French Centre National de la Recherche Scientifique (CNRS), having been awarded a Rask-Oersted Fellowship for the academic year 1947–48 at the Nordiska Institutet för Teoretisk Fysik in Copenhagen.

In retrospect, I think that in the days of the Marshall plan, Oppie was looking for a couple of European young postdocs who would benefit from a year at the Institute. Did I benefit? More than I could ever have imagined.
During my two-year stay, 1948–50, Bryce DeWitt, a postdoc at the Institute, 1949–50, asked me to marry him, and I conceived the Les Houches Summer School as my self-imposed condition for marrying a “foreigner.” Thanks to Freeman Dyson and Richard Feynman, I learned about functional integration and am still fascinated by it.