Peter Sarnak

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.

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 Matthew Kahle 

Matthew Kahle, Member (2010-11) in the School of Mathematics, writes about his interest in thinking about what it might be like inside a black hole. This illustration (Figure 1.), from Kip Thorne's Black Holes and Time Warps: Einstein's Outrageous Legacy (W. W. Norton & Company, Inc., 1994), suggests a few probabilities.

I sometimes like to think about what it might be like inside a black hole. What does that even mean? Is it really “like” anything inside a black hole? Nature keeps us from ever knowing. (Well, what we know for sure is that nature keeps us from knowing and coming back to tell anyone about it.) But mathematics and physics make some predictions.

John Wheeler suggested in the 1960s that inside a black hole the fabric of spacetime might be reduced to a kind of quantum foam. Kip Thorne described the idea in his book Black Holes & Time Warps as follows (see Figure 1).

“This random, probabilistic froth is the thing of which the singularity is made, and the froth is governed by the laws of quantum gravity. In the froth, space does not have any definite shape (that is, any definite curvature, or even any definite topology). Instead, space has various probabilities for this, that, or another curvature and topology. For example, inside the singularity there might be a 0.1 percent probability for the curvature and topology of space to have the form shown in (a), and a 0.4 percent probability for the form in (b), and a 0.02 percent probability for the form in (c), and so on.”

In other words, perhaps we cannot say exactly what the properties of spacetime are in the immediate vicinity of a singularity, but perhaps we could characterize their distribution. By way of analogy, if we know that we are going to flip a fair coin a thousand times, we have no idea whether any particular flip will turn up heads or tails. But we can say that on average, we should expect about five hundred heads. Moreover, if we did the experiment many times we should expect a bell-curve shape (i.e., a normal distribution), so it is very unlikely, for example, that we would see more than six hundred heads.