School of Natural Sciences
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.
Following the discovery in July of a Higgs-like boson—an effort that took more than fifty years of experimental work and more than 10,000 scientists and engineers working on the Large Hadron Collider—Juan Maldacena and Nima Arkani-Hamed, two Professors in the School of Natural Sciences, gave separate public lectures on the symmetry and simplicity of the laws of physics, and why the discovery of the Higgs was inevitable.
Peter Higgs, who predicted the existence of the particle, gave one of his first seminars on the topic at the Institute in 1966, at the invitation of Freeman Dyson. “The discovery attests to the enormous importance of fundamental, deep ideas, the substantial length of time these ideas can take to come to fruition, and the enormous impact they have on the world,” said Robbert Dijkgraaf, Director and Leon Levy Professor.
In their lectures “The Symmetry and Simplicity of the Laws of Nature and the Higgs Boson” and “The Inevitability of Physical Laws: Why the Higgs Has to Exist,” Maldacena and Arkani-Hamed described the theoretical ideas that were developed in the 1960s and 70s, leading to our current understanding of the Standard Model of particle physics and the recent discovery of the Higgs-like boson. Arkani-Hamed framed the hunt for the Higgs as a detective story with an inevitable ending. Maldacena compared our understanding of nature to the fairytale Beauty and the Beast.
“What we know already is incredibly rigid. The laws are very rigid within the structure we have, and they are very fragile to monkeying with the structure,” said Arkani-Hamed. “Often in physics and mathematics, people will talk about beauty. Things that are beautiful, ideas that are beautiful, theoretical structures that are beautiful, have this feeling of inevitability, and this flip side of rigidity and fragility about them.”
By Freeman Dyson
John Brockman, founder and proprietor of the Edge website, asks a question every New Year and invites the public to answer it. THE EDGE QUESTION 2012 was, “What is your favorite deep, elegant, or beautiful
explanation?” He got 150 answers that are published in a book, This Explains Everything (Harper Collins, 2013). Here is my contribution.
The situation that I am trying to explain is the existence side by side of two apparently incompatible pictures of the universe. One is the classical picture of our world as a collection of things and facts that we can see and feel, dominated by universal gravitation. The other is the quantum picture of atoms and radiation that behave in an unpredictable fashion, dominated by probabilities and uncertainties. Both pictures appear to be true, but the relationship between them is a mystery.
The orthodox view among physicists is that we must find a unified theory that includes both pictures as special cases. The unified theory must include a quantum theory of gravitation, so that particles called gravitons must exist, combining the properties of gravitation with quantum uncertainties.
By Freeman Dyson
The Evolution of Cooperation is the title of a book by Robert Axelrod. It was published by Basic Books in 1984, and became an instant classic. It set the style in which modern scientists think about biological evolution, reducing the complicated and messy drama of the real world to a simple mathematical model that can be run on a computer. The model that Axelrod chose to describe evolution is called “The Prisoner’s Dilemma.” It is a game for two players, Alice and Bob. They are supposed to be interrogated separately by the police after they have committed a crime together. Each independently has the choice, either to remain silent or to say the other did it. The dilemma consists in the fact that each individually does better by testifying against the other, but they would collectively do better if they could both remain silent. When the game is played repeatedly by the same two players, it is called Iterated Prisoner’s Dilemma. In the iterated game, each player does better in the short run by talking, but does better in the long run by remaining silent. The switch from short-term selfishness to long-term altruism is supposed to be a model for the evolution of cooperation in social animals such as ants and humans.
Mathematics is always full of surprises. The Prisoner’s Dilemma appears to be an absurdly simple game, but Axelrod collected an amazing variety of strategies for playing it. He organized a tournament in which each of the strategies plays the iterated game against each of the others. The results of the tournament show that this game has a deep and subtle mathematical structure. There is no optimum strategy. No matter what Bob does, Alice can do better if she has a “Theory of Mind,” reconstructing Bob’s mental processes from her observation of his behavior.
By David S. Spiegel
Until a couple of decades ago, the only planets we knew existed were the nine in our Solar System. In the last twenty-five years, we’ve lost one of the local ones (Pluto, now classified as a “minor planet”) and gained about three thousand candidate planets around other stars, dubbed exoplanets. The new field of exoplanetary science is perhaps the fastest growing subfield of astrophysics, and will remain a core discipline for the forseeable future.
The fact that any biology beyond Earth seems likely to live on such a planet is among the many reasons why the study of exoplanets is so compelling. In short, planets are not merely astrophysical objects but also (at least some of them) potential abodes.
The highly successful Kepler mission involves a satellite with a sensitive telescope/camera that stares at a patch of sky in the direction of the constellation Cygnus. The goal of the mission is to find what fraction of Sun-like stars have Earth-sized planets with a similar Earth-Sun separation (about 150 million kilometers, or the distance light travels in eight minutes).