Articles from the Institute Letter

Additional articles from new and past issues of the Institute Letter will continue to be posted over time and as they become available.

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.

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By Patrick Geary

Presumed Longobard settlements, first to sixth centuries © Landschaftsverband Rheinland/Rheinisches Landesmuseum

Few historical questions have so fascinated historians as the fall of the Roman Empire or, in the more fashionable modern parlance, its “transformation” into something altogether different, namely independent kingdoms ruled by successors of barbarian commanders in the West and a Greek-speaking Byzantine Empire in the East. For over two centuries, historians have particularly debated the role of barbarian invasions in this process, but in reality we have very little hard data on the nature of the barbarian “peoples” that entered the Western provinces between the fourth and sixth centuries, their numbers, their composition, or the reality of their influence on the indigenous populations of the Empire.

Were these large ethnic populations moving across Europe from Scandinavia to Italy and Spain, as nineteenth-century romantics imagined? Or were they small heterogeneous military units employed by the Empire that settled, with a minimum of force and disruption, within the administrative and fiscal mechanisms of a still-functioning Empire, as has been suggested more recently? Did these groups long maintain their distinctiveness from the local population, eschewing intermarriage and holding fast to their distinctive legal and cultural traditions, or did they rapidly integrate themselves into local elites through intermarriage and cultural transformation? Were they really distinct population groups at all or merely provincial Romans and local “barbarians” who united under ethnic labels and took advantage of opportunities to seize power from a beleaguered empire? Traditional sources with which to answer these questions—highly rhetorical accounts of the period often written centuries later, sparse administrative documents surviving in scattered fragments, and ambiguous archaeological material showing changing patterns of burial custom and settlements—simply do not provide enough evidence to reach a consensus.

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By Deva Woodly

An Obama supporter holds up a “Yes We Can” sign as President-elect Barack Obama gives his victory speech during a 2008 election night gathering in Grant Park. Courtesy of Getty Images.

I was in my final year of graduate school, writing a dissertation on the place of persuasion in the success of contemporary American social movements, when the nearly two-year-long campaign for the American president who would succeed George W. Bush began. As a student of politics, it was impossible not to be transfixed by the epic discursive battle being waged, first in the hard- fought democratic primary between Hillary Clinton and Barack Obama and finally during the general election campaign in which, Obama, having won against his formidable Democratic rival, entered a political contest with veteran politician John McCain. For the American public, this contest was the most closely followed election in decades. A Gallup poll taken in June 2008, early summer, when political attention is usually at its nadir, found that nearly two-thirds of Americans described the 2008 campaign as “exciting.” By September, Gallup found that a record 87 percent, almost nine in ten Americans, reported that they were following national politics closely. The astonished poll takers wrote, in the summary of their results, “This significantly exceeds anything Gallup has measured since it began asking this question in 1995.”*

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Slide from Nima Arkani-Hamed’s lecture, “The Inevitability of Physical Laws: Why the Higgs Has to Exist.”

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.”

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By Freeman Dyson

Freeman Dyson was awarded the 2012 Henri Poincaré Prize at the International Mathematical Physics Congress in August. On this occasion, he delivered the lecture “Is a Graviton Detectable?” a PDF of which is available at http://publications.ias.edu/poincare2012/dyson.pdf.

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.

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