Edward Witten

By Robbert Dijkgraaf 

Robbert Dijkgraaf, IAS Director and Leon Levy Professor, in the Mathematics–Natural Sciences Library in Fuld Hall

I am honored and heartened to have joined the Institute for Advanced Study this summer as its ninth Director. The warmness of the welcome that my family and I have felt has surpassed our highest expectations. The Institute certainly has mastered the art of induction.

The start of my Directorship has been highly fortuitous. On July 4, I popped champagne during a 3 a.m. party to celebrate the LHC’s discovery of a particle that looks very much like the Higgs boson—the final element of the Standard Model, to which Institute Faculty and Members have contributed many of the theoretical foundations. I also became the first Leon Levy Professor at the Institute due to the great generosity of the Leon Levy Foundation, founded by Trustee Shelby White and her late husband Leon Levy, which has endowed the Directorship. Additionally, four of our Professors in the School of Natural Sciences—Nima Arkani-Hamed, Juan Maldacena, Nathan Seiberg, and Edward Witten—were awarded the inaugural Fundamental Physics Prize of the Milner Foundation for their path-breaking contributions to fundamental physics. And that was just the first month.

Nearly a century ago, Abraham Flexner, the founding Director of the Institute, introduced the essay “The Usefulness of Useless Knowledge.” It was a passionate defense of the value of the freely roaming, creative spirit, and a sharp denunciation of American universities at the time, which Flexner considered to have become large-scale education factories that placed too much emphasis on the practical side of knowledge. Columbia University, for example, offered courses on “practical poultry raising.” Flexner was convinced that the less researchers needed to concern themselves with direct applications, the more they could ultimately contribute to the good of society.

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.

Robbert Dijkgraaf will become the ninth Director of the Institute, as of July 1, 2012.

On November 14, the Institute for Advanced Study announced the appointment of Robbert Dijkgraaf as its ninth Director, succeeding, as of July 1, 2012, Peter Goddard, who has served as Director since January 2004.

Below, Dijkgraaf speaks about his enthusiasm for the Institute and for using knowledge, creativity, and collaboration to further our understanding of a world of diverse facts, structures, ideas, and cultures.

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I am delighted to come to the Institute for Advanced Study, one of the intellectual centers of the world. The position of Director is highly distinguished, and the list of former Directors is quite intimidating. But I am particularly looking forward to combining at the highest level three elements that have been important in my professional life: the opportunity to collaborate with the very best scientists and scholars; to organize a stimulating environment for great talent from around the world; and to play an active role in science education, advocacy, and diplomacy to engage future generations.

Taking up my appointment as Director of the Institute will feel a bit like coming home. My family and I have only the best recollections of our stays in Princeton. I also expect that in many ways my life will become more focused. My present position as President of the Royal Netherlands Academy of Arts and Sciences requires giving attention to many different areas, from elementary school programs to industrial affairs, from government policy to international relations. The Institute is remarkably effective as a place for concentration and inspiration.

By Juan Maldacena 

Albert Einstein, pictured at left with J. Robert Oppenheimer at the Institute, tried to disprove the notion of black holes that his theory of general relativity and gravity seemed to predict. Oppenheimer used Einstein's theory to show how black holes could form.

The ancients thought that space and time were preexisting entities on which motion happens. Of course, this is also our naive intuition. According to Einstein’s theory of general relativity, we know that this is not true. Space and time are dynamical objects whose shape is modified by the bodies that move in it. The ordinary force of gravity is due to this deformation of spacetime. Spacetime is a physical entity that affects the motion of particles and, in turn, is affected by the motion of the same particles. For example, the Earth deforms spacetime in such a way that clocks at different altitudes run at different rates. For the Earth, this is a very small (but measurable) effect. For a very massive and very compact object the deformation (or warping) of spacetime can have a big effect. For example, on the surface of a neutron star a clock runs slower, at 70 percent of the speed of a clock far away.

In fact, you can have an object that is so massive that time comes to a complete standstill. These are black holes. General relativity predicts that an object that is very massive and sufficiently compact will collapse into a black hole. A black hole is such a surprising prediction of general relativity that it took many years to be properly recognized as a prediction. Einstein himself thought it was not a true prediction, but a mathematical oversimplification. We now know that they are clear predictions of the theory. Furthermore, there are some objects in the sky that are probably black holes.  

Black holes are big holes in spacetime. They have a surface that is called a “horizon.” It is a surface that marks a point of no return. A person who crosses it will never be able to come back out. However, he will not feel anything special when he crosses the horizon. Only a while later will he feel very uncomfortable when he is crushed into a “singularity,” a region with very high gravitational fields. The horizon is what makes black holes “black”; nothing can escape from the horizon, not even light. Fortunately, if you stay outside the horizon, nothing bad happens to you. The singularity remains hidden behind the horizon.

By Edward Witten 

Edward Witten explains how mathematicians compare knots that differ by how a missing piece is filled in (as indicated by the question mark above).

In everyday life, a string—such as a shoelace—is usually used to secure something or hold it in place. When we tie a knot, the purpose is to help the string do its job. All too often, we run into a complicated and tangled mess of string, but ordinarily this happens by mistake.

The term “knot” as it is used by mathematicians is abstracted from this experience just a little bit. A knot in the mathematical sense is a possibly tangled loop, freely floating in ordinary space. Thus, mathematicians study the tangle itself. A typical knot in the mathematical sense is shown in Figure 1. Hopefully, this picture reminds us of something we know from everyday life. It can be quite hard to make sense of a tangled piece of string—to decide whether it can be untangled and if so how. It is equally hard to decide if two tangles are equivalent.

Such questions might not sound like mathematics, if one is accustomed to thinking that mathematics is about adding, subtracting, multiplying, and dividing. But actually, in the twentieth century, mathematicians developed a rather deep theory of knots, with surprising ways to answer questions like whether a given tangle can be untangled.

But why—apart from the fact that the topic is fun—am I writing about this as a physicist? Even though knots are things that can exist in ordinary three-dimensional space, as a physicist I am only interested in them because of something surprising that was discovered in the last three decades.