Nathan Seiberg

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.

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.