Automorphic Forms

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.