- - - er 2010 Summer etc. But this case is deceptively simple, 8, – 12= – 6 3+ – 1 3+6=4, – 1 e Institute Lett 2, – 3= – Th research and emphasized the “usefulness of useless knowledge.” It was his belief that tradition has changed the course of subject, leading to profound discoveries in AS work at the Institute and demonstrates a belief in unity of mathematics that extends theory, collectively known as the Langlands program. The proof of fundamental lemma, theory, will be found to an indispensable link in a long and complex chain that may open new in theoretical physics. it and paves the way for progress in understanding underlying mathematical structures connections to physics. story of the fundamental lemma, its proof, and deep insights it provides into diverse fields simplest case of the fundamental lemma counts points with alternating signs at various distances was formulated by Robert Langlands, Professor Emeritus in the School of Mathematics, and in theory and practice.” resisted all attempts for nearly three decades, firmly establishes many theorems that had The fundamental lemma, a technical device that links automorphic representations of different different mathematical fields, and forms the basis of the School’s interaction with the School different mathematical fields, and forms the basis of School’s out of a set overarching and interconnected conjectures that link number theory repre also poignantly underscores the Institute’s continual commitment to the philosophy of Abra also poignantly underscores the Institute’s of many mathematicians associated with the Institute for Advanced Study over past thirty the center of a certain tree-like structure. As depicted in above image by former Member number theory and algebraic geometry to theoretical physics is a striking example of how math to Hermann Weyl, one of the first Professors at Institute (see articles, pages 1 and 5). This to Hermann Weyl, Flexner, founding Director of the Institute, who argued case for curiosity-driven rather than Flexner, proof of the fundamental lemma by Bao Châu Ngô that was confirmed last fall is based on Ngô’s final proof required a huge range of sophisticated mathematical tools. Ngô’s Casselman, it counts 1, 1 The The It free to take advantage of surprises, and someday an unexpected discovery, apparently leading free to take advantage of surprises, and someday an unexpected discovery, solving problems in geometry and topology the use of ideas from algebraic differential Natural Sciences, which has led to the use of ideas from physics, such as gauge fields and strings, the Institute “eschews chase for useful, minds of its scholars will be liberated, they The work years. groups, came sentation which assumed possible from Bill and from ematicians back interdisciplinary many of in geometry ham objective-driven if be nowhere, worlds I Have you moved? Non-Profit Org. Please notify us of your change of address. 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Peretsman www.ias.edu S10-02426_SummerNews.qxp:IAS Letter 8/17/10 10:37 AM Page 1 The Institute Letter InstituteIAS for Advanced Study Summer 2010 The Fundamental Lemma Measuring the Cosmos, Mapping the From Minor Irritant to Central Problem Galaxy, Finding Planets he fundamental lemma has been BY DAVID H. WEINBERG Tdescribed as a gross understate- ment.1 “The curious thing is that it is called a lemma [a subsidiary proposi- hy is the expansion of the uni- tion to be proved on the way to Wverse speeding up, instead of demonstrating a principal proposi- being slowed by the gravitational tion]. It is a theorem,” says Andrew attraction of galaxies and dark mat- Wiles, a Visitor in the School of ter? What is the history of the Milky Mathematics and an Institute Trustee. Way galaxy and of the chemical “At first, it was thought to be a minor elements in its stars? Why are the irritant, but it subsequently became planetary systems discovered around clear that it was not a lemma but other stars so different from our own ANDREA KANE rather a central problem in the field.” solar system? These questions are the Member Bao Châu Ngô’s proof of the fundamental Robert Langlands, Professor Emeri- themes of SDSS-III, a six-year program lemma was confirmed last fall. tus in the School of Mathematics, first of four giant astronomical surveys, introduced the fundamental lemma in 1979 in a lecture, “Les débuts d’une formule des and the focal point of my research at traces stable,” at the École Normale Supérieure de Jeunes Filles and published in Pub- the Institute during the last year. lications Mathématiques de l’Université Paris VII.2 The goal of the lecture was the stabi- In fact, the Sloan Digital Sky ANDREA KANE Member David H. Weinberg, Project Scientist of the lization of the Selberg trace formula, but it also introduced the fundamental lemma, a Survey (SDSS) has been a running Sloan Digital Sky Survey-III, gave a seminar on com- technical device that links automorphic representations of different groups and the theme through all four of my stays at putational cosmology and galaxy formation in March. notion of closely related transfer factors that could transport automorphic forms. This the Institute, which now span nearly led to the creation of a field of study that Diana Shelstad, a former Member in the two decades. As a long-term postdoctoral Member in the early 1990s, I joined in School of Mathematics and Langlands’s student, eventually called “endoscopy.” the effort to design the survey strategy and software system for the SDSS, a project In the theory of endoscopy, the Selberg trace formula (introduced by the late Atle that was then still in the early stages of fundraising, collaboration building, and Selberg, Professor in the School of Mathematics) is used to distinguish the internal hardware development. When I returned as a sabbatical visitor in 2001–02, SDSS (Continued on page 4) (Continued on page 6) Reflections on the Dewey Seminar Experience They are actually trying to work out concrete solutions BY ANNA MARIE SMITH to the problems related to our key concepts—abstract ideas such as “justice,” “equality,” or “democracy”—on n my discipline, political theory, we love a good story. the ground, with real live people, ticking clocks, laws IWe tell each other stories about the lives and times of that bind, and ever-shrinking budgets. great thinkers, such as Plato, Jefferson, or Gandhi. We In the past year, I participated in the Dewey Seminar enter into the fictitious worlds like the state of nature in on education in the School of Social Science. About a Hobbes, Locke, and Rousseau. Like unruly bit players on century ago, John Dewey wrote landmark works that, the cast of a low-budget science-fiction film, we wander among other things, made the case that public educa- around the authors’ otherworldly sets, poking and prod- tion can play a crucial role in producing individuals who ding at their fantastic creations. We take their concep- are well prepared to make thoughtful and knowledge- tual vehicles out for snappy test drives, we throw fancy able contributions to society, not only as wage-earners, wrenches into their scripts by conducting unauthorized taxpayers, and the heads of families, but as citizens as improvisations, and we try out alternative endings. well. Conceived by Danielle Allen, UPS Foundation We hope that our storytelling will have, in the end, Professor at the Institute, and Professor Rob Reich of some practical application. When it seems that we are Stanford University, the Dewey Seminar was designed losing credibility with our audience, we usually try to to foster research on the complex relationships between CLIFF MOORE bolster our claims about the practical relevance of our Anna Marie Smith, Rosanna and Charles Jaffin Founders’ Circle education, schools, and the state. work by spinning another yarn or two. Member in the School of Social Science, during a meeting of the The Dewey Seminar resembled a three-ring intellec- Sometimes, however, we storytelling philosophers Dewey Seminar, where discussions might center on approaches tual production.