# Atle Selberg 1917–2007

## Press Contact

Renowned Norwegian mathematician** Atle Selberg**, Professor Emeritus in the School of Mathematics at the Institute for Advanced Study, died on the evening of August 6 at his home in Princeton, NJ. He was 90.

Throughout a career spanning more than six decades, Professor Selberg made significant contributions to modular forms, Riemann and other zeta functions, analytic number theory, sieve methods, discrete groups, and trace formula. The impact of his work is evident from the many mathematical terms that bear his name: The Selberg Trace Formula, The Selberg Sieve, The Selberg Integral, The Selberg Class, The Rankin-Selberg L-Function, The Selberg Eigenvalue Conjecture, and The Selberg Zeta Function.

"Atle's passing marks a great loss, both to the Institute and to the larger scientific community," commented Peter Goddard, Director of the Institute. "His far-reaching contributions have left a profound imprint on the world of mathematics, and we have lost not only a mathematical giant, but a dear friend."

Peter Sarnak, Professor in the School of Mathematics, noted, "The 20th century was blessed with a number of very talented mathematicians, and of those, there are a few who I would say had a golden touch. In any topic about which they thought in depth, they saw further and uncovered much more -- seemingly effortlessly -- than the generations before them. Their work set the stage for many future developments. Atle was one such mathematician; he was a mathematician's mathematician."

Widely regarded as one of the world's greatest analytic number theorists, Selberg first came to the Institute for Advanced Study from Norway in 1947 at the invitation of Carl Ludwig Siegel, who noted that, at 31 years of age, Selberg "already had earned his place in the history of science in the 20th century." After a year at the Institute, Selberg took a post as Associate Professor at Syracuse University, returning to the Institute in 1949 as a permanent Member. In 1951, he was appointed Professor in the Institute's School of Mathematics, and he was named Professor Emeritus in 1987.

During the 1940s, his work centered around the theory of the Riemann Zeta Function and related problems concerning the distribution of prime numbers. The celebrated Riemann Hypothesis states that all the "non-trivial" zeros of The Riemann Zeta Function lie on the line in the complex plane consisting of numbers of the form ½ + *it*, where *t *real is a real number. This central problem remains unsolved to this day. Developing fundamental, new techniques, Selberg showed that a positive proportion of these infinitely many zeros lie on this line. These ideas led him to his powerful and novel sieving methods and in 1948 to his celebrated Selberg Formula and to the elementary proof of the Prime Number Theorem. The last took the mathematical community by surprise as such a proof had been sought since the formulation of the problem by Legendre and Gauss some 150 years before. For these works, Selberg was awarded the prestigious Fields Medal in 1950.

In the early 1950s, Selberg turned his attention to the spectral theory of automorphic forms. His 1956 paper in the *Journal of the Indian Mathematical Society* introduced, among other things, what is known today as The Selberg Trace Formula. According to the Professor Sarnak, "This is one of the most influential mathematical papers of the 20th century. It lays the foundations and many of the tools on which the modern theory of automorphic forms, with its many spectacular applications, rests." His work in automorphic forms led him in 1960 to the discovery of an unexpected phenomenon of the rigidity of lattices in higher rank Lie groups. This phenomenon was developed much further by a number of mathematicians and it is a central theme in modern geometry and group theory. Selberg continued to lecture, elaborate, and develop new aspects of the many topics that he pioneered until well into his 80s.

In 1987, nearly one hundred mathematicians from all over the world convened in Oslo, Norway, for a symposium in honor of Selberg's 70th birthday. In the preface to the collection of the 29 papers presented at the symposium and published by Academic Press in 1989, fellow mathematician Karl Egil Aubert extolled Selberg's "many-sided achievements [that] place him squarely as one of the truly great mathematicians of the 20th century."

In his more than five decades at the Institute, Selberg maintained an understated view on his highly significant accomplishments in the field. In 1990, he noted, "I think the things I have done...although sometimes there were technical details, and sometimes even a lot of calculation, in some of my early work...the basic ideas were rather simple always, and could be explained in rather simple terms...in some ways, I probably have a rather simplistic mind, so that these are the only kind of ideas I can work with. I don't think that other people have had grave difficulties understanding my work."

Enrico Bombieri, IBM von Neumann Professor in the School of Mathematics at the Institute, has described the hallmark of Selberg's style as "simplicity and elegance of method, [and] powerful results. He had an uncanny ability to see immediately what was at the core of an issue. This ability was by no means restricted to scientific matters."

In celebration of Selberg's 90^{th} birthday in June 2007, the Institute invited his close colleagues and friends to salute his lifetime of achievement. Amongst those who spoke at this event was School of Mathematics Member Nils Baas, who conveyed the congratulations of the Norwegian government and proposed a toast to "Atle Selberg - a great Norwegian." Selberg himself spoke animatedly and at length at the event, and noted of the Institute's early days, "The whole complement of people in the Institute was very small. By and large, everybody knew everybody. Even I knew everybody."

Selberg, who was born on June 14, 1917, in Langesund, Norway, was the youngest of nine children of Anna Kristina Selberg, a teacher, and Ole Michael Selberg, an educator and mathematician. His siblings became teachers and academics, including brothers Henrik and Sigmund, mathematicians who were both members of the Norwegian Academy of Sciences and Letters; Henrik was a Member at the Institute for Advanced Study in 1963-64. Selberg's childhood and youth were spent in Norway, in Voss, Bergen, and Gjøvik. At the age of 13, he began to study mathematics using his father's extensive library, where he discovered Leibnitz's series for π/*4* =* *1 - 1/3 + 1/5 - 1/7..., later describing it as "such a very strange and beautiful relationship that I determined I would read that book in order to find out how this formula came about."

In 1934, Selberg came upon a copy of the collected works of Indian mathematician Srinivasa Ramanujan, which an older brother had brought home with him from school. When he attended the Ramanujan Centenary Conference at the Tata Institute of Fundamental Research in 1988, Selberg acknowledged this as a transformative moment in his life. At age 17, Selberg wrote his first article, "On Some Arithmetical Identities." The next year, he began his education at the University of Oslo, where he submitted the paper for review to one of his professors. A year later, the article was published.

By the time Selberg obtained his Ph.D. in 1943, also at the University of Oslo, he had published eleven more articles, the later ones focusing on Riemann's Zeta Function. His paper on The Selberg Integral dates from this period and it is his only paper in Norwegian; it took more than thirty years to be recognized for its importance. He defended his dissertation in November of 1943, shortly before the German occupying forces closed down the University for the duration of the war. He had been appointed a research fellow at the University of Oslo in 1942, the year before he received his doctorate. Selberg remained in this post until 1947, when he married Hedvig Liebermann of Tirgu Mures, Transylvania, and moved to the United States. During the Second World War he worked in isolation due to the occupation of Norway by the Nazis, but after the war, his accomplishments in the theory of the Riemann Zeta Function became known.

In addition to the 1950 Fields Medal, Selberg's contributions to the field of mathematics have been widely recognized, including an honorary doctorate from the University of Trondheim (1972) and the Wolf Prize in Mathematics (1986), which is bestowed annually for outstanding achievements in agriculture, chemistry, mathematics, medicine, physics, and the arts. He was inducted into the Royal Norwegian Academy of Sciences and Letters, the Royal Danish Academy of Sciences and Letters, the Royal Swedish Academy of Sciences, the American Academy of Arts and Sciences, the Indian National Science Academy, and was named an honorary member of the London Mathematical Society. In 1987, Selberg was named a Knight Commander with Star of the Royal Order of Saint Olav.

The publication of the collected papers of Atle Selberg in two volumes (1989, 1991, Springer) was warmly welcomed by the mathematical community for Selberg's profound influence on mathematics, especially analytic number theory. The publication made available his papers up to 1947, which had previously appeared mostly in Norwegian series or journals of limited distribution.

Selberg's first wife, Hedvig, worked at the Institute for Advanced Study in the 1950s in the group headed by John von Neumann, and later at the Princeton Plasma Physics Laboratory until the 1980s; she died in 1995. He is survived by his second wife Betty Compton Selberg of Princeton; his two children from his first marriage, daughter Ingrid Maria Selberg and son-in-law Mustapha Matura of London, and son Lars Atle Selberg and daughter-in-law Julia Selberg of Middlefield, Connecticut; his two stepdaughters Heidi Faith of Mountain View, California, and Cindy Faith of Roland Park, Maryland; and his grandchildren Cayal Mathura, Maya Kristina Mathura, Atle Michael Selberg, and Katharine Rowley Selberg.

Details about a memorial will be made available at a later date.

*In lieu of flowers, donations can be made to:*

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