Atle Selberg

Atle Selberg
Atle Selberg
Born	14 6, 1917
Birthplace	Langesund, Norway
Died	Template:Death date and age Princeton, New Jersey, United States
Nationality	Norwegian
Occupation	Mathematician
Employer	Institute for Advanced Study
Known for	Selberg sieve, elementary proof of the prime number theorem, Selberg trace formula, Selberg zeta function
Awards	Fields Medal (1950), Wolf Prize in Mathematics (1986), Honorary Abel Prize (2002)
Website	[http://publications.ias.edu/selberg Official site]

Atle Selberg was a Norwegian mathematician whose contributions to analytic number theory, the theory of automorphic forms, and spectral theory placed him among the most significant mathematical figures of the twentieth century. Born on 14 June 1917 in the coastal town of Langesund, Norway, Selberg spent most of his professional career at the Institute for Advanced Study in Princeton, New Jersey, where he worked for more than five decades. He was awarded the Fields Medal in 1950, at the age of 33, for his development of the Selberg sieve and for his work toward an elementary proof of the prime number theorem — a result that had eluded mathematicians for decades. He later received the Wolf Prize in Mathematics in 1986 and an honorary Abel Prize in 2002. Selberg died on 6 August 2007 in Princeton at the age of 90, leaving behind a body of work that reshaped several branches of mathematics and influenced generations of researchers.

Early Life

Atle Selberg was born on 14 June 1917 in Langesund, a small port town in Telemark county on the southeastern coast of Norway. He grew up in an intellectually stimulating household; his family included several individuals with strong mathematical interests. His father, Ole Michael Ludvigsen Selberg, was a high school teacher of mathematics, and two of Atle's brothers, Henrik and Sigmund Selberg, also became mathematicians. This familial environment fostered an early exposure to mathematical thinking and problem-solving.

Selberg developed an interest in mathematics at a young age. As a teenager, he became fascinated by the work of the Indian mathematician Srinivasa Ramanujan, whose papers he encountered in his school library. Ramanujan's work on number theory, particularly on the partition function and related topics, left a deep impression on the young Selberg and influenced the direction of his future research. He later credited Ramanujan's collected works as a formative influence on his mathematical development.

Growing up in Norway during the interwar period, Selberg's early mathematical education was shaped by the Norwegian academic tradition, which had a notable lineage in number theory and analysis through figures such as Niels Henrik Abel and Sophus Lie. Despite the relative isolation of the Norwegian mathematical community from the larger centers of mathematical research in Germany, France, and England, Selberg demonstrated exceptional talent that would soon bring him into contact with the wider international mathematical world.

Education

Selberg pursued his higher education at the University of Oslo, where he studied mathematics. He completed his doctoral work at Oslo, receiving his Ph.D. in 1943 during the German occupation of Norway. The wartime conditions imposed significant hardships on academic life in Norway, and Selberg carried out much of his doctoral research in relative isolation from the broader international mathematical community. Despite these constraints, his thesis work on the Riemann zeta function was of exceptional quality and attracted attention when it became known to mathematicians outside of occupied Norway after the war.

His doctoral research concerned the zeros of the Riemann zeta function, a topic of central importance in analytic number theory. Selberg proved significant results about the distribution of zeros of the zeta function on the critical line, building on earlier work by G. H. Hardy and others. These results were among the first indications of the depth and originality that would characterize his subsequent career.

Career

Early Work in Norway and Postwar Period

During the years of the German occupation of Norway (1940–1945), Selberg continued his mathematical work under difficult circumstances. The isolation imposed by the war meant that he worked largely independently, developing ideas that would later prove to be of fundamental importance. After the liberation of Norway in 1945, Selberg's work began to circulate more widely in the international mathematical community, and his results on the Riemann zeta function and sieve methods drew significant attention.

In the years immediately following the war, Selberg developed what became known as the Selberg sieve, a powerful technique in analytic number theory for estimating the size of sifted sets of integers. The Selberg sieve represented a significant advance over earlier sieve methods, including those of Viggo Brun, and proved to be a versatile and widely applicable tool in number theory. The method was notable for its elegance and its ability to produce sharp upper bounds in a variety of number-theoretic problems.

Move to the Institute for Advanced Study

In 1947, Selberg traveled to the United States, where he spent time at the Institute for Advanced Study in Princeton, New Jersey. He joined the Institute as a permanent member in 1949 and was appointed a professor there in 1951, a position he held until his retirement in 1987. The Institute for Advanced Study, with its tradition of providing scholars with the freedom to pursue research without teaching obligations, proved to be an ideal environment for Selberg's work. He remained affiliated with the Institute for the rest of his life, continuing to work there as a professor emeritus after his formal retirement.

At the Institute, Selberg joined a community of distinguished mathematicians and physicists that included, at various times, Albert Einstein, John von Neumann, Kurt Gödel, and Hermann Weyl. The intellectual atmosphere at Princeton was highly stimulating, and Selberg's presence there contributed to making the Institute a leading center for research in number theory and related areas.

Selberg lived in a midcentury modern home at 35 Stonehouse Drive in Princeton, designed by architect Wynant D. Vanderpool, Jr., and built in 1952.

Elementary Proof of the Prime Number Theorem

Selberg's most famous individual achievement was his contribution to the elementary proof of the prime number theorem. The prime number theorem, which describes the asymptotic distribution of prime numbers among the positive integers, had been proved in 1896 independently by Jacques Hadamard and Charles-Jean de la Vallée Poussin using methods from complex analysis, specifically the theory of functions of a complex variable. For over fifty years, mathematicians had wondered whether it was possible to prove the theorem using only "elementary" methods — that is, methods that did not rely on complex analysis.

In 1948, Selberg established what is now called the Selberg symmetry formula (or Selberg's fundamental formula), an identity involving the logarithms of prime numbers that provided a new approach to the prime number theorem. This formula was a key breakthrough that opened the door to an elementary proof.

The subsequent history of the elementary proof became the subject of a well-known priority dispute between Selberg and the Hungarian mathematician Paul Erdős. Erdős, who was also working on the problem, used Selberg's fundamental formula to make further progress. The two mathematicians initially collaborated, but disagreements arose over how the results should be published and credited. Selberg wished to publish his contribution independently, while Erdős favored a joint publication. The dispute was documented by several mathematicians and historians of mathematics.

Ultimately, Selberg published his elementary proof of the prime number theorem in 1949 in the Annals of Mathematics, while Erdős published a separate paper with his own version of the proof shortly thereafter. The dispute caused a lasting rift between the two mathematicians and became one of the most discussed episodes in the history of twentieth-century mathematics. Despite the controversy, the achievement itself was universally recognized as a landmark in number theory. The elementary proof demonstrated that deep results about the distribution of primes could be obtained without the machinery of complex analysis, a finding that had both philosophical and technical significance for the field.

The Selberg Trace Formula

In the 1950s, Selberg developed what became known as the Selberg trace formula, a result that established a deep connection between the spectral theory of the Laplacian operator on a Riemannian manifold and the geometry of the manifold's closed geodesics. The trace formula relates the eigenvalues of the Laplacian on a compact Riemann surface (or more generally, a locally symmetric space) to geometric data about the surface, specifically the lengths of its closed geodesics.

The Selberg trace formula is considered one of the most important results in twentieth-century mathematics because of its far-reaching connections to multiple areas of the discipline. It can be viewed as a non-commutative generalization of the Poisson summation formula and has profound implications for the theory of automorphic forms, representation theory, and the study of the Riemann zeta function and its generalizations.

The trace formula provided a new way of understanding automorphic forms — functions defined on symmetric spaces that satisfy certain invariance properties — by connecting them to spectral theory. This work brought together ideas from number theory, geometry, and analysis in a manner that was unprecedented and that opened new avenues of research for subsequent generations of mathematicians.

Contributions to Spectral Theory and Automorphic Forms

Selberg's work on the trace formula led him to develop important ideas in the spectral theory of automorphic forms. He formulated the Selberg eigenvalue conjecture, which concerns the eigenvalues of the Laplacian on certain arithmetic surfaces. This conjecture, which remains unproven in its full generality, has been the focus of significant research effort and is connected to deep questions about the arithmetic of automorphic forms.

Selberg also introduced the Selberg zeta function, a zeta function associated with the closed geodesics of a compact Riemann surface. This function is analogous in many respects to the Riemann zeta function, and its study has provided important insights into the connections between geometry, dynamics, and number theory.

His work on automorphic forms and their connection to spectral theory influenced the development of the Langlands program, a vast and influential web of conjectures proposed by Robert Langlands in the late 1960s while Langlands was working at the Institute for Advanced Study. The Langlands program seeks to establish deep connections between number theory, representation theory, and algebraic geometry, and Selberg's trace formula is one of its foundational tools.

The Selberg Integral and Other Contributions

Beyond his work in number theory and automorphic forms, Selberg made contributions to other areas of mathematics. He evaluated a multidimensional generalization of the Euler beta function integral, now known as the Selberg integral. This result, first published in 1944 in a Norwegian mathematics journal, was not widely known for many years but was later rediscovered and recognized as an important result with applications in mathematical physics, combinatorics, and random matrix theory.

Selberg's published output was relatively modest in volume compared to some of his contemporaries, but it was characterized by its exceptional depth and originality. His collected works, published in two volumes, contain relatively few papers, but each one is considered significant. The Los Angeles Times described him as a mathematician with "a golden touch" who left "a profound imprint on the world of mathematics."

Personal Life

Selberg was known for his reserved and private nature. He preferred to work alone and was not given to extensive collaboration, a trait that distinguished him from many of his contemporaries in the mathematical community. His working style stood in contrast to that of mathematicians such as Paul Erdős, who thrived on collaboration and co-authorship.

Selberg lived in Princeton, New Jersey, for most of his adult life, residing in the midcentury modern house on Stonehouse Drive that he had occupied since the early 1950s. He died on 6 August 2007 of a heart attack at his home in Princeton at the age of 90.

The Institute for Advanced Study issued a statement upon his death, noting his extraordinary contributions to mathematics and his long association with the Institute.

Recognition

Selberg received numerous honors and awards throughout his career, reflecting the significance and influence of his mathematical work.

In 1950, he was awarded the Fields Medal at the International Congress of Mathematicians in Cambridge, Massachusetts. The Fields Medal, often described as the highest honor in mathematics, was awarded to Selberg for his work on generalized sieve methods and their applications to the theory of primes, as well as for his contribution to the elementary proof of the prime number theorem. He was one of only two recipients that year, the other being Laurent Schwartz.

In 1986, Selberg received the Wolf Prize in Mathematics, awarded by the Wolf Foundation in Israel. The prize recognized his contributions to number theory, automorphic forms, and related areas of mathematics.

In 2002, when the Abel Prize was established by the Norwegian government as a counterpart to the Nobel Prize for mathematics, Selberg was awarded an honorary Abel Prize in recognition of his lifetime of contributions to the field. The Abel Prize was named after the Norwegian mathematician Niels Henrik Abel, and Selberg's receipt of the inaugural honorary award was a fitting acknowledgment of his standing as Norway's preeminent mathematician of the modern era.

He was also elected to membership in several national academies of science, reflecting the international recognition of his work. His publications continue to be widely cited and studied, and his collected works remain essential references in analytic number theory and the theory of automorphic forms.

Legacy

Selberg's contributions to mathematics have had a lasting impact on several branches of the discipline. The Selberg sieve remains a fundamental tool in analytic number theory and continues to be applied and refined by contemporary researchers. The Selberg trace formula has become one of the central objects in the Langlands program, one of the most ambitious and far-reaching programs in modern mathematics. His work on the elementary proof of the prime number theorem demonstrated that deep results about prime numbers could be obtained through methods that were fundamentally different from those used in the original proofs, inspiring new approaches and perspectives in number theory.

The influence of Selberg's ideas extends beyond pure mathematics. The Selberg integral has found applications in mathematical physics, particularly in random matrix theory and the study of quantum systems. His work on the spectral theory of automorphic forms has connections to quantum chaos and the study of quantum mechanics on spaces of negative curvature.

Selberg supervised relatively few doctoral students during his career at the Institute for Advanced Study, consistent with the Institute's primary focus on research rather than formal graduate education. Nevertheless, his influence on subsequent generations of mathematicians was profound, operating primarily through his published work and through informal interactions with visitors and colleagues at Princeton. Mathematicians who worked on problems inspired by Selberg's ideas include John Friedlander, Henryk Iwaniec, and Peter Sarnak, among many others.

The New York Times, in its obituary, noted that Selberg's "theoretical work on the properties of numbers" had been recognized with the Fields Medal and other major awards, and described him as one of the foremost mathematicians of his era. The Guardian's obituary called him "one of the greatest mathematicians of the 20th century." The Los Angeles Times described him as "one of the last of the 20th century's great mathematicians" who "left a profound imprint on the world of mathematics."

Selberg's legacy is preserved through the continued study and application of his ideas, through his published collected works, and through the mathematical structures — the Selberg sieve, the Selberg trace formula, the Selberg zeta function, the Selberg integral, and the Selberg eigenvalue conjecture — that bear his name and remain active areas of mathematical research.