The Selberg trace formula established a fundamental duality between the lengths of closed geodesics on compact Riemann surfaces and the eigenvalues of the Laplacian operator. This result remains the most influential contribution from Atle Selberg, a Norwegian-American mathematician whose career spanned from the occupation of his home country during the Second World War to the Institute for Advanced Study.
War and Mathematical Isolation
Born in 1917 in Langesund, Selberg completed his doctorate at the University of Oslo in 1943. During the German occupation of Norway, he worked in isolation, having been imprisoned multiple times for his resistance efforts. Despite these conditions, he developed a proof that a positive proportion of the zeros of the Riemann zeta function lie on the critical line. His early interests were influenced by Srinivasa Ramanujan, leading to an analytical formula for the partition function.
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After the war, Selberg moved into sieve theory, introducing a method to provide auxiliary upper bounds. In 1948, he published two papers in the Annals of Mathematics, utilizing elementary methods to prove theorems regarding primes in arithmetic progression and prime density. This work provided the foundation for an elementary proof of the prime number theorem, an achievement that led to a public dispute with Paul Erdős.
Career at Princeton
Invited by C.L. Siegel to the Institute for Advanced Study in 1947, Selberg transitioned to the United States. Following a brief period as an associate professor at Syracuse University, he returned to the Institute in 1949. He maintained his position there until 1987, continuing as a professor emeritus until his death in 2007 in Princeton. Throughout his career, he worked predominantly in solitude, with Sarvadaman Chowla serving as his only coauthor.
Honors and Recognition
Selberg received numerous accolades for his work in number theory and spectral theory. He was awarded the Fields Medal in 1950, followed by the Wolf Prize in Mathematics in 1986 and the Gunnerus Medal in 2002. Additionally, he was appointed a Knight Grand Officer of the Order of Saint Olav in 1987. He held memberships in the American Academy of Arts and Sciences, the Royal Swedish Academy of Sciences, and the Royal Danish Academy of Sciences and Letters.
Fast facts
- Born: 1917, Langesund, Norway
- Died: 2007, Princeton, United States
- Fields Medal: 1950
- Wolf Prize in Mathematics: 1986
- Education: University of Oslo
- Primary Employer: Institute for Advanced Study
- Notable Work: Selberg trace formula
Questions readers ask
What is the Selberg trace formula?
It is a mathematical result that creates a duality between the lengths of closed geodesics on a compact Riemann surface and the eigenvalues of the Laplacian.
Did Selberg work with other mathematicians?
He worked almost exclusively alone; the only mathematician he coauthored with was Sarvadaman Chowla.
Achievements
- Fields medal — 1950
- Gunnerus Medal — 2002
- Knight Grand Officer of the Order of Saint Olav — 1987
- Wolf Prize in Mathematics — 1986
- Notable work: An Elementary Proof of the Prime-Number Theorem
- Notable work: An Elementary Proof of Dirichlet's Theorem About Primes in an Arithmetic Progression
- Notable work: Selberg trace formula
- Notable work: Rankin–Selberg method
- Held posts at Syracuse University, Institute for Advanced Study and Institute for Advanced Study
- Fields: number theory
