# Sigurður Helgason (mathematician)

**Sigurður Helgason** (born 1927) is an Icelandic mathematician whose research has been devoted to the geometry and analysis on symmetric spaces. In particular he has used new integral geometric methods to establish fundamental existence theorems for differential equations on symmetric spaces as well as some new results on the representations of their isometry groups. He also introduced a Fourier transform on these spaces and proved the principal theorems for this
transform, the inversion formula, the Plancherel theorem and the analog of the Paley–Wiener theorem.

Sigurður Helgason | |
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Sigurður Helgason | |

Born | 1927 Akureyri, Iceland | (age 93)

Occupation | Mathematician |

Awards | Leroy P. Steele Prize (1988) |

He was born in Akureyri, Iceland. In 1954 he earned a PhD from Princeton University under Salomon Bochner. Since 1965, Helgason has been a professor of mathematics at the Massachusetts Institute of Technology.

He was winner of the 1988 Leroy P. Steele Prize for Seminal Contributions for his books *Groups and Geometric Analysis* and *Differential Geometry, Lie Groups and Symmetric Spaces*. This was followed by the 2008 book *Geometric Analysis on Symmetric Spaces*. On May 31, 1996 Helgason received an honorary doctorate from the Faculty of
Science and Technology at Uppsala University, Sweden. ^{[1]}

He has been a fellow of the American Academy of Arts and Sciences since 1970. In 2012 he became a fellow of the American Mathematical Society.^{[2]}

## Selected worksEdit

### ArticlesEdit

- Helgason, S. (1954). "The derived algebra of a Banach algebra".
*Proceedings of the National Academy of Sciences of the United States of America*.**40**(10): 994–995. Bibcode:1954PNAS...40..994H. doi:10.1073/pnas.40.10.994. PMC 534208. PMID 16589593. - Helgason, Sigurður (1957). "Topologies of group algebras and a theorem of Littlewood".
*Transactions of the American Mathematical Society*.**86**(2): 269–283. doi:10.1090/S0002-9947-1957-0095428-5. MR 0095428. - Helgason, Sigurður (1958). "Lacunary Fourier series on noncommutative groups".
*Proceedings of the American Mathematical Society*.**9**(5): 782–790. doi:10.1090/S0002-9939-1958-0100234-5. MR 0100234. - Helgason, Sigurður (1958). "On Riemannian curvature of homogeneous spaces".
*Proceedings of the American Mathematical Society*.**9**(6): 831–838. doi:10.1090/S0002-9939-1958-0108811-2. MR 0108811. - Helgason, S. (1962). "Some results on invariant theory".
*Bulletin of the American Mathematical Society*.**68**(4): 367–371. doi:10.1090/S0002-9904-1962-10812-X. MR 0166303. - Helgason, S. (1963). "Fundamental solutions to invariant differential operators on symmetric spaces".
*Bulletin of the American Mathematical Society*.**69**(6): 778–781. doi:10.1090/S0002-9904-1963-11029-0. MR 0156919. - Helgason, S. (1963). "Duality and Radon transforms for symmetric spaces".
*Bulletin of the American Mathematical Society*.**69**(6): 782–7881. doi:10.1090/S0002-9904-1963-11030-7. MR 0158408. - Helgason, Sigurður (1964). "A duality in integral geometry; some generalizations of the Radon transform".
*Bulletin of the American Mathematical Society*.**70**(4): 435–446. doi:10.1090/S0002-9904-1964-11147-2. MR 0166795. - Helgason, S. (1965). "Radon–Fourier transforms on symmetric spaces and related group representations".
*Bulletin of the American Mathematical Society*.**71**(5): 757–763. doi:10.1090/S0002-9904-1965-11380-5. MR 0179295. - Helgason, Sigurður; Korányi, Ádám (1968). "A Fatou-type theorem for harmonic functions on symmetric spaces".
*Bulletin of the American Mathematical Society*.**74**(2): 258–263. doi:10.1090/S0002-9904-1968-11912-3. MR 0229179. - Helgason, Sigurður (1969). "Applications of the Radon transform to representations of semisimple Lie groups".
*Proceedings of the National Academy of Sciences of the United States of America*.**63**(3): 643–647. Bibcode:1969PNAS...63..643H. doi:10.1073/pnas.63.3.643. PMC 223499. PMID 16591772. - Helgason, Sigurður (1973). "Paley-Wiener theorems and surjectivity of invariant differential operators on symmetric spaces and Lie groups".
*Bulletin of the American Mathematical Society*.**79**(1): 129–132. doi:10.1090/S0002-9904-1973-13127-1. MR 0312158. - Helgason, Sigurður (1977). "Invariant differential equations and homogeneous manifolds" (PDF).
*Bulletin of the American Mathematical Society*.**83**(5): 751–774. doi:10.1090/S0002-9904-1977-14317-6. MR 0445235.

### BooksEdit

*Differential geometry and symmetric spaces*. Academic Press 1962,^{[3]}AMS 2001*Analysis on Lie groups and homogeneous spaces*. AMS 1972*Differential geometry, Lie groups and symmetric spaces*. Academic Press 1978,^{[4]}7th edn. 1995*The Radon Transform*. Birkhäuser, 1980, 2nd edn. 1999*Topics in harmonic analysis on homogeneous spaces*. Birkhäuser 1981*Groups and geometric analysis: integral geometry, invariant differential operators and spherical functions*. Academic Press 1984,^{[5]}AMS 1994*Geometric analysis on symmetric spaces*. AMS 1994,^{[6]}2nd. edn. 2008

## ReferencesEdit

**^**http://www.uu.se/en/about-uu/traditions/prizes/honorary-doctorates/**^**List of Fellows of the American Mathematical Society, retrieved 2013-01-21.**^**Auslander, Louis (1964). "Review:*Differential geometry and symmetric spaces*, by S. Helgason".*Bull. Amer. Math. Soc*.**70**(2): 227–229. doi:10.1090/S0002-9904-1964-11091-0.**^**Kulkarni, Ravi S. (1980). "Review:*Differential geometry, Lie groups and symmetric spaces*, by S. Helgason".*Bull. Amer. Math. Soc. (N.S.)*.**2**(3): 468–476. doi:10.1090/S0273-0979-1980-14772-2.**^**Howe, Roger (1989). "*Groups and geometric analysis. Integral geometry, invariant differential operators and spherical functions*, by S. Helgason".*Bull. Amer. Math. Soc. (N.S.)*.**20**(2): 252–256. doi:10.1090/S0273-0979-1989-15786-8.**^**Rouvière, François (1995). "*Geometric analysis on symmetric spaces*, by S. Helgason".*Bull. Amer. Math. Soc. (N.S.)*.**32**(4): 441–446. doi:10.1090/S0273-0979-1995-00602-6.

## SourcesEdit

- "Curriculum vitae". Massachusetts Institute of Technology. 2005.
- "Sigurdur Helgason". Massachusetts Institute of Technology.