mathematics

Sigurður Helgason (mathematician)

Sigurður Helgason (mathematician) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sigurður Helgason (mathematician) rather than just read about it. In short: Sigurdur Helgason (Icelandic: Sigurður Helgason; 30 September 1927 – 3 December 2023) was an Icelandic mathematician whose research has been devoted to the geometry and analysis on symmetric spaces. In particular, he 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.

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Reference excerpt

Sigurdur Helgason (Icelandic: Sigurður Helgason; 30 September 1927 – 3 December 2023) was an Icelandic mathematician whose research has been devoted to the geometry and analysis on symmetric spaces. In particular, he 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.

Biography Sigurdur Helgason was born in Akureyri, Iceland on 30 September 1927. In 1954, he earned a PhD from Princeton University under Salomon Bochner. Helgason became a professor of mathematics at the Massachusetts Institute of Technology in 1965, and he retired from the faculty in 2014. Helgason received the Børge Jessen Diploma Award of the Danish Mathematical Society in 1982, and the Grand Knight's Cross (Stórriddarakross) of the Icelandic Order of the Falcon in 1991. 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 31 May 1996, Helgason received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden. Helgason was elected a member of the Icelandic Academy of Sciences in 1960, the American Academy of Arts and Sciences in 1970, and the Royal Danish Academy of Sciences and Letters in 1972. In 2013, he became a fellow of the American Mathematical Society. He was made an honorary member of the Icelandic Mathematical Society when he turned 70, and a symposium was held in his honor. Helgason died on 3 December 2023, at the age of 96.

Selected works

Articles 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, Sigurdur (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. hdl:1721.1/26691. MR 0095428. Helgason, Sigurdur (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, Sigurdur (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. hdl:1721.1/26682. 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. hdl:1721.1/26683. 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. hdl:1721.1/26684. MR 0158408. Helgason, Sigurdur (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. hdl:1721.1/26685. 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. hdl:1721.1/26686. MR 0179295. Helgason, Sigurdur; 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. hdl:1721.1/26687. MR 0229179. Helgason, Sigurdur (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, Sigurdur (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. hdl:1721.1/26688. MR 0312158. Helgason, Sigurdur (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.

Books Differential geometry and symmetric spaces. Academic Press 1962, AMS 2001 Analysis on Lie groups and homogeneous spaces. AMS 1972 Differential geometry, Lie groups and symmetric spaces. Academic Press 1978, 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, AMS 1994 Geometric analysis on symmetric spaces. AMS 1994, 2nd. edn. 2008

References

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

External links Sigurdur Helgason at the Mathematics Genealogy Project Sigurdur Helgason – Publications – MIT Mathematics Gonzalez, Fulton; Ólafsson, Gestur; Anker, Jean-Philippe; Van Den Ban, Erik; Dadok, Jiri; Grinberg, Eric; Koranyi, Adam; Kuchment, Peter; Natterer, Frank; Ørsted, Bent; Orloff, Jeremy; Oshima, Toshio; Quinto, Todd; Rouvière, François; Rubin, Boris; Schlichtkrull, Henrik; Sigurðsson, Ragnar; Stanton, Robert J.; Vogan, David (November 2024). "Remembering Sigurður Helgason (1927–2023)" (PDF). Notices of the American Mathematical Society. 71 (10): 1349–1361. doi:10.1090/noti3049.

Illustrations

Worked examples

Example 1 — a first encounter with Sigurður Helgason (mathematician)

Start with the simplest possible case. Write down what Sigurður Helgason (mathematician) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sigurður Helgason (mathematician) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sigurður Helgason (mathematician) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sigurður Helgason (mathematician)

In research: Sigurður Helgason (mathematician) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sigurður Helgason (mathematician) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sigurður Helgason (mathematician) is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1927 births, 2023 deaths, 20th-century Icelandic scientists, so understanding it makes those chapters shorter.
In everyday life: Look for Sigurður Helgason (mathematician) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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