Lars Hörmander

File:Lars Hörmander 1988.jpg

Lars Hörmander, c. 1988

Lars Valter Hörmander (born January 24, 1931) is a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations"^[1]. He was awarded the Fields Medal in 1962, and the Wolf Prize in 1988. His Analysis of Linear Partial Differential Operators I - IV is considered a standard work on the subject of linear partial differential operators. Hörmander is associated with Lund University, where he completed his Ph.D. in 1955 and was professor from 1968 until his retirement emeritus in 1996, but he spent some of his most productive years at Stockholm University and various universities in the United States, including Stanford University and the Institute for Advanced Study.

Biography

Education

Hörmander was born in Mjällby, a village in Blekinge in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he attended the local elementary school, realskola in a nearby town to which he commuted by train, and the gymnasium (high school) in Lund from which he graduated in 1948.

At the time when he entered the gymnasium, the principal had instituted an experiment of reducing the period of the education from three to two years, and the daily activities to three hours. This freedom to work on his own, "[greater] than the universities offer in Sweden today", suited Hörmander "very well". He was also positively influenced by his enthusiastic mathematics teacher, a docent at Lund University who encouraged him to study university level mathematics.

After proceeding to receive a Master's degree from Lund university in 1950, Hörmander began his graduate studies under Marcel Riesz (who had also been the advisor for Hörmander's gymnasium teacher). He made his first research attempts in classical function theory and harmonic analysis, which "did not amount to much" but were "an excellent preparation for working in the theory of partial differential equations." He turned to partial differential equations when Riesz retired and Lars Gårding who worked actively in that area was appointed professor.

Hörmander took a one-year break for military service from 1953 to 1954, but due to his position in defense research was able to proceed with his studies even during that time. His Ph.D. thesis On the theory of general partial differential operators was finished in 1955, inspired by the nearly concurrent Ph.D. work of Bernard Malgrange and techniques for hyperbolic differential operators developed by Lars Gårding and Jean Leray.

Fields medal and years in the U.S.

Hörmander applied for professorship at Stockholm University, but temporarily left for the United States while the request was examined. He spent quarters from winter to fall in respective order at the University of Chicago, the University of Kansas, the University of Minnesota, and finally at the Courant Institute of Mathematical Sciences in New York. These locations offered "much to learn" in partial differential equations, with the exception of Chicago of which he however notes the Antoni Zygmund seminar held by Elias Stein and Guido Weiss to have strengthened his familiarity with harmonic analysis.

Full professorship at Stockholm University was granted Hörmander in 1957. He again visited the United States in 1960 and 1961, then as a member of the Institute for Advanced Study and in the summers as a lecturer at Stanford University. He wrote the major parts of his first book on partial differential equations while at Stanford, and completed it in 1962 in Stockholm. The same year, Hörmander was "as a complete surprise" awarded the Fields medal at the International Congress of Mathematicians which he helped organise in Stockholm.

Regarding Hörmander's part in developing the general theory of linear differential operators, Lars Gårding writes that "many people have contributed but the deepest and most significant results are due to Hörmander".^[2]

Hörmander was given a position as part time professor at Stanford in 1963, but was soon thereafter offered full professorship at the Institute for Advance Study. He first wished not to leave Sweden, but attempts to find a research professorship in Sweden failed and "the opportunity to do research full time in a mathematically very active environment was hard to resist", so he accepted the offer and resigned from both Stanford and Stockholm and began at the Institute in the fall of 1964. Within two years of "hard work", he felt that the environment at the Institute was too demanding, and in 1967 decided to return to Lund. Incidentally, he notes that his best work at the Institute was done during the remaining year.

Later years

Hörmander mostly remained at Lund University as professor after 1968, but made several visits to the United States during the two next decades. He visited the Courant Institute in 1970, and also the Institute for Advanced Study in 1971 and during the academic year 1977-1978 when a special year in microlocal analysis was held. He also visited Stanford in 1971, 1977 and 1982, and the University of California, San Diego in the winter 1990. He was briefly director of the Mittag-Leffler Institute in Stockholm between 1984 and 1986, but only accepted a two year appointment as he "suspected that the administrative duties would not agree well" with him, and found that "the hunch was right". He also served as vice president of the International Mathematical Union between 1987 and 1990. Hörmander retired emeritus in Lund in January 1996.

He received the 1988 Wolf Prize "for fundamental work in modern analysis, in particular, the application of pseudo differential and Fourier integral operators to linear partial differential equations"^[3].

Important publications

His book Linear Partial Differential Operators, which largely was the cause for his Fields medal, has been described as "the first major account of this theory". It was published by Springer-Verlag in 1963 as part of the Grundlehren series.

Hörmander devoted five years to compiling the four-volume monograph, The Analysis of Linear Partial Differential Operators, first published between 1983 and 1985. A follow-up of his Linear Partial Differential Operators, it "illustrate[d] the vast expansion of the subject" ^[4] over the past 20 years, and is considered the "standard of the field"^[5]. In addition to these works, he has written a recognised introduction to multivariate complex analysis based on his 1964 Stanford lectures, and wrote the entries on differential equations in Nationalencyklopedin.

• Linear Partial Differential Operators (Springer-Verlag, 1963)
• The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (Springer-Verlag)
• The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients (Springer-Verlag)
• The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators (Springer-Verlag)
• The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators (Springer-Verlag)
• An Introduction to Complex Analysis in Several Variables. (1966, revisions in 1973 and 1990)
• Notions of Convexity (Birkhäuser, 1994, ISBN 0817637990)

References

• Hörmander, Lars. Autobiography. Fields Medallists' Lectures. M. Sir Atiyah & D. Iagolnitzer (editors). World Scientific. ISBN 9812382593
• J. J. O'Connor & E. F. Robertson. Lars Hörmander. MacTutor archive biography. Retrieved September 20, 2005
• ^ Wolf Foundation. The 1988 Wolf Foundation Prize In Mathematics. Retrieved September 20, 2005. [6]
• ^ L. Garding. Hörmander's work on linear differential operators. Proceedings of the International Congress of Mathematicians. Stockholm, 1962 (Stockholm, 1963). As quoted by O'Connor & Robertson.
• ^ Wolf Foundation.
• ^ Unknown. "About the Author". Amazon.com entry for The Analysis of Linear Partial Differential Operators I. Retrieved September 20, 2005 [7]
• ^ Wolf Foundation.

External links

• O'Connor, John J.; Robertson, Edmund F., "Lars Hörmander", MacTutor History of Mathematics Archive, University of St Andrews
• Lars Hörmander at the Mathematics Genealogy Project