Lars Hörmander
Encyclopedia
Lars Valter Hörmander is a Swedish
mathematician
who has been called "the foremost contributor to the modern theory of linear partial differential equations". 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 completed his Ph.D.
in 1955 at Lund University
. Hörmander then worked at Stockholm University
, at Stanford University
, and at the Institute for Advanced Study
in Princeton, New Jersey
. He returned to Lund University as professor from 1968 until 1996, when he retired with the title of professor emeritus.
He was awarded the Leroy P. Steele Prize in 2006.
, a village in Blekinge
in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he attended the realskola (elementary school), 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
.
, 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.
In the theory of linear differential operators, "many people have contributed but the deepest and most significant results are due to Hörmander", according to Hörmander's doctoral advisor, Lars Gårding.
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.
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 was made a member of the Royal Swedish Academy of Sciences
in 1968.
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".
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" over the past 20 years, and is considered the "standard of the field". 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 equation
s in Nationalencyklopedin
.
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who has been called "the foremost contributor to the modern theory of linear partial differential equations". He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
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 completed his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in 1955 at Lund University
Lund University
Lund University , located in the city of Lund in the province of Scania, Sweden, is one of northern Europe's most prestigious universities and one of Scandinavia's largest institutions for education and research, frequently ranked among the world's top 100 universities...
. Hörmander then worked at Stockholm University
Stockholm University
Stockholm University is a state university in Stockholm, Sweden. It has over 28,000 students at four faculties, making it one of the largest universities in Scandinavia. The institution is also frequently regarded as one of the top 100 universities in the world...
, at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
, and at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
. He returned to Lund University as professor from 1968 until 1996, when he retired with the title of professor emeritus.
He was awarded the Leroy P. Steele Prize in 2006.
Education
Hörmander was born in MjällbyMjällby
Mjällby is a locality situated in Sölvesborg Municipality, Blekinge County, Sweden with 1,272 inhabitants in 2005. It is located on the Listerlandet peninsula....
, a village in Blekinge
Blekinge
' is one of the traditional provinces of Sweden , situated in the south of the country. It borders Småland, Scania and the Baltic Sea.The name "Blekinge" comes from the adjective bleke, which corresponds to the nautical term for "dead calm"....
in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he attended the realskola (elementary school), in a nearby town to which he commuted by train, and the gymnasium
Gymnasium (school)
A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
(high school) in Lund
Lund
-Main sights:During the 12th and 13th centuries, when the town was the seat of the archbishop, many churches and monasteries were built. At its peak, Lund had 27 churches, but most of them were demolished as result of the Reformation in 1536. Several medieval buildings remain, including 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
Docent
Docent is a title at some European universities to denote a specific academic appointment within a set structure of academic ranks below professor . Docent is also used at some universities generically for a person who has the right to teach...
at Lund University
Lund University
Lund University , located in the city of Lund in the province of Scania, Sweden, is one of northern Europe's most prestigious universities and one of Scandinavia's largest institutions for education and research, frequently ranked among the world's top 100 universities...
who encouraged him to study university level mathematics.
After proceeding to receive a Master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
from Lund university in 1950, Hörmander began his graduate studies under Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...
(who had also been the advisor for Hörmander's gymnasium teacher). He made his first research attempts in classical function theory and harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
, 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
Lars Gårding
Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
who worked actively in that area was appointed professor.
Hörmander took a one-year break for military service
Military service
Military service, in its simplest sense, is service by an individual or group in an army or other militia, whether as a chosen job or as a result of an involuntary draft . Some nations require a specific amount of military service from every citizen...
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.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
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
Lars Gårding
Lars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
and Jean Leray
Jean Leray
Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
.
Fields Medal and years in the U.S.
Hörmander applied for professorship at Stockholm UniversityStockholm University
Stockholm University is a state university in Stockholm, Sweden. It has over 28,000 students at four faculties, making it one of the largest universities in Scandinavia. The institution is also frequently regarded as one of the top 100 universities in the world...
, but temporarily left for the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
while the request was examined. He spent quarters from winter to fall in respective order at the University of Chicago
University of Chicago
The University of Chicago is a private research university in Chicago, Illinois, USA. It was founded by the American Baptist Education Society with a donation from oil magnate and philanthropist John D. Rockefeller and incorporated in 1890...
, the University of Kansas
University of Kansas
The University of Kansas is a public research university and the largest university in the state of Kansas. KU campuses are located in Lawrence, Wichita, Overland Park, and Kansas City, Kansas with the main campus being located in Lawrence on Mount Oread, the highest point in Lawrence. The...
, the University of Minnesota
University of Minnesota
The University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
, and finally at the Courant Institute of Mathematical Sciences
Courant Institute of Mathematical Sciences
The Courant Institute of Mathematical Sciences is an independent division of New York University under the Faculty of Arts & Science that serves as a center for research and advanced training in computer science and mathematics...
in New York
New York
New York is a state in the Northeastern region of the United States. It is the nation's third most populous state. New York is bordered by New Jersey and Pennsylvania to the south, and by Connecticut, Massachusetts and Vermont to the east...
. These locations offered "much to learn" in partial differential equations, with the exception of Chicago of which he however notes the Antoni Zygmund
Antoni Zygmund
Antoni Zygmund was a Polish-born American mathematician.-Life:Born in Warsaw, Zygmund obtained his PhD from Warsaw University and became a professor at Stefan Batory University at Wilno...
seminar held by Elias Stein and Guido Weiss to have strengthened his familiarity with harmonic analysis.
In the theory of linear differential operators, "many people have contributed but the deepest and most significant results are due to Hörmander", according to Hörmander's doctoral advisor, Lars Gårding.
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 UniversityLund University
Lund University , located in the city of Lund in the province of Scania, Sweden, is one of northern Europe's most prestigious universities and one of Scandinavia's largest institutions for education and research, frequently ranked among the world's top 100 universities...
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
Microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations...
was held. He also visited Stanford in 1971, 1977 and 1982, and the University of California, San Diego
University of California, San Diego
The University of California, San Diego, commonly known as UCSD or UC San Diego, is a public research university located in the La Jolla neighborhood of San Diego, California, United States...
in the winter 1990. He was briefly director of the Mittag-Leffler Institute
Mittag-Leffler Institute
The Mittag-Leffler Institute is a mathematical research institute located in Djursholm, a suburb of Stockholm. It invites scholars to participate in year-long or half-year programs in specialized mathematical subjects...
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
International Mathematical Union
The International Mathematical Union is an international non-governmental organisation devoted to international cooperation in the field of mathematics across the world. It is a member of the International Council for Science and supports the International Congress of Mathematicians...
between 1987 and 1990. Hörmander retired emeritus in Lund in January 1996.
He was made a member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...
in 1968.
He received the 1988 Wolf Prize "for fundamental work in modern analysis, in particular, the application of pseudo differential
Pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....
and Fourier integral operators to linear partial differential equations".
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-VerlagSpringer Science+Business Media
- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
in 1963 as part of the Grundlehren series.
Hörmander devoted five years to compiling the four-volume monograph
Monograph
A monograph is a work of writing upon a single subject, usually by a single author.It is often a scholarly essay or learned treatise, and may be released in the manner of a book or journal article. It is by definition a single document that forms a complete text in itself...
, 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" over the past 20 years, and is considered the "standard of the field". In addition to these works, he has written a recognised introduction to multivariate complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
based on his 1964 Stanford lectures, and wrote the entries on differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s in Nationalencyklopedin
Nationalencyklopedin
Nationalencyklopedin is the most comprehensive contemporary Swedish language encyclopedia, initiated by a favourable loan from the Government of Sweden of 17 million Swedish kronor in 1980, which was repaid by December 1990...
.
- Linear Partial Differential Operators (Springer-VerlagSpringer Science+Business Media- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
, 1963) - The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (Springer-Verlag, ISBN 978-3-540-00662-6)
- The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients (Springer-Verlag, ISBN 978-3-540-22516-4)
- The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators (Springer-Verlag, ISBN 978-3-540-49937-4)
- The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators (Springer-Verlag, ISBN 978-3-642-00117-8)
- An Introduction to Complex Analysis in Several Variables. (1966, revisions in 1973 and 1990)
- Notions of Convexity (Birkhäuser VerlagBirkhäuser VerlagBirkhäuser Verlag is the leading European publisher for architecture, landscaping and design. Since 2010 it is part of the ActarBirkhäuser Group.-Scientific publishing:Founded in 1879 by Emil Birkhäuser the publishers program focused on regional literature...
, 1994, ISBN 0817637990)