# Lars Gårding

### Born: 7 March 1919 in Hedemora, Dalarna, Sweden

Died: 7 July 2014 in Lund, Sweden

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**Lars Gårding**was the son of Jonas Ruben Gårding (1892-1985) and Matilda Elisabet Gahn (1892-1980). Jonas Ruben Gårding was an engineer working at Motala and it was in Motala that Lars grew up. We note that Hedemora is about 200 km north of Motala, Hedemora is about 160 km north west of Stockholm, while Motala is about 200 km south west of Stockholm. Lars had a younger brother Per Fredrik Gårding, born in 1920.

When Lars Gårding began his studies at the University of Lund in 1937 he intended to become an actuary. His course involved mathematics and soon he was influenced by Marcel Riesz to change topics and concentrate on mathematics. Riesz had been appointed to the chair of mathematics at the University of Lund in 1926 and by the time that Gårding studied there Riesz had turned a somewhat mediocre mathematics department into an exciting place of international fame. Gårding undertook research for a Ph.D. supervised by Marcel Riesz and was awarded a Ph.D. in 1944 for his thesis

*On a class of linear transformations connected with group representations*. D E Littlewood, reviewing the published version of the thesis, writes:-

This publication was not the first one that Gårding produced, in fact his first paper wasThe results obtained are not fundamental as are the Dirac matrices and the corresponding spinors, but an efficient technique is developed over a specialized field.

*The distributions of the first and second order moments, the partial correlation coefficients and the multiple correlation coefficient in samples from a normal multivariate population*which appeared in 1941. The Abstract states:-

C C Craig writes in a review:-When the frequency function of a statistical variable is known, one of the most important tasks of Theoretical Statistics is to find the frequency functions of some simple functions of this variable. The most important are the first and second order moments in a sample containing a certain number of values of the variable.

Two other papers appeared before his thesis, namelyThis is a well organized exposition, both concise and clear, of the derivation of the distributions named in the title.

*Conics inscribed in a triangle treated by means of complex numbers*(Swedish) (1942) and

*A general theorem concerning group representations*(1943) which is concerned with a generalization of the familiar Burnside-Schur lemma of representation theory. This 1943 paper on group character theory and the theory of infinitesimal transformations has some applications to quantum theory, relativity and nuclear physics. Indeed Gårding was at the University of Cambridge, England, in 1944 and continuing to study relativistic wave equations when, on 1 December 1944, he submitted the paper

*Relativistic wave equations for zero rest-mass*to the

*Proceedings of the Cambridge Philosophical Society*. The paper was communicated by P A M Dirac and Gårding gives two addresses on the paper, 'The Institute of Mathematics of the University of Lund, Sweden', and 'Wesley House, Cambridge'.

Gårding continued to have the University of Lund as his permanent address but made many other visits. In Lund he worked closely with Marcel Riesz and soon began to produce papers on the theory of partial differential equations. There were other doctoral students working on similar topics at Lund, for example N E Fremberg wrote a thesis

*Some applications of the Riesz potential to the theory of the electromagnetic field and the meson field*in 1946. He writes:-

Gårding was working on similar topics and publishedI want to express to Professor Marcel Riesz my sincerest gratitude for his direction of my attention to these problems, for valuable discussions and kind encouragement, and for critical examination of the manuscript. I am very grateful to ... Dr Lars Gårding for careful examination of the manuscript.

*The Solution of Cauchy's Problem for Two Totally Hyperbolic Linear Differential Equations by Means of Riesz Integrals*which he submitted to the

*Annals of Mathematics*in June 1946. In 1947 Gårding was at Princeton University, in the United States, when he submitted

*Note on continuous representations of Lie groups*.

In fact the Institute for Advanced Study at Princeton was a place where Gårding spent time on eight separate occasions between 1949 and 1977. His first visit was from October 1949 to May 1950. He sailed from Gothenburg, Sweden, to New York, USA, on the ship

*Selma Thorden*, arriving in New York on 8 October 1949. He was accompanied by his wife Eva Gårding, the two having recently married. The immigration details on arriving in the United States contains a statement which we do not understand:-

The word "Mental" is hand-written on the card. Passing over this very strange note we should say a little about Gårding's marriage at this point. He had married Eva Lundius earlier in 1949. Eva Maria Lundius (14 July 1920 - 1 January 2006) was born and grew up in Landskrona in the south of Sweden. She worked as a teacher of modern languages but studied for a Ph.D. in phonetics at Lund University which she was awarded in 1967 for the dissertationThe above mentioned alien is ordered delivered Ellis Island Hospital for further medical examination.

*Internal Juncture*(Swedish). She became a professor of phonetics in Lund University in 1980.

In 1952, after Marcel Riesz retired from his chair, Gårding was appointed as a professor at the University of Lund and, in the following year, he was elected as a member of the Royal Swedish Academy of Sciences. He spent the Winter and Spring semesters of 1957 at the University of Chicago where he taught the course "Cauchy's problem for hyperbolic equations." His third visit to the Institute for Advanced Study was from October 1958 to January 1959. For the following two years (1959-60 and 1960-61) he made similar visits. Edward Nelson writes in his historical note on analytic vectors about Gårding's 1958-59 visit [12]:-

At the University of Lund, Gårding played a major role in the development of the Mathematical Institute. This took a leading role both with his outstanding mathematical contributions and also using his skills as an organiser. For quite a number of years he was the director of the Mathematical Institute and he had this role through the difficult period in the late 1960s when there was student unrest throughout universities in many parts of the world. In particular the role of professors as academic leaders was questioned, particularly their role as head of department. There were tensions at Lund University as there were in many universities, but Gårding had a skill of damping down such unrest so the mathematics department was able to function productively through this period. In fact Gårding continued in his leadership role until 1975.When I did this work, I was a fresh Ph.D. at the Institute for Advanced Study. My wife and I lived in the brand-new Institute housing. On the other side of our apartment wall lived Lars Gårding. He was intrigued by the use of the heat equation to produce analytic vectors and told me, ruefully and quite rightly, that it was a method he should have thought of himself. He invited me to his apartment to explain to him the use of diffusion processes in deriving properties of the heat equation - at that time this technique appeared bizarre, and he wrote a paper(L Gårding, 'Vecteurs analytiques dans les représentations des groupes de Lie'Ⓣ(1960))eliminating probability theory from the proof. Our new apartments were frequently invaded by field mice that had been displaced by the construction. Gårding would balance a soup bowl on a matchstick over bait, so that he could release the mice alive and unharmed.

As a research supervisor, Gårding was skilled in suggesting topics for his students' dissertations that were challenging, but not impossible. Perhaps his most famous student was Lars Hörmander who had begun research for his doctorate at the University of Lund in 1951 advised by Marcel Riesz but, after Riesz retired in 1952, he was advised by Gårding and was awarded a doctorate in 1955 for his thesis

*On the Theory of General Partial Differential Equations*. Perhaps we have insulted several other outstanding mathematicians but suggesting the Hörmander may be Gårding's most famous student. We must mention also Bruce Kellogg who was awarded a Ph.D. by the University of Chicago in 1958 for his thesis

*Hyperbolic Equations with Multiple Characteristics*. Bruce Kellogg published a paper containing results for his thesis, with the same title as the thesis, in which he wrote:-

Another outstanding student was Charles Vidar Håkan Thomée (known as Vidar) who was born in Malmö, Sweden, on 20 August 1933 and studied at the University of Lund, graduating with his doctorate from the University of Stockholm in 1959 with his thesisThe results of this paper were submitted in a thesis to the University of Chicago in partial fulfilment of the requirements for the degree of Doctor of Philosophy. We wish to acknowledge our gratitude to Professor Lars Gårding who suggested this problem and whose encouragement and advice made this paper possible.

*Some results concerning boundary problems for partial differential equations*. He was advised by Gårding and Hörmander. He worked for many years at Chalmers University of Technology and University of Gothenburg. Another student of Gårding's who we should mention is Jan-Erik Ingvar Roos (1935-2017) who was awarded a Licentiate in 1958 advised by Gårding. Roos never submitted a thesis for a doctorate but became a professor at the University of Stockholm. Following his death on 15 December 2017, the Department of Mathematics at the University of Stockholm made an announcement which stated:-

Gårding had a career during which he made impressive research contributions to several different areas of mathematics. He also made important contributions to teaching, both at undergraduate level and with books aimed at students who only had a high school mathematics background. For example he publishedHis influence on the Department's development has been enormous, both in research and through supervision of doctoral students.

*Encounter with mathematics*in 1977 writing in the Preface:-

For a longer extract from the Preface and extracts from reviews of this book see THIS LINK.Trying to make mathematics understandable to the general public is a very difficult task. The writer has to take into account that his reader has very little patience with unfamiliar concepts and intricate logic and this means that large parts of mathematics are out of bounds. When planning this book, I set myself an easier goal. I wrote it for those who already know some mathematics, in particular those who study the subject the first year after high school. Its purpose is to provide a historical, scientific, and cultural frame for the parts of mathematics that meet the beginning student.

In 1988 he published

*Algebra for computer science*co-authored with Torbjörn Tambour. We include this book in a list of some of Gårding's books for which we give some additional information such as extracts from reviews and extracts from Prefaces; see THIS LINK.

After he retired from his professorship at Lund, Gårding made a major contribution to the history of mathematics with the outstanding book

*Mathematics and mathematicians. Mathematics in Sweden before 1950*published in Swedish in 1994 and in English in 1998. Between these two dates he published another important historical work, namely

*Some points of analysis and their history*(1997). Richard Beals writes [2]:-

He also wrote historical articles such asThis lively book is a guided tour through some of the highlights of twentieth-century analysis.

*Why is there no Nobel prize in Mathematics?*(1985),

*History of the mathematics of double refraction*(1989), and

*Niels Henrik Abel and solvable equations*(1994).

It was not only mathematics which interested Gårding. He was very interested in and knowledgeable about art, literature and music and he played both violin and piano. In 1987, he published a book on bird singing and sounds, a result of his interest in bird watching. He also published papers such as

*A Simple Model for the Interplay of Predators, Rodents and Food*(2000), and

*Interactions driving the population cycle of Arctic small rodents*(2005). His interest in more philosophical issues resulted in papers such as

*Models in science and otherwise*(1999),

*A philosophical dialog. Mathematics, life, and death*(2000)

*Von Neumann with the Devil*(2010).

Gårding was involved in several of the International Congress of Mathematicians both as a speaker and as an organiser. He was on the Organising Committee of the International Congress of Mathematicians in Stockholm in 1962. The scientific programme for this Congress was drawn up in close cooperation with the International Mathematical Union, which for this purpose nominated a Consultative Committee which included Gårding as a member. He was an invited speaker at the International Congress of Mathematicians held in Edinburgh, Scotland, in 1958 when he gave the lecture

*Some Trends and Problems in Linear Partial Differential Equations*. Again in 1970 he was an invited speaker at the International Congress of Mathematicians held in Nice delivering the lecture

*Sharp fronts and lacunas*.

Many mathematicians have a prize named after them but this is usually set up after their death. The Eva and Lars Gårding Prize, however, was set up by Eva and Lars Gårding during their lifetimes. It:-

Eva Gårding died on 1 January 2006 and because of the huge support the Gårdings gave each other in their academic studies, despite the very different fields in which they worked, we choose to end this biography of Lars Gårding with a quote from [5] concerning Eva Gårding:-... is intended for the promotion of research and for rewarding scientific excellence in the fields of Mathematics and Linguistics. It is awarded for Mathematics during odd years and for Linguistics during even years. The prize was first awarded in2003, and in2014the prize was worth220,000kr.

Eva Gårding had great personal splendour. She had a special ability to inspire students and get them irresistibly involved in scientific thinking. She was a very good teacher as well as a very good writer. For her, a scientific report was not only a work of science, but equally importantly, a work of art. This is apparent from her own scientific writings, where precision, clarity and elegance are characteristic features. She was also a very sharp critic. Her perspicacity allowed her to immediately discover flaws in reasoning and weak points in an argument, and she was wise enough not to be overly consenting and complying. Eva Gårding was also a very good friend. Her house was always open to phonetics students, colleagues, guest researchers and scientists visiting from all parts of the world.

**Article by:** *J J O'Connor* and *E F Robertson*

**List of References** (13 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

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