Died: 22 May 2009 in London, England

**Walter Ledermann** was born in Berlin into a Jewish family. His father was William Ledermann and his mother was Charlotte Apt. He entered the Köllnisches Gymnasium in Berlin in 1917, progressing to the Leibniz Gymnasium in the same city in 1920. There he learnt classics, studying Latin for nine years and Greek for six years. The school also taught French but, as was usual at this time, not much science. Although very little mathematics was taught in German schools in general, Walter had the advantage that the Leibniz Gymnasium taught more mathematics than other schools as a mark of respect for Leibniz after whom the school was named. He writes [2]:-

Although I was fond of the classics, especially Greek with its wonderful literature, I was fascinated by mathematics immediately after my first lesson at the age of eleven, and I decided there and then to make mathematics my career.

In 1928, when he was seventeen years old, Ledermann graduated from the Leibniz Gymnasium and received the necessary certificate which allowed him to study at any German university. He remained in Berlin and entered the University there (now the Humboldt University) to study for the State Examination which was the qualification necessary to enter secondary school teaching.

At the University of Berlin Ledermann was taught by many famous mathematicians including Schur, Schmidt, von Mises, Planck, Schrödinger, Hopf, Feigl, and others. As one might imagine given this array of famous names, he found it a stimulating experience. Ledermann was most influenced by Schur and, in [3], he describes attending Schur's lectures:-

Schur was a superb lecturer. His lectures were meticulously prepared... [and] were exceedingly popular. I remember attending his algebra course which was held in a lecture theatre filled with about400students. Sometimes, when I had to be content with a seat at the back of the lecture theatre, I used a pair of opera glasses to get at least a glimpse of the speaker.

His main subjects were mathematics and physics, but he also had to study chemistry to a lower level and to take an oral examination on philosophy. Ledermann spent one semester in 1931 in Marburg but other than this all his courses were taken in Berlin.

On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which exempted non-Aryans who had civil service appointments before the end of World War I. Although Schur qualified under this clause he was initially expelled from his professorship and was only reinstated after the intervention of Schmidt. He was not, however, allowed to lecture but he examined Ledermann before the end of 1933.

Ledermann was nearing the end of his studies for the *Staatsexamen* (which would allow him to teach) when Hitler came to power and began passing the anti-Jewish legislation. In order to complete the course he had to write a dissertation and be given an oral examination. Schur gave him the topic for his dissertation *On the various ways of expressing an orthogonal matrix in terms of parameters* and conducted his oral examination in November 1933. At the oral he was examined by Schur and also by Bieberbach who was wearing Nazi uniform.

It was quite clear to Ledermann that he had to leave Germany to escape the Nazi persecution of the Jews. He had already made many strenuous efforts to find a way to leave Germany before his oral. He won a scholarship from the International Student Service in Geneva to study at the University of St Andrews in Scotland and they supplied him with the necessary papers to allow him to travel to St Andrews in January 1934. Ledermann writes [5]:-

I have a deep affection for St Andrews. For it was the students and the people of St Andrews who saved my life by helping me to escape from the Nazis. It became my home in a real sense.

At first it appeared that he might get caught up in the University of St Andrews' regulations. The problem was that the State Examination was just that, in that it was awarded by the Ministry of Education, not by the University of Berlin. Therefore it was not a degree but a diploma. Fortunately Ledermann was spared the stupidity of having to take the undergraduate courses at St Andrews, and, as Ledermann writes in [2]:-

... for the first time in the five hundred years history of the university a person with a German state examination was admitted as a research student.

Ledermann's doctoral studies were supervised by Turnbull and he was awarded his Ph.D. in 1936. Under Turnbull, Ledermann studied the problem of finding the canonical form for a pair of real or complex *n* × *n* matrices under simultaneous equivalence. He also worked on the problem of classifying the stabiliser of the pencil which is a linear combination of the two matrices.

In 1937 he became a temporary lecturer in Dundee but the most fruitful work he undertook during this period was as a private assistant to Professor Sir Godfrey Thomson at the University of Edinburgh. Thomson headed the Moray House Group at Edinburgh which was undertaking research into intelligence testing. Ledermann was able to use his expert knowledge of matrix theory to put the work of this group onto a sound mathematical footing. As well as matrix theory he was involved in using statistical methods and he retained this interest in his later research publications. The quality of the work Ledermann undertook at this time is clearly shown from the fact that Edinburgh awarded him a D.Sc. for it in 1940. While in Edinburgh Ledermann also worked with Max Born and A C Aitken.

In 1938 Ledermann returned to St Andrews. The letter [6] from Turnbull to Ledermann offers him a one year post as an assistant, capable of renewal, at St Andrews for the academic year 1938-39 at a salary of £350. Turnbull writes [6]:-

It will involve from8to10lectures or tutorials a week in mathematics pure and applied; and the Court understand that I have looked out for one who is competent to teach particularly in Analysis and Applied Mathematics. Your special knowledge of the applications of mathematics to problems involving matrices and statistics is an advantage; but you will realise, Walter, in view of the needs of this Department, that proficiency in teaching Applied Mathematics and the functions connected therewith will be important.

In fact Ledermann remained at St Andrews from 1938 until 1946. During this time he became a British citizen (1940) and also undertook some war work. In [5] he describes his association with Freundlich during this period:-

... Freundlich was very close to me. He was a fatherly friend of whom I have many fond memories, most especially because he introduced me to my wife[Ruth Stefanie(Rushi)Stadler], whom I married in1946. During the war Freundlich and I taught navigation at the Initial Training Wing of the RAF which was stationed in St Andrews. We also published a joint paper in the Monthly Notices of the Royal Astronomical Society in1944... . We had other interests in common apart from mathematics: Freundlich was a keen cellist, and we frequently played chamber music where I played the violin or viola. Once we went on holiday together to the West coast of Scotland, when Mrs Freundlich was unable to come. ... he was a tall impressive man, and when we walked side by side through the streets of St Andrews people would say: "Here come the Sun and Moon".

Ledermann's wife Ruth (known as Rushi) was a psychotherapist and the newly married couple decided to move to a bigger city so that Ruth might be able to pursue her work. Ledermann accepted a lectureship at the University of Manchester in 1946. He writes [4]:-

The sixteen years we spent in Manchester were stimulating and fruitful.

Particularly of note during his time in Manchester was the fact that he was secretary to the first British Mathematical Colloquium which he organised in Manchester at the request of Hodge, Henry Whitehead, and Max Newman.

By the time he left Manchester, Ledermann had been promoted to a Senior lecturer. In 1962 he accepted a readership at the newly established University of Sussex and three years later he was promoted to professor. He retired in 1978 and was made Emeritus Professor. For many years after retiring he continued to teach at Sussex, giving both tutorials and seminars.

As well as work in matrix theory which we have commented on above, Ledermann was especially known for his work in homology theory, group theory, and number theory. As a result of Schur's teaching he [4]:-

... developed a liking for "concrete" mathematics and a distaste for "abstraction" for its own sake.

This is evident, even in his work in what is usually thought of as one of the most abstract of topics, homology theory.

First I [EFR] will make some comments on Ledermann's book *Introduction to the Theory of Finite Groups* (1949). It is the book from which I learnt group theory and, although it was not the only influence on my choice of research topic, it was a major factor in my decision to work on group theory problems for my doctoral dissertation. The topics covered in the book look fairly standard but one has to remember that in the 1940s there were few group theory texts and the concept of standard material for such courses did not exist. The little book (152 pages) discusses the group axioms, isomorphisms, cyclic groups, coset decompositions, Lagrange's theorem, permutation groups, normal subgroups, quotient groups, homomorphisms, the first and second isomorphism theorems, and the Jordan-Hölder theorem. The simplicity of the alternating groups is proved and the Sylow theorems, *p*-groups and finitely generated abelian groups are discussed. Ledermann succeeds admirably in meeting his own aims in that he:-

... never hesitated to sacrifice completeness for breadth or to reject more modern methods when[he]considered alternative presentations to be more intelligible.

Other books which Ledermann has written for undergraduates include *Complex numbers* (1960), *Integral calculus* (1964), *Multiple integrals* (1966), *Introduction to group theory* (1973), and *Introduction to group characters* (1977). This last volume, which still shows Schur's influence, strikes a good balance between the abstract approach to representation theory emphasising modules, and the concrete approach built around matrices. It is an outstanding text from which to teach the topic.

Among the editorial work undertaken by Ledermann is his editorship of the *Journal of the London Mathematical Society* (1968-71) and of the *Bulletin of the London Mathematical Society* (1974-77). He is the chief editor of the *Handbook of Applicable Mathematics* which consists of nine volumes, an index volume and a number of guide books. This project, again very much in line with Ledermann's approach to mathematics, is designed for the "professional adults" who:-

... find themselves needing to understand a particular mathematical idea... will then be able to turn to the appropriate article in the core volume ... and find out just what they want to know.

Ledermann has received honours for his work which include election to the Royal Society of Edinburgh (1944) and an honorary doctorate from the Open University (1993).

His son writes (May 2009):

It is with great sadness that I am writing to let you know that my father passed away peacefully this morning. His energy, enthusiasm and dry humour were with him almost to the end. We shall miss him very much but we all have wonderful memories of his long and fulfilled life.

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

**July 2009**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Ledermann.html]