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Leonard Roth

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29 Aug 1904

Edmonton, London, England

28 Nov 1968

Pittsburgh, Pennsylvania, USA

Leonard Roth was born in Edmonton in the London Borough of Enfield. His father was a merchant in this district in the north east of London. Leonard showed great promise at school, but his school years were not very happy ones. Perhaps it is surprising that mathematics was not his first love at school, and he found himself attracted towards the humanities. Nevertheless his achievements in mathematics made him decide to try for a mathematics scholarship to study at Clare College, Cambridge. He was successful and matriculated at Cambridge in 1923.

He graduated from Cambridge with a First Class degree in 1926. Among his teachers were Pars who taught him dynamics, and Littlewood who taught the foundations of function theory. However Roth had a very low opinion of the examinations set by Cambridge :

He left a witty and moving description of the agony involved in [the Mathematical Tripos] examination in a manuscript which appeared posthumously. Although he probably never intended this to be published, it is nevertheless a delightful and fascinating source of information about mathematical life in Cambridge ... with vivid anecdotes about the great men of the time such as Forsyth and Littlewood .

This quote is intriguing enough that one must look at Roth's experiences as detailed in . The Tripos examinations consisted of six papers for Schedule A, each of 3 hours, sat 9.00 to 12.00 and 13.30 to 16.30 on each of Monday, Tuesday, Wednesday. Schedule B consisted again of six papers in exactly the same pattern on the following week. Roth writes :

The typical Schedule A question was a triple-decker: first the candidate would be asked to prove a theorem; then would come a problem based more or less on the theorem; and thirdly, another problem even less based than the first. In fact, despite all appearances to the contrary, this last might break fresh ground: that was the sting in the tail. Everybody knew that only complete answers to questions really counted, and that postscript mattered more than the rest. Hence a certain general foreboding. A candidate, even a well-prepared one, might go into the examination on the Monday morning and find himself unable to do a single complete question; if, unduly depressed by this failure, he had the same experience on the Monday afternoon, then it was all over save the post-mortem.

After giving an example of a typical triple-decker question, Roth writes:

In questions of this type one might polish off the first two parts in no time at all, only to waste up to an hour on the third. And that way madness lies.

He survived the Mathematical Tripos, obtaining a First Class degree. As he recalled :

Luck certainly played a considerable part in the examination. I myself had some of each sort, though admittedly more good than bad.

He then undertook research under H F Baker gaining great inspiration from Baker 's Principles of geometry. After completing his doctoral studies, he was appointed as a Demonstrator in Imperial College of Science and Technology, London. Soon after arriving there he was awarded a Rockefeller Research Fellowship which enabled him to spend the academic year 1930-31 in Rome. It was an extremely profitable time for Roth, for there he met many of the great Italian mathematicians and he learnt a great deal from Castelnuovo , Enriques , Levi-Civita and Severi . He was greatly influenced by the work of these mathematicians and his future research directions were very much laid down at this time.

Not only did Roth benefit mathematically from his personal contact with the Italian geometers while he was in Rome, but also at that time he met Marcella Baldesi, the only daughter of the reformist socialist Member of Parliament Gino Baldesi, and soon they were married :

[Marcella] was a very lively and intelligent person with a passionate interest in music - she was an excellent pianist - and together they made a most wonderful couple.

After Roth returned to London from Rome in 1931, he was appointed as an Assistant Lecturer in Mathematics at Imperial College. In 1938 he was promoted to Lecturer in Mathematics, in 1946 he was given a Senior Lectureship, and four years later he was promoted to Reader. He remained at Imperial College until 1965 when he went to Pittsburgh in the United States, first as a visiting professor, then from 1967 as Andrew Mellon Professor of Mathematics.

Almost all his work was on geometry where he extended work begun in the Italian school. He therefore worked with classical methods and concepts but applied these to a wide range of problems in different areas. To succeed in this approach he required an intimate knowledge of the vast range of literature that existed, and also an ability to see connections in results on apparently distinct topics. In many ways it can be said that he knew the work of the Italian mathematicians better than they did themselves, partly since he was able to view the whole development from the outside. He also took an approach to geometry which produced large amounts of experimental material and this is highlighted in the review of his most famous book from which we quote below. Also highlighted in this review is the informality of Roth's approach, which again was typical of all his work.

The famous book we referred to is Introduction to Algebraic Geometry which Roth wrote with Jack Semple, and it was published in 1949. Zariski , reviewing the book, wrote:

The ground covered in this book is very extensive: almost every topic of the classical algebro-geometric theory of algebraic curves, surfaces and varieties is discussed, or at least briefly mentioned, in the course of the exposition. Since it is obviously impossible to write a treatise in one volume on the whole of algebraic geometry (even if the transcendental and topological theories are excluded), one would be tempted to conclude a priori that the present book must be something in the nature of an encyclopedia article. This, however, is not the case. While the book is much less than a treatise in which the subject matter is not only presented but also developed step by step with complete rigor, it is also much more than a formal report of results. For one thing, the authors have included in the text a very large number of special but important examples (special curves, surfaces, special transformations), and these examples are discussed in great detail. It is in these examples that the ideas and methods of the general theories are put to work on concrete situations. This wealth of experimental material will be welcome even by the specialist, but it will be really invaluable to the beginner who wishes to acquire a geometric insight and develop a geometric technique. In the second place, the theorems which belong to the general theoretical topics of the book are not merely stated. There is a definite attempt to prove them or at least to justify them to the reader.

Another text written by Roth is Algebraic threefolds, with special regard to problems of rationality (1955). J A Todd in a review of this book writes:

This book gives an account of the birational properties of algebraic threefolds. The emphasis is very much on the algebro-geometric treatment, though for a number of results reference is made to transcendental and topological methods and to theorems of Lefschetz . ...
Chapter VI is concerned with threefolds with infinite groups of birational self-transformations, a topic to which the author has contributed much of what is known.

In addition we should mention Roth's other books: Elements of probability (1936), written with Hyman Levy , and Modern elementary geometry (1948). The obituary lists about 90 papers in addition to the books we have mentioned.

Roth and his wife both died tragically in a car accident in the United States. He was sixty-four and very active mathematically up to the time of his death, so much so that many interesting unpublished results were found among his papers and they were published posthumously. The article was among the unpublished papers found at that time and was almost certainly not intended for publication. It is, however, a joy to read and is filled with Roth's humour.

Source:School of Mathematics and Statistics University of St Andrews, Scotland