# Claude Ambrose Rogers

### Quick Info

Born
1 November 1920
Cambridge, England
Died
5 December 2005
Islington, London, England

Summary
Ambrose Rogers wrote extensively on Number Theory and on Sphere-packing problems.

### Biography

Ambrose Rogers' father, Leonard Rogers (1868-1962), was a distinguished doctor who served in the Indian Medical Service. Leonard Rogers married Una Elsie North (1883-1951) in 1914 in Calcutta, West Bengal, India. They had three children, Gordon Leonard Rogers (1916-2006), who became a physicist specialising in optics (he was a professor at the University of Aston in Birmingham, England), and twins Claude Ambrose Rogers, the subject of this biography, and Stephen Clifford Rogers (1920-1995). The twins were born in Cambridge in 1920, the year in which their father returned from India. Leonard Rogers retired from the Indian Medical Service in 1921 and the family lived in London where Leonard took on many duties such as lecturing on tropical medicine, serving on the India Office medical board, and being the medical adviser to the secretary of state.

Ambrose, as he was known, spent his early childhood years in London and then was a boarder at Berkhamsted School, an independent school in Hertfordshire, England about 45 km northwest of London. He graduated from Berkhamsted School in 1938 and, later that year, matriculated at University College, London where he studied mathematics. He was awarded his B.Sc. from the University of London in 1941 having spent part of his undergraduate studies in Bangor where University College had been evacuated to since the bombing of London made that city too dangerous for students. However, the years of World War II had more impact on Rogers than just going to Bangor, for he did war work as an Experimental Officer with the Ministry of Supply from 1940 until the end of the war in 1945. Other mathematicians such as David Kendall also held similar posts with the Ministry of Supply.

After the war ended, Rogers returned to his mathematical studies. He was appointed a lecturer in mathematics at University College, London in 1946 and worked for his doctorate at Birkbeck College, London while lecturing at University College. His thesis advisors at Birkbeck were Lancelot Stephen Bosanquet and Richard G Cooke and in his first paper, Linear transformations which apply to all convergent sequences and series, published in 1946 in the Journal of the London Mathematical Society, he thanked them writing:-
I am indebted to Dr L S Bosanquet and Dr R G Cooke for advice and encouragement during the preparation of this note.
In fact he had done this work in a part-time capacity while he was undertaking war work. It was a slightly disappointing start to Rogers publishing career for he published an Addendum in the same volume of the Journal explaining that the results of his first paper could be obtained more easily from a theorem of Stefan Banach and also reported that the first of his theorems was proved by the American mathematician Ralph Palmer Agnew (1900-1986) in 1939. However, this was only the first of a remarkable number of papers that Rogers published while a research student: A note on a theorem of Blichfeldt (1946); (with Harold Davenport) Hlawka's theorem in the geometry of numbers (1947); A note on a problem of Mahler (1947); A note on irreducible star bodies (1947); Existence theorems in the geometry of numbers (1947); (with J H H Chalk) The critical determinant of a convex cylinder (1948); A problem of Hirsch (1948); The product of the minima and the determinant of a set (1949); The product of n homogeneous linear forms (1949); The successive minima of measurable sets (1949); On the critical determinant of a certain nonconvex cylinder (1949); (with Harold Davenport) A note on the geometry of numbers (1949); and (with J H H Chalk) The successive minima of a convex cylinder (1949). This list is quite exceptional but these were only the papers which appeared in print before 1949. Others such as The signatures of the errors of simultaneous Diophantine approximations and The asymptotic directions of n linear forms in n + 1 integral variables, were submitted by Rogers to the Proceedings of the London Mathematical Society on 6 September 1947 but did not appear in print until 1951.

Rogers was awarded a Ph.D. in 1949 for his thesis The Transformation of Sequences by Matrices in which he studied divergent series. A paper with the same title as his thesis, containing 21 theorems, was submitted to the Proceedings of the London Mathematical Society in August 1948 but only appeared in print in 1951. Rogers writes:-
This paper forms part of a thesis approved for the Degree of Ph.D. in the University of London.
Before the award of his Ph.D., Rogers began working with Harold Davenport and, in addition to their joint papers given above, they published two joint papers On the critical determinants of cylinders and Diophantine inequalities with an infinity of solutions. Both were submitted for publication in June 1949 and appeared in print in 1950. The second of these is a major work with 34 pages and the authors give the following abstract:-
Many results in the geometry of numbers assert, in effect, that inequalities of a certain type are soluble in integers, the constant on the right of the inequality being the best possible. Recent work of Mahler often enables one to prove that such an inequality has infinitely many solutions. In this paper we develop the theory of inequalities with infinitely many solutions, and investigate more deeply some of the questions which naturally arise.
Through Davenport, Rogers became interested in packing and covering. After being awarded a Commonwealth Fund Fellowship, in the autumn of 1949 he went to the Institute for Advanced Study in Princeton. His paper A note on coverings and packings (1950) was written while he was in Princeton. While in Princeton, he met the Russian born Israeli mathematician Aryeh Dvoretzky (1916-2008) and, working together, they solved a conjecture which had stood for twenty years. They proved:
Absolute convergence is equivalent to unconditional convergence of series of points of a Banach space if and only if the space is finite-dimensional.
The result was published in their joint paper Absolute and unconditional convergence in normed linear spaces (1950).

Rogers married Joan Marion North (1920-1999), the daughter of metallurgist Frank Wevil North and Gladys May Paybody, in 1950. Joan North was an author of children's books; they had two daughters, Jane Nee Rogers born 1955 and Petra Nee Rogers born 1956. Returning to his position at University College London, Rogers was awarded a D.Sc. in 1952. At University College, Rogers was promoted to reader before he moved to Birmingham University in 1954 as the Mason Professor of Pure Mathematics [1]:-
In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened. In particular, with Geoffrey Shephard, he produced sharp bounds for the volume of a difference body, a problem which had been open for 30 years.
After four years in Birmingham, Rogers returned to London, this time as the Astor Professor of Mathematics at University College. He succeeded Harold Davenport as the Astor Professor of Mathematics who had moved to Cambridge in 1958. His cousin and brother-in-law Jeremy North writes:-
Visiting Jo and Ambrose in Gray Close, Hampstead, was always amazing. Ambrose had the largest desk piled high with totally incomprehensible research papers. Towards the end of his life he had to have a second desk which quickly became covered too. At the last Birthday party I attended there, the room was full of mathematical colleagues who might have been on a different planet. Over these occasions my sister presided invariably with a cold collage for lunch or a polite tea. Domestic considerations were never high on their agenda.
Rogers held this post until he retired in 1986, when he became professor emeritus and remained at University College.

Rogers produced a remarkable mathematical output having published around 180 papers and books. As related above, his early work was on number theory, and he wrote on Diophantine inequalities and the geometry of numbers. Jointly with Paul Erdős, he wrote The covering of n-dimensional space by spheres (1953) and Covering space with convex bodies (1961), writing many other articles on coverings and packings including Covering space with equal spheres with Donald Coxeter. His later work covered a wide range of different topics in geometry and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions. Rogers has written three important books, Packing and Covering in 1964, Hausdorff Measures in 1970, and (with John E Jayne) Selectors (2002). Much of Packing and Covering was written in 1961 while Rogers was spending a year in Canada at the University of British Columbia in Vancouver. The Publisher of the book, Cambridge University Press, writes:-
Professor Rogers has written this economical and logical exposition of the theory of packing and covering at a time when the simplest general results are known and future progress seems likely to depend on detailed and complicated technical developments. The book treats mainly problems in $n$-dimensional space, where $n$ is larger than 3. The approach is quantitative and many estimates for packing and covering densities are obtained. The introduction gives a historical outline of the subject, stating results without proof, and the succeeding chapters contain a systematic account of the general results and their derivation. Some of the results have immediate applications in the theory of numbers, in analysis and in other branches of mathematics, while the quantitative approach may well prove to be of increasing importance for further developments.
We quote briefly from reviews of Packing and Covering. R P Bambah writes:-
This monograph, based on the work of the author and his collaborators, gives a very elegant and readable account of many important results in the theory of packings and coverings, lattice as well as non-lattice. The author has managed in most cases to include proofs of best known results without affecting the readability of this book, which is entirely self-contained.
Donald Coxeter writes [2]:-
This little book maintains the high standard of scholarly exposition which has distinguished the Cambridge Tracts for several generations.
E A Maxwell writes [3]:-
The style is brisk, clear and interesting, without any fuss, and I have no doubt at all that the book will be found a most valuable source of information in a very readable setting.
Edwin Hewitt, reviewing Hausdorff Measures writes:-
This beautifully written and beautifully printed little book contains an astonishing amount of information. ... this book is a combination, unique as far as the reviewer knows, of pedagogy and description of an active if special field of analysis as it now exists. A bright student can pick up this book and read it by himself. When he has finished the book, he will be able to ask questions of his own and will also have a useful guide to the literature. Mature mathematicians who need facts about Hausdorff measures for their own purposes - surface theory, harmonic analysis, and so on - will find clear statements, clear proofs, and abundant references for further pursuit.
Here is an extract from the authors' Preface to Selectors:-
For sometime we have thought that the theory of measurable selectors was somewhat lacking, in that one knows little about the topological properties of measurable functions. It is surprising that in very general circumstances upper semi-continuous set-valued maps do have selectors that, although not continuous, are of the first Baire class; that is, are point-wise limits of sequences of continuous functions. In the book we are mainly concerned with proving results showing the existence of selections of the first Baire class. We give a number of geometrically interesting examples, and some unexpected consequences for functional analysis.
Rogers wrote an obituary of Harold Davenport for the London Mathematical Society which was published in 1972, the year after Rogers wrote a survey of Davenport's work. He was one of the editors of The collected works of Harold Davenport published in four volumes in 1977.

Rogers was elected a fellow of the Royal Society in 1959 and served on the Council of the Royal Society for two spells, first from 1966 to 1968 and then again in 1983-84. He also served as the 55th President of the London Mathematical Society from 1970 to 1972 and was honoured with the award of the London Mathematical Society's Junior Berwick Prize in 1957 and the De Morgan Medal in 1977. An extract from the citation for the De Morgan Medal reads:-
In the period after the Second World War, Rogers rapidly emerged in the forefront of the renaissance of the geometry of numbers. His work on the lattice constants of cylinders both convex and non-convex, on the reducibility of star bodies and on the implications for the existence of infinitely many lattice points in automorphic bodies, and on the successive minima of general sets with respect to a lattice remains definitive. Since about 1958, Professor Rogers' main research interests moved to the theory of Hausdorff measures, of analytic sets and of general convex bodies, to all of which he has made important contributions.
He was a plenary speaker at the British Mathematical Colloquium in 1964 giving the lecture The Brunn-Minkowski theorem and related inequalities. On two other occasions he was a specially invited morning speaker at the British Mathematical Colloquium, in 1955 and 1984. He was also invited to address the special meeting of the Mathematical Association to celebrate its centenary. He began his talk Length, area and volume as follows (see [5]):-
I will discuss the elementary concepts of Length, Area and Volume from an advanced standpoint. Although my readers will have acquired a clear intuitive understanding of these basic geometric quantities, they may not realise that these quantities are very far from elementary. Many deep investigations have been necessary to refine the concepts and many basic problems remain with only inadequate answers.

### References (show)

1. S Abbott, Review: Hausdorff Measures, by C A Rogers, The Mathematical Gazette 83 (497) (1999), 362.
2. H S M Coxeter, Review: Packing and Covering, by C A Rogers, SIAM Review 7 (2) (1965), 303-305.
3. E A Maxwell, Review: Packing and Covering, by C A Rogers, The Mathematical Gazette 50 (373) (1966), 343.
4. D Larman, Ambrose Rogers, London Math. Soc. Newsletter 344 (January 2006).
5. C A Rogers, Length, area and volume, The Mathematical Gazette 57 (400) (1973), 100-110.
6. H Zassenhaus, Review: Packing and Covering, by C A Rogers, Amer. Math. Monthly 73 (4) (1966), 436.