# Robert Palmer Dilworth

### Quick Info

Hemet, California, USA

California, USA

**Bob Dilworth**was an American mathematician who worked in lattice theory.

### Biography

**Robert Dilworth**(known as Bob to his friends and colleagues) was brought up on a ranch in California at the foot of the San Jacinto Mountains. This gave him a love of the outdoors which he kept throughout his life. He studied for his Bachelor of Science degree at the California Institute of Technology. Caltech was to play a very important role in Dilworth's life, for it was the Institution that he was associated with for almost the whole of his career. He received his B.S. degree in 1936 and remained at Caltech to undertake postgraduate work for his doctorate.

At Caltech, Dilworth's doctoral studies were supervised by Morgan Ward. Ward had himself been a student of E T Bell and Bell was still on the Faculty at Caltech at this time. Like Bell, Ward was someone who valued all aspects of mathematics. He was as interested in teaching elementary mathematics as he was in the latest research problem on which he was working. His ideas of what constituted "a mathematician" rubbed off on Dilworth, as Chase relates in [3]:-

Professor Ward helped instil in Professor Dilworth his profound respect for the teaching of mathematics, at all levels, even very elementary levels.Dilworth obtained his doctorate in 1939 and was then awarded a Sterling Research fellowship to study at Yale. He held this Fellowship at Yale during the academic year 1939-40 and he was then appointed as an Instructor there. Dilworth married Miriam White on 23 December 1940 after taking up the Instructor appointment. He held this position from 1940 until 1943 when he returned to Caltech as Assistant Professor of Mathematics. At this point Dilworth was back at Caltech and he was to remain there for the rest of his career.

Of course 1943 was in the middle of World War II and Dilworth was involved in military service. In July 1944 he became a member of an analysis unit at the 8th Air Force headquarters at Brampton Park in England. Dilworth wrote:-

This unit was to serve as a liaison between the main operational analysis unit located at the headquarters of the 8th Air Force near London and the command of the 1st Air Division. ... In the spring of 1945, in collaboration with the Division navigator, an elaborate experiment was carried out to evaluate the intrinsic accuracy of radar bombing. A special radar target placed in the Wash on the east coast of Britain was used in this exercise.Back at Caltech, Dilworth was promoted to Associate Professor in 1945 and then full Professor in 1950. He held this position for the rest of his career until he retired in 1982.

Let us now turn to Dilworth's research contributions. He worked in lattice theory and it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right. He began his studies in the 1930s by reading the first contributions to lattice theory which were by Dedekind. Dilworth himself remarked that although Dedekind's papers were excellent introductions to the subject it was unclear what his motivation had been. By the time Dilworth began his research, the motivation behind much of lattice theory was to develop methods to attack problems in group theory. This is well explained by Dilworth himself writing in 1959 (see [1]):-

The theory of groups provided much of the motivation and many of the technical ideas in the early development of lattice theory. Indeed, it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory. Two decades later, it seems to be a fair judgement that, while this hope has not been realised, lattice theory has provided a useful framework for the formulation of certain topics in the theory of groups ... and has produced some interesting and difficult group-theoretic problems ...Dilworth then goes on to explain where the main thrust in developing lattice theory subsequently come from and one has to say that, although he modestly does not say so, he played the major role in this development himself:-

On the other hand, the fundamental problems of lattice theory have, for the most part, not come from this source but have arisen from attempts to answer the intrinsically natural questions concerning lattices and partially ordered sets; namely, questions concerning the decompositions, representations, imbedding, and free structure of such systems ... As the study of these basic questions has progressed, there has come into being a sizable body of technical ideas and methods which are peculiarly lattice-theoretic in nature. These conceptual tools are intimately related to the underlying order relation and are particularly appropriate for the study of general lattice structure.The main topics in lattice theory to which Dilworth contributed are: Chain partitions in ordered sets, in particular his chain decomposition theorem for partially ordered sets; Uniquely complemented lattices; Lattices with unique irreducible decompositions; Modular and distributive lattices, in particular his covering theorem for modular lattices; Geometric and semimodular lattices; and Multiplicative lattices, where he studied, among other topics, abstract ideal theory, and the representation and embedding theorems for Noether lattices and $r$-lattices.

One important aspect of Dilworth's research was that he always attacked the big problems in lattice theory. He always had a stock of open problems in the subject which he used to direct his research and that of his students. For example in 1959 he writes of about the big problems of the subject [1]:-

... the construction of a set of structure invariants for certain classes of Boolean algebras, the characterisation of the lattice of congruence relations of a lattice, the imbedding of finite lattices in finite partition lattices, the word problem for free modular lattices, and a construction of a dimension theory for continuous, non-complemented, modular lattices, have an intrinsic interest independent of the problems associated with other algebraic systems. Furthermore, these and other current problems are sufficiently difficult that imaginative and ingenious methods will be required in their solution.Let us now turn to Dilworth as a teacher. We have already mention the influence of his supervisor Morgan Ward on him. R Freese and J B Nation write (see [1]):-

When [Dilworth] lectured, he rarely used abbreviations and his handwriting was nearly perfect. Students had to write as fast as they could, using several abbreviations, to keep up with him. When he got stuck he would step back from the blackboard, stare at the problem and whistle "Stars and Stripes Forever".Teaching and examining mathematics played an important part in Dilworth's career. He was appointed to the College Board Advanced Mathematics Committee in 1954. The task of this Committee was to set policy and administer the Advanced Mathematics Examination. Dilworth was Chairman of this Committee from 1957 to 1961. He also became involved with a project to develop mathematics education in African countries. He was Director of the Testing and Evaluation group for this project from 1962 to 1969 and he described its role:-

The objective was to develop a core of mathematics educators in each of the participating countries who would be able to produce curriculum materials in mathematics which would be appropriate for the needs of each of the countries. During six summer sessions from 1962 to 1968 the representatives of the African countries involved met with mathematics educators from the United States and Britain to develop specimen mathematics texts covering primary and secondary years. It was the responsibility of the Testing and Evaluation group to see that there were African personnel in each of the countries trained in modern testing methods by developing tests and other evaluative materials ...In addition Dilworth served on numerous other bodies concerned with the teaching and examining of mathematics. For example the Board of Examiners in Mathematics, the School Mathematics Study Group Advisory Board, the Miller Mathematics Improvement Program, and several programs set up by the National Science Foundation.

Finally we should say a little about Dilworth other than his mathematical interests. As a young man he was an exceptionally good athlete, competing in the decathlon. Later in life he complained that running was damaging his knees and he took up swimming which he did regularly. He kept himself very fit and [3]:-

He never dawdled, but always walked with a spring in his step, and got wherever he was going very fast.Another of his interests was music and [2]:-

... he often commented that if he were not accepted to CalTech, he would have made music his life. He loved playing Chopin on the piano late at night in total darkness. He insisted that it improved his mathematical abilities. He played several other instruments as well when necessary for the CalTech Orchestra; however the piano was his relief from the pressures of mathematics.Crawley [4] writes:-

... he was an electrifying teacher and colleague. And apart from his intellectual power as a mathematician, I think this was primarily a product of two traits: Bob Dilworth loved a challenge, and he was tenacious in confronting one; and he had great mathematical taste.Bogart in [2] write that Dilworth:-

... had a keen sense of humour and was known as a warm and approachable person.

### References (show)

- K P Bogart, R Freese and J P S Kung (eds.),
*Robert P Dilworth : The Dilworth theorems*(Boston, MA, 1990). - K P Bogart, Obituary : R P Dilworth [1914-1993],
*Order***12**(1) (1995), 1-4. - P J Chase, Recollections of Professor Dilworth, in
*K P Bogart, R Freese and J P S Kung (eds.), Robert P Dilworth : The Dilworth theorems*(Boston, MA, 1990), xxi-xxii. - P Crawley, Recollections of R P Dilworth, in
*K P Bogart, R Freese and J P S Kung (eds.), Robert P Dilworth : The Dilworth theorems*(Boston, MA, 1990), xix-xx. - Mathematical publications of Robert P. Dilworth, in
*K P Bogart, R Freese and J P S Kung (eds.), Robert P Dilworth : The Dilworth theorems*(Boston, MA, 1990), xxiii-xxv.

### Additional Resources (show)

Other websites about Robert Dilworth:

Written by J J O'Connor and E F Robertson

Last Update September 2001

Last Update September 2001