# Alexander Doniphan Wallace

### Born: 21 August 1905 in Hampton, Virginia, USA

Died: 16 October 1985 in New Orleans, USA

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**Alexander Doniphan Wallace**was born in Hampton, southeastern Virginia, U.S.A. Hampton, lying on Chesapeake Bay which is the largest inlet in the Atlantic Coastal Plain of the eastern United States, was a town when Wallace was born but three years later it became a city. Wallace was a student at the University of Virginia and he was awarded a B.S. in 1935. He continued to take postgraduate courses at the University of Virginia and was awarded a Master's Degree in 1936 following which he undertook research with Gordon Whyburn as his thesis advisor. He submitted his doctoral dissertation

*On the Interior and Related Transformations*to the University of Virginia in 1939 and was awarded a Ph.D.

In fact Wallace had written six papers during his time as a research student and three of these were published in 1939 with the next three appearing in 1940. The 1939 papers were: (with D W Hall)

*Some invariants under monotone transformations; On non-boundary sets*; and

*Some characterizations of interior transformations*. The 1940 papers were:

*Monotone coverings and monotone transformations*which relates monotonic transformations to monotonic coverings of the space, that is, coverings with connected sets;

*On 0-regular transformations*which defines and develops some of the properties of a new transformation; and

*Quasi-monotone transformations*in which he shows, among many other things, that the property of being a rational curve or a regular curve is invariant under quasi-monotone transformations.

Wallace spent the year 1940-41 as an Instructor in Mathematics and assistant to Lefschetz at Princeton, then was appointed as an Assistant Professor at the University of Pennsylvania. He spent the years 1941-47 there building an impressive expertise in algebraic topology [2]:-

Wallace was invited to join the faculty at Tulane University as Professor and Head of the Department of Mathematics in 1947. There he played a major role in developing a graduate programme over 16 years in which his research interests moved from studying Leray's work, to locally compact connected groups, and then in 1952 to topological semigroups. By the time he gave an invited address to the American Mathematical Society on the topic in 1953 he had already written six papers on the topic. For example he published a two part paperIn1947he gave an invited address to the American Mathematical Society in which he introduced a significant modification of the Alexander cochain complex. This notion, subsequently developed by Spanier in his dissertation, is frequently referred to as Alexander-Wallace-Spanier-Kolmogorov cohomology. He began to develop a set of lecture notes on algebraic topology which, although formally unpublished, represents a substantial achievement in research and scholarship. These notes, through his students, had a great influence on research in topology and its applications in topological algebra.

*A note on mobs*(1952, 1953) which looks at idempotents in semigroups and subgroups of semigroups. Three further 1953 papers are

*Cohomology, dimension and mobs, Indecomposable semigroups*, and

*Inverses in euclidean mobs*. In the second of these he showed that an indecomposable continuum which is a semigroup with identity must be a group, while the third showed that a unit of a continuum semigroup in

^{n}always lies on the boundary.

His style while at Tulane is illustrated in [1] with the following description:

In 1963 Wallace moved to the University of Florida at Gainesville. He spent the rest of his career there until he retired in June 1973. He took up the professorship at Florida in the same year as Kermit Sigmon became a graduate student. Sigmon wrote his thesis onEach day, in the early afternoon, he swept into the department making the rounds of all his students and colleagues with his usual greeting, "Ah, Mr Koch"; or, "Ah, Professor Mostert; a theorem a day brings promotion and pay!. What have you got today?" He did not wait for people to come to see him about their results, or about their problems. He was always there looking for something new from anyone and everyone.

*Topological Means*with Wallace as advisor. He expressed his thanks at the acknowledgements:-

This is typical of the care and effort that Wallace put into helping all the students in his department, those he supervised and those supervised by others.The author wishes to express his gratitude to Professor A D Wallace, Chairman of his supervisory committee, for his continued interest and efforts on behalf of the author in preparing him for a career in mathematics. ...

Wallace married Willie-Catherine and they had a daughter Sandra. They moved to New Orleans after Wallace retired and there he wrote [1]:-

The authors of [2] write:-... frequent letters-to-the-editor in the local newspapers, dealing with matters political and social.

The personal influence which A D Wallace had on his students and colleagues alike was enormous. His style was relaxed, yet combined with great personal dignity. An expert teacher, he once enunciated the "principle of the bite sized chunk", his technique of communicating, both in written and in oral form. It expresses the idea of presenting small, well-chosen segments of a subject. ... For those of us who were touched by his personality, he was our hero, our leader, and friend. The bite sized chunks he dispensed were little gems, pointing in the direction of larger, nobler goals. For all of those mathematicians having in the remotest sense been connected with topological semigroups, however, Alexander Doniphan Wallace will always be remembered as the great pioneer of this line of mathematical research.

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

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