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Indeed he did rapidly decide that he wanted to pursue a career in mathematics and undertook research for his doctorate under Birkhoff 's supervision. His doctorate was awarded in 1926 for a thesis entitled Ordinary Linear Homogeneous Differential Equations of Order n and the Related Expansion Problems. In 1925 he had been appointed as an instructor in mathematics at Columbia University,and he spend two years there before being appointed to Harvard as an instructor in 1927. In this latter year he married Emmy Portman; they had three children but eventually the marriage ended in divorce in 1962.

During the period that he was an instructor Stone's interests followed very much those of his research supervisor Birkhoff . He published eleven papers on the theory of orthogonal expansions between 1925 and 1928. For example he published An unusual type of expansion problem (1924), A comparison of the series of Fourier and Birkhoff (1926), Developments in Legendre polynomials (1926), and Developments in Hermite polynomials (1927). In these papers a special role is played by expansions in terms of the eigenfunctions of linear differential operators.

In 1928 Stone was promoted to associate professor at Harvard. Although he would return to Harvard again in 1933, Stone first accepted a post as associate professor at Yale from 1931 to 1933. Back at Harvard as an associate professor in 1933, he was promoted to full professor there in 1937.

During these years Stone's research took a number of different directions. From 1929 he worked on self-adjoint operators in Hilbert space and included his results in the major publication of his 662 page book Linear transformations in Hilbert space and their applications to analysis (1932). Halmos writes about the book in 1990 when it was reprinted by the American Mathematical Society :

This book is a classic by now. ... In the language of the 1990's the book belongs to functional analysis (a subject we used to call topological algebra - didn't we?). One of the most famous books in that subject is Stefan Banach 's Théorie des opérations linéaires - it too was published in 1932 (in Warsaw, when Banach was 30). Banach begins with a chapter on Lebesgue integration followed by a chapter on metric spaces (but I couldn't find a mention of either Hilbert or Stone in his book). Stone's book is very different in spirit, in aim, in tone, in style, in timeliness, and in applicability. The work was begun in 1928, when Stone was 25; it was published when he was 29. A leisurely foreword tells the reader that he hopes to offer "a detailed treatment which would start with the foundations and carry the development as far as possible in every direction ... . In order to compress the material into the compass of six hundred odd pages, it has been necessary to employ as concise a style as is consistent with completeness and clarity of statement ... ." After the foreword, he dives right in; there is a short meditation on what the word "space" might mean, followed immediately (on page 2) by the axiomatic definition of abstract Hilbert space. (The only reference to Banach concerns a 1924 paper in the Fundamenta.) The work consists of ten chapters, of which the last (Applications) is more than a third of the book.

Because of lack of space, results from Stone's 1930 article Linear transformations in Hilbert space III : Operational methods and group theory, which had appeared in the Proceedings of the National Academy of Sciences, had to be omitted. This article included the celebrated Stone- von Neumann uniqueness theorem.

In 1932 he proved results on spectral theory , arising from group theoretical methods in quantum mechanics , which had been conjectured by Weyl . Then in 1934 he published two papers on Boolean algebras : Boolean algebras and their applications to topology and Subsumption of Boolean algebras under the theory of rings. Both appeared in the Proceedings of the National Academy of Sciences. These papers contain what today is called Stone-Cech compactification theory. He made this study while attempting to understand more deeply the basics underlying his results on spectral theory. One particularly important result proved by Stone during this period was a substantial generalisation of Weierstrass 's theorem on uniform approximation of continuous functions by polynomials. This result is now known as the Stone-Weierstrass theorem.

During World War II Stone undertook secret war work being attached to the Office of Naval Operations during 1942-43 and then the Office of the Chief of Staff of the War Department for the rest of World War II. Then in 1946 he left Harvard to take up the chairmanship of the mathematics department at the University of Chicago. This did not happen easily for Stone negotiated for a year with Robert Maynard Hutchins, President of the University of Chicago, before taking up the appointment. One problem was certainly the protests from the faculty members at Chicago who wished their present chairman, Ernest P Lane, to continue in office. In Stone explains why he decided to leave Harvard. A major motivation was:

... my conviction that the time was also ripe for a fundamental revision of graduate and undergraduate mathematical education.

Stone did an outstanding job in returning this famous research school to the eminence it had previously known. First he had to overcome the resistance to his appointment from his own Department, then he had to persuade the University authorities to accept his proposals. However :

Stone was a man of forceful character and unquestioned integrity.

His first argument with the university administration was over his wish to offer an appointment to Hassler Whitney . Stone won the argument, the offer was made to Whitney , but he turned it down preferring to remain at Harvard. Stone next decided to appoint André Weil but he :

... was a somewhat controversial personality, and I found a good deal of hesitation, if not reluctance, on the part of the administration to accept my recommendation.

Again Stone won the argument and this time the offer was accepted. This was :

... an important event in the history of . . . American mathematics.

Stone continued to appoint leading mathematicians. Next came Mac Lane , then Zygmund followed by Chern . This last appointment proved the hardest to get past the Chicago administration and Stone threatened to resign in a bid to get his own way. He wrote :

This was the stormiest incident in a stormy period.

From 1952 Stone stepped down as head of department in favour of Mac Lane but he remained at Chicago until he retired in 1968. He then accepted a professorship at the University of Massachusetts where he worked full-time until 1973, then half-time until 1980. We noted above that he was divorced from his first wife in 1962 and he subsequently married Ravijojla Kostic.

Stone received many honours for his outstanding achievements. He was elected to the National Academy of Sciences (United States) in 1938, in the following year he was American Mathematical Society Colloquium Lecturer and he was president of the Society in 1943-44. He was honoured by being named Josiah Willard Gibbs lecturer for 1956, delivering a lecture on Mathematics and the future of science at Rochester, New York, on 27 December 1956. Stone was elected president of the International Mathematical Union in 1952-54 and he was president of the International Committee of Mathematical Instruction from 1961 to 1967. In this capacity he founded the Inter-American Committee on Mathematical Education (Comitê Interamericano de Educaçao Matemática - CIAEM) in 1961.

Stone's interests, which included cooking, are described in :

Of all Stone's many interests his love of travel was surely dominant. He began to travel when he was quite young and was on a trip to India when he died. ... Marshall Stone was a man with a very broad outlook and a wide range of interests who seems to have thought rather deeply about a number of issues. ... here was an unusually thoughtful man with a high degree of penetration and insight. ... he seemed well endowed with a quality which I can only describe as wisdom.

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