... an atmosphere of gentleness and courtesy.Although Allen shared his parents' views of the social injustices in the United States, he did not accept their ideology nor their belief that Communism was a panacea :-
This did not seem to vitiate his affectionate relationships with his parents ...Shields was an undergraduate at the City College New York where his lecturers included Emil Post and Lee Lorch. After enlisting in the United States Army in 1944 and serving in Berlin from 1945 to 1946 after the end of World War II, he completed his undergraduate studies in New York and was awarded his bachelor's degree in 1949. Shapiro, who knew him during his final undergraduate year, gives this description :-
A tall, slim, redhead, he usually wore army clothing and plaid flannel shirts. He was very tidy and always had his homework done. With his rustic appearance and Anglo-Saxon name he seemed rather exotic in that population of typically urban, predominantly Jewish students.His doctoral studies were at the Massachusetts Institute of Technology under Witold Hurewicz's supervision and he received his doctorate in 1952 for his thesis On Additive Properties of Real Numbers. Despite the fact that Hurewicz was Shields' official supervisor, he was unofficially supervised by Raphaël Salem. At that time, Salem spent half the year in Paris and half at M.I.T. Shapiro, who was a close friend of Shields and his roommate while they were undertaking research at M.I.T., writes in  of the good and bad aspects of this time:-
One professor, a champion of austere formalism, often needed to draw a diagram to see what was going on, but in order not to "corrupt" us he would hide it with his body, then erase it before the class could see it. This became a standing joke between Allen and me, and perhaps it influenced the evolution of his own lecturing style into the model of clarity and vividness that was his hallmark. ... One of my fondest memories from M.I.T. years is running into Allen in the library at ungodly hours of the night. We shared reverential feelings for "classical" mathematics and had acquired the habit of browsing in sources like Fundamenta Mathematica and Crelle's Journal. We compared notes on what we were reading or working on. I'll never forget that atmosphere - we were the only living souls around; it seemed truly amazing that we had access to the departed, to Banach and Poincaré, to the Urysohn metricisation theorem as seen through the eyes of Urysohn, and rather clearer than in out topology course.Shields' first post was as a Research Instructor in Mathematics at Tulane where he learnt a lot of functional analysis and wrote papers on topological semigroups. This post, funded by the Office of Naval Research, had a reduced teaching load. Here he began to collaborate with Paul Mostert on topological semigroups. Over the following years they published joint papers such as: On a class of semigroups on En (1956), On continuous multiplications on the two-sphere (1956), On the structure of semigroups on a compact manifold with boundary (1957), Semigroups with identity on a manifold (1959), and One-parameter semigroups in a semigroup (1960).
In 1955, Shields was appointed to the University of Michigan at Ann Arbor, taking up his appointment in the following year, and remained there for the rest of his career except for two years (1959-61) spent in New York. In  Duran recounts the story, told to him by George Piranian, of how Shields was appointed to the University of Michigan.
In 1955, on the first day of the American Mathematical Society Summer Meeting in Ann Arbor, George [Piranian] asked Chairman T H Hildebrandt for leave of absence for the Winter Term of 1956. Immediately Hildebrandt declared that he could not grant the request unless George found a replacement. On his way from Hildebrandt's office to one of the lecture sessions, George ran into Allen Shields, whom a year earlier he had met at the Summer Meeting at Laramie. Allen was cooperative, and George dashed back to report that he had found a substitute and that, after fifteen more minutes the substitute would present a ten-minute paper. ... Young as he was, Allen had already mastered the art of beginning his blackboard work in the upper left-hand corner and ending neatly at the lower right, with one minute to spare. Hildebrandt was so impressed that on the spot he offered Allen a one-term appointment. Later, the department persuaded both Shields and Hildebrandt to extend the arrangement.Soon after George Piranian returned from his leave, he began working with Shields and they published the joint paper The sets of Luzin points of analytic functions (1957). Shields worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis and operator theory. F W Gehring described his work in these words:-
Allen's standards were high, his taste impeccable, and his ideas deep. He was one of the world's most versatile practitioners in the art of applying functional analysis to gain insight on and solve problems of classical function theory. He had a profound effect on the development of this subject through his research and his personal contacts with colleagues and students.S Axler, in dedicating the second part of the twelfth volume of The Mathematical Intelligencer to Shields, writes :-
Allen was one of the world's leading authorities on spaces of analytic functions. He directed more Ph.D. theses than anyone in the history of the Mathematics Department of the University of Michigan, where he served as chair for a few years [beginning in 1975]. In 1979 the University of Michigan awarded Allen its Distinguished Faculty Achievement Award. ... Allen's 'Years Ago' columns [in The Mathematical Intelligencer] sparkle with insight and demonstrate his unusual knowledge of history as well as mathematics.J H Shapiro was one of Shields' many PhD students. He also notes that Shields advised the greatest number of mathematics PhD theses at the University of Michigan, but as a student was able to give insight into the style of Shields' supervision. He writes :-
... I want you to know that beneath that gentle exterior lurked a very tough advisor. If you were his student, then under his guidance, you settled on your own topic, within which you found your own problem. And there was no guarantee that you could crack that problem. Allen kept to the sidelines but always made himself available to guide us to the literature, offer opinions, suggest options, lighten the mood with gentle wisecracks, and generally shout encouragement. Any student who got a degree with him really earned it, and in the process learned how to find the dreaded "post-thesis" research problem, and the ones after that, too.Shields married Smilka Zdravkovska who had come to the United States with her daughter Bojana from Skopje, Yugoslavia. She writes :-
I probably fell in love with Allen the first time I heard him lecture (it was on power series, in the University of Michigan Math Club). Here was this tall, lean, red-haired boy (it was only later that I found out he wasn't a junior faculty member in the Mathematics Department, but rather its acting chairman) ...P Duren wrote joint papers with Shields. In , he gives a feel for what it was like working with him. Duren writes:-
Doing mathematics with Allen was fun. He was creative and powerful, yet childlike in his enthusiasm. His lectures were models of clarity and inspiration. He was a wonderfully gregarious mathematician who found excitement in mathematical discussions with colleagues and students. Throughout his career, he always had a strong tendency to collaborate. The record shows that of his 80 research papers, all but 15 were written jointly with other mathematicians. He had 44 different coauthors.H S Shapiro tells us that, in addition to mathematics, there were many other skills that Allen Shields possessed  :-
Looking back on Allen's life one sees that "the elements were mixed in him" and he could have done well in several professions. He loved history and, like his father, had a phenomenal memory. He was a fine amateur linguist [fluent in Spanish, German, Russian and French]. He could have been a journalist - he loved to travel about the country (later, the world) drinking in local atmosphere and history, hitching or offering rides, always curious to talk with people he met en route ...Tragically Shields was struck down with cancer in the summer of 1988. A conference was held in his honour in August 1989 which he attended, although he was gravely ill, and joked with the participants. He died a few weeks later.
Article by: J J O'Connor and E F Robertson