# Deane Montgomery

### Quick Info

Weaver, Minnesota, USA

Chapel Hill, North Carolina, USA

**Deane Montgomery**was an American mathematician who worked in algebraic and geometric topology.

### Biography

**Deane Montgomery**'s parents were Richard Montgomery and Florence Hitchcock. He was brought up on a farm. He studied at Hamline University, Saint Paul, Minnesota. This was the first institution of higher education in Minnesota, founded by Methodists. He was awarded a B.S. in 1929 and then went to the University of Iowa where he was awarded a Master's Degree in 1930 and his doctorate in 1933. His thesis advisor at Iowa was Edward W Chittenden and his thesis was on point-set topology. His first few papers reflect the material of his thesis:

*Sections of point sets*(1933); and

*Properties of Plane Sets and Functions of Two Variables*(1934).

On 14 July 1933 Montgomery married Katharine Fulton. In 1933-34 he was National Research Council Fellow at Harvard. He said in an interview with Albert Tucker:-

We had a private study group consisting of Norman Steenrod, myself, Garrett Birkhoff, and M R Hestenes. Just about that group. Solomon Lefschetz's book was in existence, and we thought of reading that, but we found it was too hard for us. So we began with Oswald Veblen's Analysis Situs. We read all of that, and we read some other things too.In 1934-35 Montgomery was National Research Council Fellow at Princeton having decided to make a change from Harvard:-

The Institute had started the year before, and all the mathematics of the Institute was combined with that of the University in Fine Hall. It was a little crowded, but I found the whole thing an extremely pleasant experience ... At that time my main interest was in a form of set theory. I was especially interested in Borel sets, analytic sets, and projective sets ...In 1935 he was appointed as an assistant professor at Smith College being promoted to associate professor in 1938 and then full professor in 1941. In fact he was offered an assistant professorship at Harvard in 1938 but he chose to remain at Smith College. He was a Guggenheim fellow at the Institute for Advanced Study during 1941-42. After returning to Smith College in 1942, he was back at Princeton in 1943 to teach Army students. He undertook other war work at that time working with von Neumann on numerical analysis:-

I never worked very hard on that; by that I mean that I tried to do it in a conscientious way, but in my spare time I always thought about something else. In general, my experience with war work was like that: I didn't really want to think about it much, but I tried to be, conscientious about it even while I tried to keep on with what I was really interested in.In 1946 Montgomery left Smith College and accepted an appointment as an associate professor at Yale University. He was at Yale for two years but the atmosphere there was not particularly good:-

When I was at Yale it was emerging from quite an old period. There were some good mathematicians there, certainly, but there wasn't an overwhelming number. There was a great feud going on at Yale, which had gone way back in history before many of the participants then were there even. I was the only one on speaking terms with everyone in the department when I was there.Montgomery returned to the Institute for Advanced Study in 1948 as a permanent member, becoming a professor in 1951. He remained at the Institute until he retired in 1980. He and his wife Kay (Katharine) had one daughter Mary and one son Richard. In 1988 Deane and Kay Montgomery moved to Chapel Hill to be near to their daughter Mary Heck and their two granddaughters.

His research interests included algebraic and geometric topology and he made major advances to transformation groups. For many years he was at the centre of activity in topology at the Institute for Advanced Study. His interests turned from point-set topology to transformation groups quite early in his career and he published a series of papers on the topic in collaboration with Leo Zippin. These include:

*Periodic one-parameter groups in three-space*(1936);

*Translation Groups of Three-Space*(1937);

*Compact Abelian transformation groups*(1938);

*Non-Abelian Compact Connected Transformation Groups of Three-Space*(1939);

*A theorem on the rotation group of the two-sphere*(1940);

*Topological group foundations of rigid space geometry*(1940);

*Topological transformation groups. I*(1940); and

*A theorem on Lie groups*(1942).

The problem which interested them was Hilbert's Fifth Problem:-

A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.This problem had been investigated by many mathematicians such as Chevalley, Gleason, Iwasawa, Kuranishi, Pontryagin, von Neumann and Yamabe. Montgomery solved Hilbert's Fifth Problem in dimension three in 1948. By 1952 Montgomery had solved Hilbert's Fifth Problem with the assumption of finite dimensionality and this restriction was finally removed by Yamabe who began working as Montgomery's assistant in 1952. In 1955, in collaboration with Leo Zippin, Montgomery published the monograph

*Topological transformation groups*. Iwasawa, himself a contributor to the solution of Hilbert's Fifth Problem, wrote in a review:-

Almost two decades have passed since publication of the first edition of Pontryagin's "Topological groups" which has been since considered as one of the standard reference books in the field. In the meantime, the theory of topological groups has made outstanding progress, culminating in the solution of Hilbert's fifth problem by Gleason and by the authors of the present book. The authors give here a detailed account of those important results on locally compact topological groups obtained in this period, suggesting at the same time further future developments in the theory. The book is divided roughly into three parts; in the first two chapters, the authors give those results on topological groups which were obtained in what they call the classical period (ending around 1935), thus preparing for the succeeding chapters. The next two chapters are devoted to the study of the structure of locally compact groups which leads to a solution of Hilbert's problem. In the last two chapters, the authors discuss the properties of transformation groups acting on various spaces. Here the results are not yet as conclusive as those of the preceding chapters, but even so, the reader may find there fundamental theorems on transformation groups which the reader himself may use as a basis for further investigation of this yet unexploited territory in mathematics.In [3] Fintushel comments on important work which Montgomery undertook later in his career:-

In a long series of papers written in the late 1960s and early 1970s, [Montgomery and C T Yang] used the study of group actions on homotopy 7-spheres to showcase and test the growing new techniques of differential topology, especially index theory and surgery theory. At a time when much work in topology consisted in building these machines, their papers demonstrated the beauty of applying this theory to unfurl complexities of symmetry and structure.Montgomery received many honours for his contributions. He was awarded honorary doctorates from Hamline University (1954), Yeshiva University (1961), Tulane University (1967), the University of Illinois (1977), and the University of Michigan (1986). He served the American Mathematical Society as Vice-President (1952-53) and President (1961-62). He was awarded the Leroy P Steele Prize for Lifetime Achievement by the American Mathematical Society in 1988 for his lasting impact on mathematics, particularly mathematics in America. He also served as President of the International Mathematical Union (1974-78). Montgomery was elected a member of the National Academy of Sciences of the United States in 1955 and the American Academy of Arts and Sciences in 1958.

As to Mongomery's character, and the influence he had on his students and colleagues, we quote from Fintushel [3]:-

He was justly famous for his efforts in helping young topologists and, more generally, making all visitors feel welcome at the Institute. He was especially ardent at searching for students like myself from smaller, less prestigious graduate programs and encouraging their careers. In 1979-1981, I was fortunate to serve as Deane Montgomery's assistant. ... During our weekly sessions I learned much about this remarkable man. ... there are certain basic aspects of these conversations that I will never forget - Deane's love of mathematics and his joy at the success of others, his gentleness and personal humility, his abhorrence of pretence in any form, his pride in the Institute and conviction to uphold its standards. Most of all, he absolutely never gave false praise.Armand Borel expresses similar views in [2]:-

He was always seeking out and encouraging young mathematicians. He and his wife Kay would regularly and very warmly receive the visiting members at their home. Maybe remembering his own beginnings in an out of the way place, he had a special interest, and talent, in finding out people with considerable potential among some applicants from rather isolated places about whom not much was available.

### References (show)

- Bibliography of Deane Montgomery,
*Contemp. Math.***36**(Amer. Math. Soc., Providence, RI, 1985), 13-16. - A Borel, Deane Montgomery (1909-1992),
*Notices Amer. Math. Soc.***39**(7) (1992), 684-686. - R Fintushel, A tribute to Deane Montgomery,
*Notices Amer. Math. Soc.***52**(3) (2005), 348-349. - F Raymond and R Schultz, The work and influence of Deane Montgomery,
*Contemp. Math.***36**(Amer. Math. Soc., Providence, RI, 1985), 1-11. - 1988 Steele Prizes awarded at centennial celebration in Providence,
*Notices Amer. Math. Soc.***35**(7) (1988), 965-970.

### Additional Resources (show)

Other pages about Deane Montgomery:

Other websites about Deane Montgomery:

### Honours (show)

Honours awarded to Deane Montgomery

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update November 2006

Last Update November 2006