**Leo Zippin**'s parents were Bella Salwen and Max Zippin who were immigrants to the United States arriving in New York in 1903. Bella and Max were Jewish and they had lived in the city of Chernigov, situated in northern Ukraine around 140 km north of Kiev. The years around 1900 were difficult in the Ukraine, with problems which would lead eventually to the Russian Revolution. There was also extreme anti-Semitism and for Max Zippin, who loved the Yiddish culture and supported a workers' revolution, it was a difficult place to live. Once settled in New York, Max wrote plays for the Yiddish theatre (not successfully since none was ever performed) and he wrote for the *Jewish Daily Forward*:-

Launched as a Yiddish-language daily newspaper on22April1897, the 'Forward' entered the din of New York's immigrant press as a defender of trade unionism and moderate, democratic socialism.

Leo was brought up in New York where he began his education. He entered the DeWitt Clinton High School but the Russian Revolution of 1917 dramatically changed the family's life. Max, the staunch supporter of the workers, was of course delighted and could not resist taking the family back to witness the historic events. What they found in Russia was extreme poverty, illness and a civil war. It was not the joyous revolution that Max had hoped to rejoice in, and after a while they decided to return to the United States. This, however, proved much harder than they realised since all routes leading west were blocked. If you can't go west then you have to go east and that is precisely what the Zippins did boarding the Trans-Siberian railway [1]:-

The family story has them as far as Harbin, Manchuria, nearly out of money. Bella marched up to the local potentate's residence and introduced herself as an English teacher to the startled man. She taught members of the household until the family was able to move on. The trip took, perhaps, close to two years.

Back in the United States, the family settled in Philadelphia where Max found a job. Leo entered Central High School, Philadelphia, on 19 September 1919. The Russian adventure had quite an effect on Leo and certainly from that time on he did not want to follow his father's political ideas, rejecting Communism. Taking two years out seems to have done little, however, to slow down his educational achievements and his performance at the prestigious Central High School was outstanding. He entered the University of Pennsylvania in 1922, was awarded the Freshman Entrance Prize in 1923 and graduated with his first degree in 1925. He continued with graduate studies at the University of Pennsylvania advised by the topologist John Robert Kline (1891-1955). John Kline had studied at the University of Pennsylvania under R L Moore and was appointed to his chair in 1920 when Moore moved to Texas. John Kline was the most influential mathematician in the Philadelphia area during the 1920s and Zippin prospered under his advice. His doctoral studies were funded first by his appointment as an assistant instructor in mathematics in 1927 and then for two years as a Harrison Fellow. He was awarded his Ph.D. on 19 June 1929 for his thesis *A Study of Continuous Curves and their Relation to the Janiszewski-Mullikin Theorem*. He published the main results of this thesis in a paper of the same name in the *Transactions* of the American Mathematical Society in 1929.

Despite the depression, which made it almost impossible for Zippin to gain employment, he did succeed in gaining a National Research Fellowship which funded his visit during 1929-30 to the University of Texas. John Kline had written to R L Moore on 22 April 1929, before Zippin had been awarded his doctorate, including the comment:-

Zippin has applied and asked to come to Texas to work with you.

Zippin's paper *Generalization of a theorem due to C M Cleveland *was read at the 1931 Easter meeting of the American Mathematical Society at the University of California, Berkeley. The paper was published in the *American Journal of Mathematics* in 1932. Again a National Research Fellowship funded him during the year 1931-32 which he spent at Princeton. This was an important year for Zippin who was able to work with some outstanding mathematicians such as Oswald Veblen and James Alexander. He married Frances Levinson, a teacher at the James Madison High School in Brooklyn in 1932; they had two daughters Nina and Vivian. In March 1932 the American Mathematical Society met at Columbia University in New York. Zippin attended the meeting and delivered a talk* On the Rutt-Nöbeling theorem*. His abstract begins:-

A new and thoroughly independent proof is given that a locally compact continuous curve C, containing two points x and y such that no N points of C separate x and y, contains at least N+1independent arcs xy.

In addition, his paper *Irreducible continuous curves* was also read to the meeting. In September of that year the Society again met at Columbia University with Zippin attending and delivering his lecture *Characterization of the closed 2-cell*.

If jobs had been hard to get in 1929, they were even harder to get in 1933 when Zippin's fellowship ended. However, the Institute for Advanced Study had just opened in Princeton and Zippin was offered a position as research assistant to James Alexander. He gladly accepted and he undertook joint work with Alexander leading to the publication of their joint paper *Discrete abelian groups and their character groups* (1935). Zipppin had published papers prior to this, in addition to that already mentioned, with *The Moore-Kline problem* (1932) and *Independent arcs of a continuous curve* (1933).

An important event for Zippin's career occurred when, in 1934-35, Deane Montgomery was a National Research Council Fellow at the Institute for Advanced Study in Princeton. The two mathematicians became firm friends and over the following years undertook important joint research projects. In 1936 Zippin left the Institute for Advanced Study when he was appointed as an instructor at New York University. After spending the first four years of their married life living far from each other, Zippin and his wife were at last together. Two years later, in 1938, he was appointed to Queens College, a part of the City University of New York. He had a heavy teaching load at Queens College but continued to undertake research, collaborating with Deane Montgomery who was, at this time, at Smith College, Northampton, Massachusetts. When the United States entered World War II in December 1941, Zippin was nearly 37 years old; too old for active service. He volunteered for war work and was given leave by Queens College in 1942 to work at the Ballistics Research Laboratory in Aberdeen, in Maryland. Isaac Schoenberg writes about his time at the Aberdeen Proving Ground [2]:-

Several mathematicians, among them O Veblen, Mina Rees and Leo Zippin, were concerned with making mathematicians available for the war effort. Leo Zippin, as a Corporal, followed the development of the first electronic computer, the ENIAC, at the Moore School of the University of Pennsylvania, which was to be moved to the Ballistics Research Laboratory in Aberdeen, Maryland. Leo Zippin arranged my going to the Ballistics Research Laboratory for the duration[1943-46].

In late 1945, Zippin returned to his duties at Queens College but he also continued with the work he had undertaken at the Aberdeen Proving Ground being attached to the Institute for Mathematics and Mechanics at New York University. He slowly returned to his joint research with Deane Montgomery, and together they made progress towards a solution to Hilbert's fifth problem [1]:-

They met at least once a month, sometimes in New York, sometimes Princeton, talked on the phone and exchanged letters. ... Montgomery and Zippin conducted their collaboration while walking. ... In Princeton the countryside was nearby; in New York they took long walks in Central Park. ... they were "wonderful friends."

In 1952 they published three joint papers which, together with a theorem of Andrew Gleason, prove that every locally Euclidean group is a Lie group. This solves Hilbert's Fifth Problem. Two of the three papers have the title *Small subgroups of finite-dimensional groups* while the third is entitled *Four-dimensional groups*. The theorem which Zippin and Montgomery proved was:-

If G is a separable, metric, locally compact, finite-dimensional, connected and locally connected topological group, and if all proper subgroups of G are generalized Lie groups, then G contains an invariant closed generalized Lie subgroup H such that G/H is finite-dimensional and has no small subgroups.

Kenkichi Iwasawa writes in a review that this:-

... theorem, together with a theorem of Gleason that a finite-dimensional group without small subgroups is a Lie group, gives an important result that every finite-dimensional(separable, metric, locally compact)group is a generalized Lie group. It then follows immediately that every locally euclidean group is a Lie group, namely, the solution of Hilbert's fifth problem.

Zippin wrote the monograph *Topological transformation groups* (1955) with Montgomery. A review by Kenkichi Iwasawa begins:-

Almost two decades have passed since publication of the first edition of Pontryagin's "Topological groups"[1938]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.

Although Zippin continued to make an important contribution to Queens College up to his retirement in 1971, his last research paper appeared in 1956. After that he had only one further publication, namely the book *Uses of infinity* (1962). He writes in the Preface:-

Most of this book is designed so as to make little demand on the reader's technical competence in mathematics; he may be a high school student beginning his mathematics now or one who has put away and forgotten much of what he once knew. On the other hand, the book is mathematical except for the first chapter - that is to say, it is a carefully reasoned presentation of somewhat abstract ideas. ... I hope that the reader will believe me when I say that professional mathematicians do not profess to understand better than anybody else what, fro a philosophical point of view, may be called the "meaning of infinity." This is proved, I think, by the fact that most mathematicians do not talk about this kind of question, and those who do, do not agree.

At Queens College, Zippin was rightly considered a celebrity. Joseph Malkevitch, a student of Zippin's in the early 1960s, wrote about his:-

... special excitement in taking a course with someone who had solved one of the world-famous Hilbert problems.

Zippin was honoured with election to the American Academy of Arts and Sciences in 1970. In April 2008 the following resolution was made:-

... the Board of Trustees of The City University of New York approve the naming of Room508of Kiely Hall at Queens College as "The Leo Zippin Mathematics Common Room."

They gave the following explanation:-

Leo Zippin was a renowned mathematician who taught at Queens College from1938until1971. He was a founding member of the Queens College Mathematics Department and the first Executive Officer of the Mathematics Ph.D. program at 'The CUNY Graduate Center'. Professor Zippin was an inspiration to many students who pursued graduate studies and careers in mathematics. In1995, Room508of Kiely Hall was dedicated to the memory of Professor Zippin, who died that year.

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