# Lipman Bers

### Quick Info

Riga, Russia (now Latvia)

New Rochelle, New York, USA

**Lipman Bers**was a Latvian-born American mathematician who worked on pseudoanalytic functions, Riemann surfaces and Kleinian groups.

### Biography

**Lipman Bers**, always known as Lipa, was born into a Jewish family. His parents Isaac Bers and Bertha Weinberg were teachers, his mother being head at an elementary school in Riga where teaching was in Yiddish while his father was head at the Yiddish high school in Riga. Born in 1914, Lipa's early years were much affected by the political and military events taking place in Russia. Latvia had been under Russian imperial rule since the 18th century so World War I meant that there were evacuations from Riga. The Russian Revolution which began in October 1917 caused fighting between the Red Army and the White Army and for the next couple of years various parts of Russia came first under the control of one faction then of the other. Lipa's family went to Petrograd, the name that St Petersburg had been given in 1914 when there was strong anti-German feeling in Russia, but Lipa was too young to understand the difficulties that his parents went through at this time.

At the end of World War I in 1918, Latvia regained its independence although this was to be short-lived. Lipa spent some time back in Riga, but he also spent time in Berlin. His mother took him to Berlin while she was training at the Psychoanalytic Institute. During his schooling mathematics became his favourite subject and he decided that it was the subject he wanted to study at university. He studied at the University of Zürich, then returned to Riga and studied at the university there.

At this time Europe was a place of extreme politics and, in 1934, Latvia became ruled by a dictator. Lipa was a political activist, a social democrat who argued strongly for human rights. He was at this time a soap-box orator putting his views across strongly both in speeches and in writing for an underground newspaper. Strongly opposed to dictators and strongly advocating democracy it was clear that his criticism of the Latvian dictator could not be ignored by the authorities. A warrant was issued for his arrest and, just in time, he escaped to Prague. His girl friend Mary Kagan followed him to Prague where they married on 15 May 1938.

There were a number of reasons why Bers chose to go to Prague at this time. Firstly he had to escape from Latvia, secondly Prague was in a democratic country, and thirdly his aunt lived there so he could obtain permission to study at the Charles University without having to find a job to support himself. One should also not underestimate the fact that by this stage his mathematical preferences were very much in place and Karl Loewner in Prague looked the ideal supervisor.

Indeed Bers did obtain his doctorate which was awarded in 1938 from the Charles University of Prague where he wrote a thesis on potential theory under Karl Loewner's supervision. At the time Bers was rather unhappy with Loewner [1]:-

Lipa spoke of feeling neglected, perhaps even not encouraged, by Loewner and said that only in retrospect did he understand Loewner's teaching method. He gave to each of his students the amount of support needed ... It is obvious that Lipa did not appear too needy to Loewner.In 1938 Czechoslovakia became an impossible country for someone of Jewish background. Equally dangerous was the fact that Bers had no homeland since he was a wanted man in Latvia, and was a left wing academic. With little choice but to escape again, Bers fled to Paris where his daughter Ruth was born. However, the war followed him and soon the Nazi armies began occupying France. Bers applied for a visa to the USA and, while waiting to obtain permission, he wrote two papers on Green's functions and integral representations. Just days before Paris surrendered to the advancing armies, Bers and his family moved from Paris to a part of France not yet under attack from the advancing German armies. At last he received the news that he was waiting for, the issue of American visas for his family.

In 1940 Bers and his family arrived in the United States and joined his mother who was already in New York. There was of course a flood of well qualified academics arriving in the United States fleeing from the Nazis and there was a great scarcity of posts, even for the most brilliant, so he was unemployed until 1942, living with other unemployed refugees in New York. During this time he continued his mathematical researches. After this he was appointed Research Instructor at Brown University where, as part of work relevant to the war effort, he studied two-dimensional subsonic fluid flow. This was important at that time since aircraft wings were being designed for planes with jet engines capable of high speeds.

Between 1945 and 1949 Bers worked at Syracuse University, first at Assistant Professor, later as Associate Professor. Gelbart wanted to build up the department at Syracuse and attracting both Bers and Loewner was an excellent move. Here Bers began work on the problem of removability of singularities of non-linear elliptic equations. His major results in this area were announced by him at the International Congress of Mathematicians in 1950 and his paper

*Isolated singularities of minimal surfaces*was published in the

*Annals of Mathematics*in 1951. Courant writes:-

The nonparametric differential equation of minimal surfaces may be considered the most accessible significant example revealing typical qualities of solutions of non-linear partial differential equations. With a view to such a general objective, [Bers] has studied singularities, branch-points and behaviour in the large of minimal surfaces.Abikoff writes in [1] that this paper is:-

... a magnificent synthesis of complex analytic techniques which relate the different parameterisations of minimal surfaces to the representations of the potential function for subsonic flow and thereby achieves the extension across the singularity.Bers then became a member of the Institute for Advanced Study at Princeton where he began work on Teichmüller theory, pseudoanalytic functions, quasiconformal mappings and Kleinian groups. He was set in the right direction by an inequality he found in a paper of Lavrent'ev who attributed the inequality to Ahlfors. In a lecture he gave in 1986 Bers explained what happened next:-

I was in Princeton at the time. Ahlfors came to Princeton and announced a talk on quasiconformal mappings. He spoke at the University so I went there and sure enough, he proved this theorem. So I came up to him after the talk and asked him "Where did you publish it?", and he said "I didn't". "So why did Lavrent'ev credit you with it?" Ahlfors said "He probably thought I must know it and was too lazy to look it up in the literature".When Bers met Lavrent'ev three years later he asked him the same questions and, indeed, Ahlfors had been correct in guessing why Lavrent'ev had credited him. Bers continued in his 1986 lecture:-

I immediately decided that, first of all, if quasiconformal mappings lead to such powerful and beautiful results and, secondly, if it is done in this gentlemanly spirit - where you don't fight over priority - this is something that I should spend the rest of my life studying.It is ironic, given Bers strong political views on human rights, that he should find that Teichmüller, a fervent Nazi, had already made stunning contributions. In one of his papers on Teichmüller theory, Bers quotes Plutarch:-

It does not of necessity follow that, if the work delights you with its grace, the one who wrought it is worthy of your esteem.In 1951 Bers went to the Courant Institute in New York, where he was a full professor, and remained there for 13 years. During this time he wrote a number of important books and surveys on his work. He published

*Theory of pseudo-analytic functions*in 1953 which Protter, in a review, described as follows:-

The theory of pseudo-analytic functions was first announced by [Bers] in two notes. These lecture notes not only contain proofs and extensions of the results previously announced but give a self-contained and comprehensive treatment of the subject.The author sets as his goal the development of a function theory for solutions of linear, elliptic, second order partial differential equations in two independent variables (or systems of two first-order equations). One of the chief stumbling blocks in such a task is the fact that the notion of derivative is a hereditary property for analytic functions while this is clearly not the case for solutions of general second order elliptic equations.

Another classic text was

*Mathematical aspects of subsonic and transonic gas dynamics*published in 1958:-

It should be said, even though this is taken for granted by everybody in the case of Professor Bers, that the survey is masterly in its elegance and clarity.In 1958 Bers address the International Congress of Mathematicians in Edinburgh, Scotland, where he lectured on

*Spaces of Riemann surfaces*and announced a new proof of the measurable Riemann mapping theorem. In his talk Bers summarised recent work on the classical problem of moduli for compact Riemann surfaces and sketched a proof of the Teichmüller theorem characterizing extremal quasiconformal mappings. He showed that the Teichmüller space for surfaces of genus g is a (6g-6)-cell, and showed how to construct the natural complex analytic structure for the Teichmüller space.

Bers was a Guggenheim Fellow in 1959-60, and a Fulbright Fellow in the same academic year. From 1959 until he left the Courant Institute in 1964, Bers was Chairman of the Graduate Department of Mathematics.

In 1964 Bers went to Columbia University where he was to remain until he retired in 1984. He was chairman of the department from 1972 to 1975. He was appointed Davies Professor of Mathematics in 1972, becoming Emeritus Davies Professor of Mathematics in 1982. During this period Bers was Visiting Miller Research Professor at the University of California at Berkeley in 1968.

Tilla Weinstein describes in [1] Bers as a lecturer:-

Lipa's courses were irresistible. He laced his lectures with humorous asides and tasty tidbits of mathematical gossip. He presented intricate proofs with impeccable clarity, pausing dramatically at the few most critical steps, giving us a chance to think for ourselves and to worry that he might not know what to do next. Then, just as the silence got uncomfortable, he would describe the single most elegant way to complete the argument.Jane Gilman [1] describes Bers' character:-

Underneath the force of Bers' personality and vivacity was the force of his mathematics. His mathematics had a clarity and beauty that went beyond the actual results. He had a special gift for conceptualising things and placing them in the larger context.In [1] Bers life is summed up by Abikoff as follows:-

Lipa possessed a joy of life and an optimism that is difficult to find at this time and that is sorely missed. Those of us who experienced it directly have felt an obligation to pass it on. That, in addition to the beauty of his own work, is Lipa's enduring gift to us.We have yet to say something about Bers' great passion for human rights. In fact this was anything but a sideline in his life and one could consider that he devoted himself full-time to both his mathematical work and to his work as a social reformer. Perhaps his views are most clearly expressed by quoting from an address he gave in 1984 when awarded an honorary degree by the State University of New York at Stony Brook:-

By becoming a human rights activist ... you do take upon yourself certain difficult obligations. ... I believe that only a truly even-handed approach can lead to an honest, morally convincing, and effective human rights policy. A human rights activist who hates and fears communism must also care about the human rights of Latin American leftists. A human rights activist who sympathises with the revolutionary movement in Latin America must also be concerned about human rights abuses in Cuba and Nicaragua. A devout Muslim must also care about human rights of the Bahai in Iran and of the small Jewish community in Syria, while a Jew devoted to Israel must also worry about the human rights of Palestinian Arabs. And we American citizens must be particularly sensitive to human rights violations for which our government is directly or indirectly responsible, as well as to the human rights violations that occur in our own country, as they do.Bers received many honours for his contributions in addition to those we have mentioned above. He was elected to the American Academy of Arts and Sciences, to the Finnish Academy of Sciences, and to the American Philosophical Society. He served the American Mathematical Society in several capacities, particularly as Vice-President (1963-65) and as President (1975-77). The American Mathematical Society awarded him their Steele Prize in 1975. He received the New York Mayor's award in Science and Technology in 1985. He was an honorary life member of the New York Academy of Sciences, and of the London Mathematical Society.

### References (show)

- W Abikoff, C Corillon, I Kra, T Weinstein and J Gilman, Remembering Lipman Bers,
*Notices Amer. Math. Soc.***42**(1995), 8-25. http://www.ams.org/notices/199501/bers.pdf - D J Albers and C Reid, An interview with Lipman Bers,
*College Math. J.***18**(4) (1987), 266-290. - L Bers, The migration of European mathematicians to America, in
*A century of mathematics in America***I**(Providence, RI, 1988), 231-243. - I Kra and B Maskit, Lipman Bers, complex analyst, in
*Lipa's legacy, New York, 1995*(Providence, RI, 1997), 389-415. - L Nirenberg, Louis Lipman Bers and partial differential equations, in
*Lipa's legacy, New York, 1995*(Providence, RI, 1997), 455-461.

### Additional Resources (show)

Other websites about Lipman Bers:

### Honours (show)

Honours awarded to Lipman Bers

Written by J J O'Connor and E F Robertson

Last Update April 2002

Last Update April 2002