# Bers, Lipman

# BERS, LIPMAN

Riga, Latvia, 22 May 1914; *d*. New Rochelle, New York, 29 October 1993)

*mathematics, complex analysis, partial differential equations*.

The central theme of Bers’s work was the theory of complex analytic functions, which are essentially infinitely long complex polynomials. In his early work he most often used complex analytic techniques, in the guise of partial differential equations, to study physical problems. Later, in the body of work he assessed as his most important, he and Lars Ahlfors further developed the theory of quasiconformal mappings and related areas. They then used these mappings and the spaces they define to solve Riemann’s problem of moduli, initiated the modern theory of Kleinian groups, and participated in the foundational work on iterations of rational functions.

**Family and Early Life.** The members of Bers’s immediate family were all intellectuals. Both his parents were educators in the Yiddish language school system in Riga, Latvia. His wife, Mary Kagan Bers, taught disabled children. In the early 2000s their daughter, Ruth, was a psychoanalyst and professor emeritus of psychology at the City University of New York; and their son, Victor, was a professor of classics at Yale University.

Bers, known informally to all as Lipa, spent his early years in Petrograd during the Russian Revolution and was raised in Riga and Berlin between the world wars. Even from his youth, he mixed his love for mathematics with political and social involvement. After a short period of study at the University of Zurich in Switzerland, he returned to Riga. His political activism—Latvia was ruled by a right-wing dictator—led to an arrest warrant. He fled to Prague, Czechoslovakia, where he continued his graduate studies under the direction of Charles Loewner. His dissertation was submitted, in some haste, to Charles University in Prague in 1938; he then moved to France just ahead of the Nazi takeover of Czechoslovakia. His dissertation was never published. In Paris, he worked on Green’s functions and integral representations.

**From Europe to America.** He and Mary left Paris in 1940 ahead of the German invasion of France and, after a stay in the unoccupied part of that country, departed war-torn Europe for New York City. There, they were welcomed by his mother, then a psychoanalyst, and his stepfather, Benno Tumarin, later a theatrical director and teacher at the Julliard School in Manhattan.

After a period of living as a refugee in New York, Bers joined the Advanced Research and Instruction in Applied Mathematics Program at Brown University, in Providence, Rhode Island, during 1942. At Brown he was involved in war-related research; in particular, he started his work on two-dimensional subsonic fluid flow—to laypersons, the study of what allows planes to fly. With respect to a suitable choice of coordinates, the potential *u* for the flow is the real part of a complex function *f = u + iv*. The equations satisfied by *u* and *v* are generalizations of the Cauchy-Riemann equations that characterize complex analytic functions. A direct outgrowth of this work

was the development, paralleled by Ilia Vekua in the Soviet Union, of the theory of classes of pseudo-analytic functions. Each class is defined by replacing the numbers 1 and *i*, in the definition of *f* given above, by suitable functions. There is a Cauchy theory for these classes. Much of Bers’s work during World War II was not published immediately and appeared, if at all, after the end of the war.

**Minimal Surfaces and Quasiconformality.** In 1945, Bers moved to Syracuse University in New York State. About that time, he started to work on removability of singularities of the solutions to nonlinear elliptic partial differential equations. In particular, he worked on the minimal surface equation. These are the equations satisfied by a soap film or bubble, although there are other types of solutions. Bers’s first widely recognized scientific result was the proof that the equation has no isolated singu larities. One physical interpretation of the nonexistence of singularities is that a pinprick will destroy a soap bubble. The technique used in the proof is to continually reparamatrize the surface so that the singularity becomes removable. This is the same technique later used in particle physics to “gauge away singularities.” This work led to his first address to the International Congress of Mathematicians in 1950 at Cambridge, Massachusetts.

While he was staying at the Institute for Advanced Study from 1949 to 1951, his interests started to move from partial differential equations to geometric analysis— the area of overlap between geometry and such analytic techniques as partial differential equations and complex analysis. His interest quickly settled on the study of quasiconformal mappings of surfaces and, in particular, of planar regions. These are mappings that do not distort shape too much; they take infinitesimal circles into infinitesimal ellipses of uniformly bounded eccentricity. The basic notion of a quasiconformal map was first given by Helmut Grötzsch. He proved that, among all maps of a rectangle onto another which sends vertices to corresponding vertices, the “best” map is the affine stretching of the first rectangle to fit over the second without overlap.

Around 1940, Oswald Teichmüller extended the result of Grötzsch to more general Riemann surfaces, such as donuts and pretzels. He showed that one could piece together affine stretchings to achieve a map between almost any two (topologically finite) Riemann surfaces which may be distorted one onto the other. The restriction is that punctures (that is, missing points) cannot be stretched into holes, which are not punctures, and vice versa. Indeed, he showed that the piecewise affine stretches are the best maps between two given surfaces. He also showed that the best maps are unique. These two results are known as Teichmüller’s existence and uniqueness theorems. The proofs were not widely accepted until Ahlfors produced technically formidable proofs in 1954. At roughly the same time, Bers’s work on quasiconformal mappings began appearing.,

**Teichmüller Theory and Moduli.** It was, however, the proof of the Teichmüller theorems produced by Bers in 1958 that led directly to the later theory. In the late 1950s, Ahlfors and Bers worked in close contact on problems related to the geometric implications of quasiconformal mappings in both surface theory and complex analysis. In their only joint paper, entitled “Riemann’s mapping theorem for variable metrics,” which appeared in the *Annals of Mathematics* in 1960, they gave a new proof of (as it later became known) the measurable Riemann mapping theorem, or, in the original terminology, the Riemann mapping theorem for variable metrics. In his 1958 lecture at the International Congress of Mathematicians in Edinburgh, Scotland, Bers announced that work and many of its consequences—including the solution to Riemann’s problem of moduli. A short description, of both the moduli problem and the Ahlfors-Bers solution, follows.

Much of Riemann’s early work was devoted to the study of solution sets of algebraic equations in two variables, i.e. the pairs of points *(z,w)* at which a given polynomial *P(z,w)* =0. That solution set is called the Riemann surface of the algebraic function *P*. Without focusing on details here, it is important to note that Riemann gave an interesting count of the dimension of the space of Riemann surfaces *S* that are topologically equivalent to *S* but are not the same (holomorphically equivalent) as *S*. Justifying that count is called “Riemann’s problem of moduli.”

Planar sets may be described as those that may be globally parametrized using the complex variables *z* and On an infinitesimal level, arc length may be measured as

The quantity, or more precisely the form, *ds*^{2} is called a Riemannian metric. The Euclidean metric is simply *ǀdzǀ*^{2}.

The function, more precisely the form, *Λ(z)*, is a pointwise rescaling of the metric and does not, on the infinitesimal level, distort Euclidean circles. Any distortion of such circles, arising by changing the metric, is carried by the form *µ(z)*, which is called a Beltrami coefficient. A mapping of plane domains or Riemann surfaces is effected by a quasiconformal mapping that satisfies the differential equation

This is called the Beltrami equation. The existence of solutions of this equation, under successively weaker conditions on *µ*, goes back to Carl Friedrich Gauss. The key contribution of Ahlfors and Bers was to show that, if *µ* depends on parameters, then so does *w*—indeed, *w* depends on the parameters to the best extent possible— and to realize the implications of this fact.

Bers used the solution of the Beltrami equation to prove the Teichmüller theorems. In a one and one-half page paper, entitled “Correction to Spaces of Riemann Surfaces as Bounded Domains” which appeared in the Bulletin of the American Mathematical Society in 1962 Bers—correcting an earlier error—produced what he considered to be his finest work. He embedded Teichmüller’s space of deformations of Riemann surfaces into *N*-dimensional complex space ** C^{N}** . The embedding uses the solution of the Beltrami equation together with classical notions from complex analysis such as Schwarzian derivatives and quadratic differentials and gives an elegant solution to the moduli problem of Riemann.

**Later Research and Academic Positions.** In 1951, Bers became a professor at the Courant Institute of New York University in New York City. He was considered one of the legendary teachers of mathematics. His lecture notes, informally published by that institute, helped set the direction of modern partial differential equations and the study of complex analysis on surfaces and in several variables.

In 1964, Bers moved to Columbia University, also in New York City, and then started working on Kleinian groups, the groups of motions of hyperbolic 3-space **H** ^{3}.

Again, his methods were complex analytic—he studied the groups through their action on the sphere at infinity for**H** ^{3}. He did some of the earliest work on the Eichler cohomology for Kleinian groups and found sharp bounds on the area of the conformal boundary of hyperbolic 3-orbifolds. He gave an analytic proof of the Nielsen-Thurston classification of homeomorphisms of surfaces. Late in his career, he worked on both Kleinian groups and the parallel theory of iteration of rational functions and studied their common roots. After retiring from Columbia in 1984, he was Distinguished Professor at the Graduate Center of the City University of New York. During Bers’s career he supervised forty-eight PhD dissertations, and many of his students carried forward his mathematical legacy.

**Personality, Vision, and Commitments.** Throughout much of his career, Bers was in gentlemanly collaboration and competition with Ahlfors—a typical contest, at a given conference, might have been who had the most students in attendance.

Bers had a broad knowledge and vision of mathematics—it just seemed to come naturally to him. Marc Kac wrote of a fascinating technical question:

I first heard the problem posed this way some ten years ago from Professor Bochner. Much more recently, when I mentioned it to Professor Bers, he said, almost at once: “You mean, if you had perfect pitch could you find the shape of a drum.” (Kac, p. 3)

His was an active presence in defining the societal role of scientists, in particular, mathematicians. Among his other activities in the mathematics community, he was active in the American Mathematical Society from 1957 and was its president in 1975–1976. He expressed his pride at being elected to the American Philosophical Society—he took great delight in that select group especially since it had been founded by Benjamin Franklin, whom he much admired. He was the 1971 Colloquium Lecturer for the American Mathematical Society as well as the first G. H. Hardy Lecturer of the Royal Society in 1967.

It is not only as a researcher that Bers is renowned. He was deeply committed to human rights, helping to found the National Academy of Sciences (U.S.A.) Committee on Human Rights and the American Mathematical Society’s Committee on Human Rights of Mathematicians.

## BIBLIOGRAPHY

### WORKS BY BERS

With Ahfors, L.V. “Riemann’s mapping theorem for variable metrics,” Annals of Math. 72 (1960): 345–404.

“Correction to ‘Spaces of Riemann surfaces as Bounded Domains.” *Bulletin of the American Mathematical Society* 67 (1961): 465–466.

Kra, Irwin, and Bernard Maskit, eds. *Selected Works of LipmanBers: Papers on Complex Analysis*. 2 vols. Providence, RI: American Mathematical Society, 1998. This is a collection of Bers’s papers on complex analysis that Bers asked Kra and Maskit to edit. Additionally, it contains a list of his students and the texts of articles by Kra and Maskit, William Abikoff, and Frederick Gardiner and Linda Keen.

### OTHER SOURCES

Abikoff, William. “Remembering Lipman Bers.” *Notices of theAmerican Mathematical Society* 42 (1995): 8–25. This obituary includes contributions by Cathleen S. Morawetz, Carol Corillon and Irwin Kra, Tilla Weinstein, and Jane Gilman. It also contains a bibliography listing videotapes of some of Bers’s lectures as well as interviews with him and his survey articles about his work.

Bass, Hyman, and Irwin Kra. “Lipman Bers: May 22, 1914–October 29, 1933: A Biographical Memoir.” *Proceedings of the American Philosophical Society* 140 (1966): 206–219.

Kac, Marc. “Can one hear the shape of a drum?” *AmericanMathematical Monthly* 73:4 (1966): 1–23.

Keen, Linda. “Lipman Bers: A Mathematical Mentor.” *AWM* [Association for Women in Mathematics] *Newsletter*, July 1984.

O’Connor, J. J., and E. F. Robertson. “Lipman Bers.” Available from http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Bers.html

*William Abikoff*

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