August 28, 1984
Purdue Professor Solves 68-Year-Old Math Problem
West Lafayette, Ind. A Purdue University professor has solved one of the most famous problems of mathematics.
Louis de Branges, professor of mathematics, has proved the Bieberbach conjecture, which has challenged mathematicians since it was proposed in 1916.
Allan H. Clark, dean of the School of Science, terms the discovery the equivalent in mathematics of finding a new particle in physics. "It's a stunning feat of intellectual mountain climbing--like conquering Everest. Undoubtedly it is the best piece of mathematical work to come out of Purdue."
The Bieberbach conjecture is a statement about the coefficients of power series for a function of a complex variable with certain properties. Bieberbach, himself, verified the conjecture only for the second coefficient. Others have verified the conjecture for the third, fourth, fifth, and sixth coefficients.
De Branges has shown that the conjecture is true for all coefficients. His work was confirmed by Soviet mathematicians during a visit he made this spring to the Steklov Institute in Leningrad.
"Experts on the subject are astounded by the brevity and ingenuity of de Branges' solution," said Professor Joseph Lipman, acting head of the Department of Mathematics at Purdue.
De Branges' proof depended upon deep properties of certain special functions. Together with Professor Walter Gautschi, of Purdue's Department of Computer Sciences, he was able to verify the conjecture by numerical methods for the first 30 coefficients using Purdue's Cyber 205 supercomputer. De Branges was able to complete his proof using an inequality of Richard Askey of the University of Wisconsin and George Gasper of Northwestern University.
De Branges was born in Paris, France in 1932. He was educated at St. Andrew's School of Middleton, Del., and the Massachusetts Institute of Technology. He received a doctorate in mathematics from Cornell University in 1957.
De Branges came to Purdue in 1962. He was an Alfred P. Sloan Fellow from 1963 to 1966, and a Guggenheim Fellow during 1967-68. He is the author of two mathematical monographs and more than 60 research articles.
(Note: Enclosed is a more complete summary of de Branges' work.)
One of the most famous problems of mathematics has been solved by Professor Louis de Branges of Purdue University.
The problem, known as the "Bieberbach conjecture", was formulated by the German mathematician Ludwig Bieberbach in 1916, and has defied the concentrated efforts of numerous prominent mathematicians ever since. It concerns certain transformations of the unit disc (i.e. the region bounded by a circle whose radius is one unit of length) into other planar regions. Such transformations, called "univalent functions", distort shapes, but they preserve angles between curves.
The Bieberbach conjecture has played a central role in the development of the subject of univalent functions.
Part of the attraction of the conjecture is its relatively elementary statement, easily understood by anyone who has been exposed to complex variable calculus. A point on the unit disc can be represented by a complex number z , and a univalent function f transforms the point into the point represented by the complex number f(z) given by an infinite series
f ( z ) = z + a2z2 + a3z3 + a4z4 +
where the coefficients a2 , a3 , a4 , ... are fixed complex numbers, which then specify f . "Univalence" means that two different points are never transformed into the same point. After studying particular examples, Bieberbach guessed (conjectured) that no matter which such f we consider, each coefficient an must be of absolute value at most n ; i.e. | an | <n. Further, equality (for any n ) can only hold when f has the special form
f(z) = z / (1-z2) = z + 2z2 + 3z3 + ...
or its rotations.
Bieberbach himself verified his conjecture only for the coefficient a2 . The next coefficient a3 was successfully treated in 1923 by the Czech mathematician Charles Loewner. Loewner's basic idea was to make the transformation evolve in time, according to some differential equation. This was one of the first appearances in mathematical analysis of "semi-groups," a technical tool which has found wide applicability throughout mathematics.
Not till 1955 was the conjecture verified for a4 , by Paul Garabedian and Menachem Schiffer. This was considered to be an outstanding achievement of Schiffer's "variational method," which has since become a major technique in the study of differential equations, and in mathematical physics. Both Loewner and Schiffer came to the United States as refugees in the 1940's, and played an important role in American mathematical life. Garabedian is from Cincinnati.
The conjecture was verified for a6 by R. Pederson (U.S.A.) and M. Ozawa (Japan) in 1968, and for a5 by Pederson and Schiffer in 1972. Beyond this, the question remained open.
This year de Branges showed conclusively that the conjecture is indeed true for all the coefficients. Experts in the subject are astounded by the brevity and ingenuity of de Branges' methods. His proof makes contact with the early work (referred to above) through its use of the Loewner differential equation. However, crucial new ideas are involved. De Branges states that his method should be useful for related problems in pure and applied mathematics involving optimization, or "best choices."
De Branges' ideas led him to a specific inequality involving certain special functions studied more than 150 years ago. The validity of this inequality was tested in February on a computer by Professor Walter Gautschi of Purdue. The computations were sufficient to verify the Bieberbach conjecture up to the twenty-fifth coefficient. This was an encouraging sign that the conjecture was true generally (though no computer could verify that!). The general solution was obtained in March after Gautschi contacted Richard Askey of the University of Wisconsin to see if more was known about the inequality: it turned out that Askey, together with George Gasper of Northwestern University had established it in 1976!
That Askey and Gasper should have, for entirely different reasons, studied just that inequality, was one of two remarkable coincidences which appeared in connection with the proof of the Bieberbach conjecture. Tbe otber was that de Branges was scheduled for a visit to the Soviet Union in April, May and June under the exchange agreement between the Academy of Sciences of the USA and the Academy of Sciences of the USSR.
It should be mentioned here that de Branges had obtained a stronger result than the Bieberbach conjecture. He had proved a conjecture of I. M. Milin, which implies a conjecture of M. S. Robertson, which implies the Bieberbach conjecture. Since Professor Milin was in Leningrad, it was possible to schedule a series of seminar lectures to present the proof of the Bieberbach conjecture to Milin and his students.
At the time these lectures were scheduled, few people had given serious attention to the Robertson and Milin conjectures. A climate of skepticism had been created by the erroneous proof of the Bieberbach conjecture which had been announced in a Soviet journal. In fact several false proofs of the Bieberbach conjecture litter the historical landscape; and it was the general expectation that some subtle error would be found in the present argument.
To their surprise, the participants of the Leningrad Seminar in Geometric Function Theory became convinced of the validity of the argument during five sessions which took place in April and May. Each session lasted late into the evening and was interrupted only by a break for tea. Two members of the seminar, E. V. Emel'ianov and I. M. Milin, submitted written reports confirming the proof and presenting variants which they considered advantageous. In June, Professor de Branges worked with the seminar leader, Professor G. V. Kuz'mina, to consolidate the findings of the seminar. The resulting argument was accepted for release by Academician L. D. Faddeev, the director of the Leningrad Branch of the V. A. Steklov Mathematical Institute. The 21 page preprint is available both in Russian and in English.
In July, de Branges presented the proof of the Bieberbach conjecture at universities in Wurzburg, Hannover, and Amsterdam. News of the proof quickly spread to the world mathematical community. Many experts have now confirmed the correctness of the argument presented in the preprint. De Branges is now preparing a final version of the argument for journal publication.
Louis de Branges de Bourcia was born in Paris, France, on Aug. 21, 1932. He came to the United States with his mother and two sisters in 1941. He was educated at St. Andrew's School of Middletown, Del., and the Massachusetts Institute of Technology, and at Cornell University, where he received a doctorate in mathematics in 1957. His thesis advisor was Professor Harry Pollard, now at Purdue University. De Branges taught at Lafayette College, Easton, Pa., and at Bryn Mawr College, before coming to Purdue as an associate professor in 1962. He was promoted to professor in 1963. He was an Alfred P. Sloan Fellow from 1963 to 1966 and a Guggenheim Fellow during 1967-68. He is the author of two mathematical mongraphs and more than sixty research articles in mathematical journals.
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