Lars Valerian Ahlfors

Lars Ahlfors

Lars Valerian Ahlfors (born April 18, 1907 in Helsinki , † October 11, 1996 in Pittsfield , Massachusetts ) was a Finnish-American mathematician . In 1936 he was awarded the Fields Medal for special services to mathematics. Ahlfors wrote several excellent textbooks in the fields of analysis and function theory . His book Complex Analysis in particular is considered to be one of the best on function theory to this day.

life and work

Ahlfor's father was a professor of mechanical engineering at the Polytechnic Institute in Helsinki, and his mother died when he was born. The family was Swedish-speaking . In 1924 he began his mathematics studies at the University of Helsinki , with Ernst Leonard Lindelöf and Rolf Nevanlinna , which he graduated in 1928 (in the same year he accompanied Nevanlinna to the ETH Zurich ) and where he received his doctorate in 1930. In the same year he started teaching at the Swedish-speaking University (Abo Akademi) in Turku. During this time he also made several trips to Central Europe, including a. to Paris. From 1933 to 1936 he was an assistant professor in Helsinki. In 1935 he accepted a position at Harvard University , with a three-year probationary period. As early as 1936 he was honored at the International Congress of Mathematicians (ICM) in Oslo together with Jesse Douglas with one of the first Fields medals . In 1938 he was offered a professorship in mathematics at the University of Helsinki, which he accepted despite the looming Second World War . Finnish universities soon closed due to the war against the Soviet Union. Ahlfors himself had been retired from military service as unfit. In 1944 he received an offer from the University of Zurich , which he was not able to accept until 1945 due to the chaos of the war. Since he and his wife did not feel comfortable as foreigners in Switzerland so shortly after the war, he immediately accepted the offer from Harvard University in 1946, where he stayed until his retirement in 1977 (from 1964 as "William Caspar Graustein Professor" for mathematics ). From 1948 to 1950 he was head of the mathematics faculty.

After his retirement he was a. a. 1978 visiting professor at Columbia University , 1979 at the University of Michigan , 1980 at the University of Minnesota and 1983 at the University of California, San Diego .

Ahlfors gave three plenary lectures at the International Congress of Mathematicians (ICM): 1978 (Quasiconformal mappings, Teichmüller spaces and Kleinian Groups), 1936 (Geometry of Riemann surfaces), 1962 (Teichmüller Spaces). Ahlfors was elected to the National Academy of Sciences in 1953. In 1986 he was honorary president of the ICM. In 1981 he received the Wolf Prize for Mathematics.

He had been married to Erna Lehnert since 1933, who originally came from Vienna and with whom he had three daughters.

Ahlfors worked a. a. About value distribution theory in the sense of his teacher Nevanlinna, quasi-conformal images (to which he gave the name), Teichmüller theory (with Lipman Bers he was significantly involved in the strict establishment of the theoretical building by Oswald Teichmüller ), conformal geometry, meromorphic curves, Riemann surfaces and Kleinsche Groups (partly with Lipman Bers ). As early as 1929 he attracted attention when he proved a conjecture by Denjoy (which says that an entire function of order k has at most 2k finite asymptotic values). Ahlfors also dealt a lot with the type problem of non-compact Riemann surfaces, namely to specify criteria as to whether they are of the parabolic or hyperbolic type (conformally equivalent to the entire complex plane or to the unit disk). He also examined other conformal invariants, e.g. For example, he and Arne Beurling investigated the extreme length of curve families in one area. Ahlfors regarded the theorems of Picard and Bloch as special cases of the type problem and in 1935 also gave the Nevanlinnasche value distribution theory a geometric interpretation through special conformal metrics and in the same year a further geometric interpretation in his theory of overlapping surfaces (according to Constantin Carathéodory he received above all for this work the Fields Medal). In the 1960s he proved his finiteness theorem for Kleinian groups ( discrete subgroups of , the group of Möbius transformations ): finitely generated Kleinian groups represent Riemann surfaces of finite gender (compacted by adding a finite number of points). A loophole in Ahlfor's proof was closed by Bers. Klein groups also played an important role in William Thurston's program on three-dimensional hyperbolic manifolds . Before that, Ahlfors had already given a new proof (with Eichler cohomology) of the corresponding finiteness theorem for Fuchsian groups in 1964 . ${\displaystyle PSL(2,\mathbb {C} )}$

His PhD students include Dale Husemoller , Paul Garabedian , Albert Marden , Halsey Royden , Robert Osserman , George Springer, and Henry Otto Pollak .

literature

• Ahlfors “Collected Papers”, 2 volumes, Birkhäuser, 1982
• Ahlfors Complex Analysis , 1979 (first 1953)
• Ahlfors Contributions to the Theory of Riemann Surfaces , Princeton, Annals of Mathematics Studies 1953
• Ahlfors Riemann Surfaces 1960
• Ahlfors Conformal Invariants 1973
• Olli Lehto On the Life and Work of Lars Ahlfors , Mathematical Intelligencer, 1998, No. 3
• Donald J. Albers, Gerald L. Alexanderson Fascinating Mathematical People: Interviews and Memoirs , Princeton University Press 2011

Web links

Commons : Lars Valerian Ahlfors  - Collection of pictures, videos and audio files

• Literature by and about Lars Valerian Ahlfors in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Lars Valerian Ahlfors. In: MacTutor History of Mathematics archive .
• Ahlfors on the Harvard University website
• Lars Valerian Ahlfors in the Mathematics Genealogy Project (English)
• List of publications
• Frederick Gehring, Irwin Kra, Steven Krantz, Robert Osserman The Mathematics of Lars Valerian Ahlfors , Notices of the American Mathematical Society February 1998, PDF file (478 kB)
• Lars Valerian Ahlfors (1907–1996) , Notices of the American Mathematical Society, February 1998, PDF file (396 kB)
• Ahlfors Fundamental Polyhedrons and Limit Point Sets of Kleinian Groups , Proceedings of the National Academy of Sciences, Vol. 55, 1966, p. 251, online here [1]
• Gehring on Ahlfors in Biographical Memoirs National Academy of Sciences, PDF file (972 kB)
• Author profile in the database zbMATH

Remarks

1. ↑ in Ahlfors on the theory of overlapping surfaces Acta Mathematica vol. 65, 1935, p. 157.
2. ↑ C. Carathéodory, report on the awarding of the field medals, in Comptes Rendus du Congrès International des Mathématiciens, Oslo, 1936. Tome I: Procès-Verbaux et Conférences Générales ( Memento des Originals of March 7, 2014 in the Internet Archive ) Info: Der Archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. , Pp. 308-314. 
3. ↑ stands for special projective linear group , for coefficients from the complex numbers. The corresponding discrete subgroups with real coefficients are referred to as Fuchs groups ${\displaystyle PSL}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle PSL(2,\mathbb {R} )}$
4. ↑ Lars Valerian Ahlfors in the Mathematics Genealogy Project (English)