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Karl Löwner and His Student Lipman Bers - Pre-war Prague Mathematicians

Martina Bečvářová and Ivan Netuka
European Mathematical Society
Publication Date: 
Number of Pages: 
Heritage of European Mathematics 10
[Reviewed by
Jeff Ibbotson
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An old adage says that a traditional Chinese curse is “May you live in interesting times”. Karl Löwner (or Charles Loewner, as he became known in this country) and Lipman Bers clearly lived in interesting times. Bers was born in Latvia in 1914 and Löwner, his doctoral advisor, in 1893 in Prague, in the Czech Republic. Each lived through the turbulent times of the Nazi Party and the Soviet Socialist Party. Each was Jewish but not particularly religiously observant and each eventually fled the oppression created by the Nazis. The path they took ended in the USA after many perilous twists and turns and formed an important part of the development of the mathematics community in post WW2 times.

The bulk of this book is devoted to Karl Löwner. His claim to fame is founded largely upon his solution of the \(n = 3\) case of the Bieberbach Conjecture as well as his work on monotone matrix (and operator) functions and in interpolation. Much of his work on monotone operators is connected with the development of Lie semigroups for infinite-dimensional vector spaces. In fact, a quote from J. D. Lawson (p. 159) indicates that “He [Löwner] thus appears to be deserving of being called the father of the Lie theory of semigroups.” The early chapters document Löwner’s early mathematical education through the Staatsgymnasium and the German University of Prague and the beginnings of his teaching career. Throughout he appears a staid but solid scholar whose first dedication is to his mathematics.

The chapters following feature lengthy and detailed descriptions of his work in geometric function theory. Much of the writing is in German; I found the tendency of the authors to begin sentences in English and end them in (quoted) German a bit distracting. Nonetheless, the subject matter is beautiful and will appeal to any devotee of classical complex analysis. In following chapters we are also treated to a (shorter) presentation of his work on matrix functions, complex interpolation and fluid dynamics. There are also lists of Löwner’s students and their dissertations, public lectures delivered by Löwner and reviews of his work . Only the last 50 or so pages are devoted to Lipman Bers and his own relationship to Löwner. Bers’s doctoral dissertation, long considered lost, was found by his son in 2006 and is here published for the first time. It involves the Dirichlet problem in complex potential theory.

Perhaps the most exciting part of both stories is told in the footnotes (which are quite extensive). We are treated to numerous elements of the story behind each man’s (and his family’s) escape from occupied Europe and journey to America. J. E. Littlewood attempts to garner a position for Löwner at Cambridge only to have the plans scuttled at the last moment by Germany’s invasion of Poland and Britain’s declaration of war on Germany. Löwner is arrested for failing to complete paperwork correctly and thrown in jail. Einstein and von Neumann both write letters on his behalf and finally he finds a position at the Louisville University Speed Scientific school teaching elementary mathematics. He and his wife are able to leave Europe but his first teaching of advanced level mathematics in this country is done at a Kentucky Brewery before the arrival of the morning shift!

Bers faced similar hardships in his early career in the USA but both men were eventually “saved” by two important institutions — Brown University and its Advanced Research and Instruction in Mechanics Program, and Syracuse University. At Brown both men were put to work on applied mathematics related to the war effort. In fact, the list of names there is an astounding testimony to the ability of this program to rescue some of Europe’s foremost mathematicians and scientists and put them to work helping to build the USA’s prominent post-war dominance in those fields. Names like Feller, Prager, Bergman, Friedrichs, Hurewicz, Murnaghan, Sokolnikoff, von Mises and Zygmund certainly made me sit up and take note! Syracuse University, as well, played a large role in bringing such powerhouses as Selberg, Samelson, Halmos, Erdős (as well as, of course, both Löwner and Bers) and others to prominence on the post-war scene for American Universities. It served as a critical appointment to both men for 4 or 5 years. There they lived in the same house, sharing family life together as much as they shared mathematical ideas. From there, Löwner went to Stanford and a position he held until his death. Bers went to the Institute for Advanced Study, Courant Institute, and Columbia University.

This commemorative book is really only a sketch of the lives of two important mathematicians of the 20th century. The book is full of great detail concerning the outward facts of each man’s life and career but it is not until the last chapter, on Bers, that we encounter writing that conveys more of the human feeling behind the choices they made. Bers’s lifelong activism is mentioned at some length, but the true effect of the man is not conveyed in terms which will captivate the reader. Given the fact that the AMS is publishing a collection of essays regarding his life and activity, there is every hope that more is coming regarding this aspect. This book will be of use to historians of mathematics, especially those looking to write about Karl Löwner and his legacy. Those looking for the human dimensions of the story might wish to look elsewhere.

Jeff Ibbotson is the Smith Teaching Chair at Phillips Exeter Academy. He spends much of his time reading, playing ping pong and raising beagles.

  • Brief overview of Karl Löwner's life
  • Karl Löwner: Distinguished teacher and scholar
  • Publications of Karl Löwner before 1939
  • Bibliography of Karl Löwner
  • List of reviews by Karl Löwner
  • Lecture courses and seminars attended by Karl Löwner
  • Lecture courses and seminars delivered by Karl Löwner
  • Karl Löwner and dissertations at the German University in Prague
  • Karl Löwner lectures to the mathematical community
  • Brief overview of Lipman Bers' life
  • Lipman Bers: The final doctoral student of Löwner in Prague
  • L. Bers' dissertation on potential theory
  • Harmonisches Maß im Raume
  • Karl Löwner and Lipman Bers according to Marian Tracy's memory
  • Karl Löwner and Lipman Bers: Ruth Bers Shapiro recalls their friendship
  • Name index
  • Subject index