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

This monograph is devoted to two distinguished mathematicians, Karl Lowner (1893-1968) and Lipman Bers (1914-1993), whose lives are dramatically interlinked with key historical events of the 20th century. Karl Lowner, Professor of Mathematics at the German University in Prague (Czechoslovakia), was dismissed from his position because he was a Jew, and emigrated to the USA in 1939 (where he changed his name to Charles Loewner). Earlier, he had published several outstanding papers in complex analysis and a masterpiece on matrix functions. In particular, his groundbreaking parametric method in geometric function theory from 1923, which led to Lowner's celebrated differential equation, brought him worldwide fame and turned out to be a cornerstone in de Branges' proof of the Bieberbach conjecture. Unexpectedly, Lowner's differential equation has gained recent prominence with the introduction of a conformally invariant stochastic process called stochastic Loewner evolution (SLE) by O. Schramm in 2000. SLE features in two Fields Medal citations from 2006 and 2010. Lipman Bers was the final Prague Ph.D. student of Lowner. His dissertation on potential theory (1938), completed shortly before his emigration and long thought to be irretrievably lost, was found in 2006. It is made accessible here for the first time, with an extensive commentary, to the mathematical community. This monograph presents an in-depth account of the lives of both mathematicians, with special emphasis on the pre-war period. Lowner's teaching activities and professional achievements are presented in the context of the prevailing complex political situation and against the background of the wider development of mathematics in Europe. Each of his publications is accompanied by an extensive commentary, tracing the origin and motivation of the problem studied, and describing the state-of-art at the time of the corresponding mathematical field. Special attention is paid to the impact of the results obtained and to the later development of the underlying ideas, thus connecting Lowner's achievements to current research activity. The text is based on an extensive archival search, and most of the archival findings appear here for the first time.