Loewner's torus inequality can be written as follows:
View an introduction pdf html to Loewner's torus inequality. See also the entry under Rodin.
A considerable amount has been written about Loewner's Mathematics in the biographies available in the literature and on the internet (see also item 12 below). On the human side, what one finds in the biographies is invariably brief. This side of him is perhaps as much his heritage as the Mathematics. To this day, some colleagues who knew him personally, regret that there are not more mathematicians like him nowadays. Others still remember fondly their contacts with Loewner, nearly half a century later. We aim to bring out this aspect of Loewner the man in the notes below.
1. Childhood. Originally named Karel (later Karl) Löwner, he was born into an Orthodox Jewish family, in the village of Lány, near Prague, which was at the time the capital of Bohemia. Karel's father, Sigmund (or Zygmund) Löwner, owned a general store in Lány which supplied the needs of all the villagers, as well as those of the local Count, whose castle dominated the village. Inspite of this, the family was on a strict budget, as the goods were sold at a very small markup. Thus caring for others and helping them, the elder Löwner planted the seeds of the middah (character trait) of rachmanus (see below) that would later blossom in his offspring. The middah of rachmanus can be loosely translated as concern for others, pity, or lovingkindness; see gemara Talmud Bavli, maseches Beitzah, page 32b; as well as Maseches Yevamos, page 79a. Karel lost his father close to bar mitzvah age, six years prior to the onset of World War 1. In accordance with Jewish law, he said the traditional Kaddish prayer for his father every day for a year following his loss.
In line with Sigmund's wishes, Karel studied in a Prague gymnasium where German was not only taught, but was also the language of instruction. The attendant infringement upon traditional Jewish learning, may have seemed a small price to pay for the entry ticket to Europe's most refined ivory towers. Sigmund's infatuation with all things German has been noted by biographers, see editor's introduction to [Lo88].
2. The Holocaust. Whether or not the infatuation went sour, a generation later, as many as four of Sigmund's children were destined to perish in the Holocaust. Meanwhile, his son Viktor (Karel's only sibling to survive the war) married out (well before the war), fabricated a genealogical tree (perhaps to save his life) purporting to prove that his father's family was not Jewish in the first place, and switched to an unrelated surname (derived from the name of his hometown). A helpful website on mathematicians under the Third Reich is maintained by Prof. Thomas Huckle at Munich, detailing also Karel's narrow escape from the Nazis' clutches. See also Beryl Wein's highly informative history books for background, root causes, and lessons of the Holocaust.
Karel Löwner held several positions in Germany in the 1920's, and later in Prague, before fleeing for his life from the Nazis. According to his curriculum vitae from 1939, his students in Europe were E. Lammel, F. Kraus, O. Dobsch, and Ch. Bers. The latter is Lipman Bers, destined to become his colleague, neighbor, and ultimately editor [Lo88].
3. Berlin. We will spare the reader the litany of mathematical celebrities who surrounded Löwner at Berlin University, typically found in internet McBiographies. A name conspicuous through its absence from such lists is that of Ludwig Bieberbach (see below). It needs to be mentioned that among Löwner's colleagues in Berlin in the 1920's was Albert Einstein, who played a pivotal role in securing his first job in the US, see item 7. Bieberbach was a major influence on Löwner. He may have been the source of the invitation for Löwner to leave his central European university, where he defended his thesis under Georg Pick in 1917, and come to Berlin in 1922. Löwner's famous proof of the Bieberbach conjecture in the first highly nontrivial case, that of the third coefficient, was published the following year~\cite{Lo23}. The manuscript was sent to Bieberbach for confirmation when it was submitted for publication. In accepting the work, Bieberbach wrote in the margin that it was an outstanding contribution. Löwner's Habilitation was mainly reviewed by Bieberbach in 1923 \cite [pp. 203, 367]{Bie}.
4. Aryan versus Jew. Bieberbach's reputation as a mathematician is solid. Alas, he was also a Nazi sympathizer, capable of arriving to a university lecture while sporting a brownshirt uniform, of harrassing his Jewish colleagues during the 1930's, as well as of pursuing a sinister thesis concerning the existence of a dichotomy of Aryan versus Jewish Mathematics. Ostensibly, his dichotomy centers on purely didactic and pedagogical qualities; yet the Aryan aversion to the Jewish trait of rachmanus (pity; see item 1 above) is well-known (see Pomerantz ). The disappointment with Bieberbach is particularly acute for those who wish to believe that toiling in Mathematical pursuits provides a vaccine of sorts against indecency \cite[vol.~2, p.~267, lines~19-25]{Goodman}. As a reaction against Bieberbach's indecency (see item 12), his name is routinely doctored out of lists of mathematical celebrities at Berlin University during Löwner's employment there. The name only appears as an afterthought at the end of Bers' introductory essay in [Lo88], in the context of a brief discussion of his conjecture.
5. Paradox. Yet, there is a bit of a paradox in attempting to
obscure Bieberbach's mathematical influence on fellow mathematicians.
The fact is that O. Teichmuller's name has been canonized in the
name of the space covering the moduli space of Riemann surfaces,
inspite of his Nazi sympathies (which were even more extreme than
Bieberbach's). As it happens, Bers himself actually took part in such
a canonisation, although the author of these lines personally heard
him express regrets about his role in it, in a lecture at Columbia
University in the 1980's.
Louis de Branges succeeded in proving Bieberbach's 1916 conjecture,
over half a century later \cite{de}. He feels that Löwner would
not have undertaken the extremely difficult project of carrying out
the necessary estimates for the case of the third coefficent, were it
not for the guiding and directing force of Bieberbach's intuition. De
Branges further wrote in the fall of 2006:
[Löwner] uses an original construction which was not
appreciated by his colleagues, and indeed is not even cited in the
1978 book by H. Grunsky \cite{Gru}. No one seems to know how
Löwner had the confidence to undertake the proof of what was
generally considered to be a wild and improbable conjecture by a
mathematician who was noted for imprecision in his publications.
An introduction to the proof of the conjecture may be
found in~\cite{Kor}.
6. Emigration. In an effort to leave the continent and escape
its Nazi menace, Löwner applied for university positions both in
England and in America. In a testimony to a trait of decency and
unselfishness that will have characterized an entire lifetime, Richard
von Mises wrote from Istanbul, in a 1939 letter of recommendation for
Löwner:
During his activity at the University of Berlin, [Löwner] was,
among all the instructors in Mathematics, the one who had the
strongest influence upon the students, stimulated them to independent
research, and helped them in his unselfish way. Much of the work
involved in the published theses of his pupils is not only due to his
influence, but can in a true sense be considered as his work.
See also item 11 below.
7. America. Loewner's first job in the United States, already under the suitably Americanized spelling of the surname sans umlaut, was at the University of Louisville, Kentucky, arranged by John von Neumann in 1939. However, the deal was not clinched until Loewner's former colleague Einstein agreed to transfer an offprint of the original edition of his renowned work on relativity theory~\cite{Ein} to a lawyer and collector at the University of Louisville named W. Bullitt. The autographed offprint is currently in the Bullitt collection of the University of Louisville~\cite{Da}. Loewner's phenomenal generosity immediately made itself felt in Louisville. Thus, he conceded to a request by undergraduate students to offer an advanced Mathematics course, which had to be taught without remuneration. The only available location turned out to be the local beer brewery. Loewner did in fact teach his course there, at a vigorous 7 a.m. time slot, before the arrival of the first shift of brewery workers.
8. Character. Loewner's daughter recalls that one of his main character strengths was the ability to listen to people and empathize with their viewpoint, even when disagreeing with them. This sometimes led people of many different persuasions to assume, by projection, that he was in agreement with them, whether or not this may have been the case. At the same time, he would always seek out a point of agreement, and back them up on it. This may have been one of the secrets of his popularity, and certainly helped navigate the political tensions of the postwar decade.
9. Syracuse. After a period of employment at Brown University during World War 2, Loewner obtained a permanent position at Syracuse University. One of his students here was P.M. Pu. When a nephew, Paul Gráf, a Holocaust survivor, arrived from Europe in 1947, the Loewners not only welcomed him into their home, but adopted him.
For a time, it seemed as though Syracuse would become one of the leading research departments in Mathematics. However, certain tensions arose within the department, which may have been exacerbated by the parallel political tensions of the decade. Some of the faculty were called in for questioning by the House of Representatives' Committee on Un-American Activities (HUAC). Certain appointments were made against the wishes of the star mathematicians in the department. In the end, many of the star faculty ended up leaving Syracuse. Loewner left for Stanford University, where he would stay for the rest of his life.
10. Identity. Though unobservant as an adult, Loewner "deeply felt his Jewish identity" [Lo88, p. ix, line 18]. "A person of Jewish faith", as Loewner is revealingly described by his colleague A. Gelbart \cite[p.~1571]{Gelb} in his testimony before the HUAC, he was a decent man and baal rachmanus (someone possessing the trait of rachmanus, see item 1 above), by all accounts.
A telling circumstance is that, inspite of professional links with leftist academics, Loewner himself was never cross-examined by the HUAC. This circumstance suggests that, contrary to an editorial claim in [Lo88, p. ix, line 20], he was a largely apolitical man, as well. Artfully planted in the midst of a rhetorical flourish, there is a claim in [Lo88, p. ix, line 20] that Loewner was "a man of the left". Yet, Loewner himself would have likely shunned the Procrustean bed of the left-right political dichotomy, his sympathy for H. Wallace's ideas, and his appeal for clemency for E. and J. Rosenberg, notwithstanding.
The aforementioned editorial claim in [Lo88, p. ix, line 20] appears to be a personal projection by an admiring student, see item 8 above. Indeed, the teacher, unlike the student, never called himself a menshevik, \cf \cite[p.~20]{ReidC}, \cite{AR}. Nor is there any evidence that Loewner shared Bers' sense of belonging to a strain of marxism euphemistically described as "the [J.] Martov tradition", in an interview Bers gave to Constance Reid, of Hilbert--Courant \cite{Re} fame, see~\cite[p.~21]{ReidC}. While the aforementioned editorial claim has been recycled by internet biographies (of both men), one of his colleagues at the time feels today that Loewner was never engaged politically.
11. Students. Loewner felt particular resposibility toward students who were not among his strongest ones. His personal investment in his students was proverbial; a colleague of his reports, only partly in jest, that "it became generally known that the quality of his students' dissertations tended to vary inversely as the quality of the student". Some of his students' theses were almost entirely Loewner's own work, in America as in Europe, see item 6 above. With consistent generosity, the Loewners always had Charles' graduate students for dinner on holidays, while his wife Elisabeth (née Alexander) was still alive. He lost her in 1956. Their daughter recalls that her father insisted on always keeping the only telephone in the house, in his bedroom, "so as to be able to discuss his students' and colleagues' problems whenever they called, even after he had gone to bed". When a Chinese student found himself without suitable housing, Loewner took him into his home in Syracuse for a while.
12. Mathematics. One of Loewner's central scientific contributions is his proof of the case n=3 of the celebrated Bieberbach conjecture, exploiting what is known today as the Loewner differential equation. The latter turned out to be instrumental in the proof of the conjecture~\cite{de}, see item 3. The stochastic Loewner equation provided the inspiration for the work of Wendelin Werner, a 2006 Fields medalist. Two of Loewner's most prolific students, Adriano Garsia and Lipman Bers, have each on the order of a hundred publications, in widely divergent fields. Both have started their own schools, each numbering dozens of students, \cf \cite{Ga}. Loewner's seminal role in systolic geometry is discussed in the introduction. As early as 1955, Bers urged Loewner to publish a book based on his course at the University of California at Berkeley, and even proposed a publisher. The fact that the book \cite{Loe} was only completed posthumously, is surely a reflection of Loewner's dedication to his students, whom he continued to supervise even after retiring officially from Stanford. In addition to his published articles, invariably path-breaking, a lot of his work was published, with his encouragement, by his students, for which he never claimed credit. Thus, it would be difficult to gauge the full extent of his influence as one of last century's great mathematicians. But first and foremost, he was a decent man and a baal rachmanus, see items 4 and 10.
13. Additional historical remarks may be found in the entry under Berger.
14. Loewner prize. In the hallowed tradition of
P. Erdös, we are hereby offering a prize of $50 for
a proof of the conjecture that the surface of genus 3 satisfies
the Loewner inequality (see above). For European contestants only,
the prize is upped to €50.
Weigh also other
prizes.