**Oswald Teichmüller**'s parents were Gertrude Dinse and Adolf Julius Paul Teichmüller. Although Oswald was born in Nordhausen im Harz, his parent's home was in the small village of Sankt Andreasberg, famed as a winter sports resort, about 25 km north of Nordhausen. When Oswald was only a few days old his mother took him from Nordhausen to their family home in Sankt Andreasberg. Paul Teichmüller, Oswald's father, was a weaver. He was 33 years old when Oswald was born and his wife Gertrude was 39 years old. When Oswald was one year old World War I broke out and in 1915 Paul Teichmüller's weaving business closed when he left home to fight in the war. At some stage during the war Paul Teichmüller was wounded but he did not return to Sankt Andreasberg until the war ended in 1918. At this stage he opened his weaving business again but, when Oswald was twelve years old, his father died.

Segal writes in [4] about Teichmüller's early talents:-

According to his mother, when Teichmüller was three-and-a-half, she discovered that he knew how to count, and he also learned to read on his own, his first self-instruction being from labels on tin cans. When his father returned in1918, young Oswald read to him fluently and recited a poem previously unknown to his mother.

Teichmüller attended the local school in Sankt Andreasberg until the death of his father. At this time he was twelve years old and was far too advanced to be getting anything useful from the lessons at the local school. Gertrude took him to Nordhausen, having arranged for him to live there with his aunt. He attended the Gymnasium in Nordhausen until he was seventeen years old, then entered the University of Göttingen in the summer semester of 1931 to study mathematics. He was certainly not a typical student; his mathematical work was brilliant but he had few friends and appeared to be an ungainly, uncomfortable country lad who was a complete misfit in the Göttingen scene [4]:-

Peter Scherk, a student at the time, and Hans Lewy, a young instructor, both of whom would become well-known mathematicians(and both forced emigrés in1933), told anecdotes of the ungainly student's brilliance.

After one semester studying at Göttingen, Teichmüller joined the NSDAP (Nationalsozialistiche Deutsche Arbeiter Partei) commonly called the Nazi Party. Three weeks later he joined the SA (Sturm Abteilung: Storm Section, the military wing of the Nazi Party). Although support for the Nazi Party at this time was not large, it was rapidly becoming stronger mainly due to the economic situation resulting from the Depression. Göttingen was, in fact, a stronghold for the Party and a large proportion of the student population were moving to give their support. It is rather difficult to see whether Teichmüller joined the Nazis because he already shared the far right views of the Party, or whether he rapidly became indoctrinated after joining. Certainly the awkward country lad without friends now found friends in the Party and he moved from being an outsider to being one of the student leaders. He soon became the deputy leader of the Mathematics and Natural Sciences branch of the student's Nazi Party.

On 2 November 1933 Teichmüller led the student boycott of Edmund Landau's lectures. As background to this we note that Hitler had come to power on 30 January 1933 and on 7 April of that year the 'Law for the reorganisation of the Civil Service' was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Before any official word reached Göttingen from the Ministry, the Dean wrote to Landau on 28 April asking him not to give his summer lecture courses and these were given instead by Landau's assistant Werner Weber. Having received no further advice from the university authorities, Landau decided to give his autumn lectures as advertised. Landau described in unemotional terms what happened on the first day of lectures (see [4]):-

On211.15November, about80, as I wished to leave my office and go to the large lecture theatre to begin my lecture, the entrance hall was filled with about100to(students who let me pass through unhindered. In the lecture hall was one person. Clearly therefore, there was a boycott with sentries at the door who had prevented)without forcethose students who wanted to work from setting foot in the lecture room.

Teichmüller, as the leader of the boycott, went to Landau's office and discussed what had happened. Landau requested that Teichmüller put his views in writing and he did so. A translation of part of this letter is given in [6]:-

Through yesterday's action a completely new situation has now been created. In order to restore peace in our institute it is necessary, above all, to clear up the fundamentals behind it. You spoke of your belief that what happened yesterday was an anti-Semitic demonstration. My standpoint was, and continues to be, that an anti-Jewish individual action might rather be directed against everyone else than against you. I am not concerned with making difficulties for you as a Jew, but only with protecting - above all - German students of the second semester from being taught differential and integral calculus by a teacher of a race quite foreign to them. I, like everyone else, do not doubt your ability to instruct suitable students of whatever origin in the purely abstract aspects of mathematics. But I know that many academic courses, especially the differential and integral calculus, have at the same time educative value, inducting the pupil not only to a conceptual world but also to a different frame of mind. But since the latter depends very substantially on the racial composition of the individual, it follows that an Aryan student should not be allowed to be trained by a Jewish teacher.

Chowdhury comments [6]:-

I find it an extraordinary piece of writing, shamelessly upholding an indefensible attitude and an ignominious action, wherein the brilliant but thoroughly indoctrinated mind of the writer shines through.

We note that Teichmüller had some success in converting others to the Nazi beliefs. For example he convinced Landau's assistant Werner Weber so that he joined the Nazi Party in 1933.

Helmut Hasse was appointed to fill Weyl's chair at Göttingen in 1934. The appointment of Hasse was not approved by certain hard-line Nazis since, although he was a strong supporter of Hitler, he did not believe in politicizing mathematics. However, despite Hasse being in a very different area of mathematics from Teichmüller, it was Hasse who Teichmüller chose as his thesis supervisor. It would have made more sense from a mathematical perspective for Teichmüller to have asked Gustav Herglotz to be his supervisor but Herglotz had no specific connection with the Nazi Party while Hasse did. Unsurprisingly Teichmüller put political considerations ahead of mathematical ones but when he handed Hasse a first draft of his doctoral thesis *Operatoren im Wachsschen Raum* Ⓣ, Hasse sent it to Manfried Köthe in Münster and it was Köthe's comments which led to the final polishing of the thesis. Teichmüller submitted his thesis on 10 June 1935 and he was examined by Hasse, Herglotz and the physicist Robert Pohl on 26 June 1935. In November he was officially awarded the degree.

After his doctoral examination in June 1935, Hasse requested that the university authorities appoint Teichmüller as an assistant in the Department. Hasse stated in his letter that Teichmüller had "extraordinary mathematical gifts" and "promises to become a mathematician of importance." As regards his teaching abilities Hasse notes that his lecturing style was "painfully exact, in high degree suggestive, and impressive sort." Teichmüller was appointed as an assistant and there is some evidence that for a while his mathematical interests became more important to him than his political ones. He held the position of Rottenführer (Team Leader with command over paramilitary troops) in the SA but some within the organization criticised him for doing little or nothing to further the Nazi cause, being too involved with mathematics. Even some of those with the same extreme Nazi views as Teichmüller described him as "disagreeable," "inexperienced in the ways of the world," and "eccentric."

Teichmüller's attitude towards Hasse was also rather strange - before Hasse's appointment he had written that Hasse was "a great algebraist," "a German nationalist," "apparently still incapable of fitting in with the regime. At present small and hateful. A call for him to come here is in the offing; we cannot approve of it." His views remained the same over the years that he worked in the Göttingen department headed by Hasse, yet Teichmüller chose him as his supervisor and was influenced mathematically by him to work on algebraic problems (considered by some Nazis to be Jewish mathematics). In October 1936 Teichmüller began to work towards habilitating in Berlin with Ludwig Bieberbach (also an outstanding mathematician and staunch Nazi sympathiser). He was publishing papers in Bieberbach's journal *Deutsche Mathematik* which was set up as a racialist publication intended to promote 'German style mathematics' as opposed to 'Jewish style mathematics'. Teichmüller's habilitation thesis *Untersuchungen über konforme und quasikonforme Abbildungen* Ⓣ was not influenced by Hasse, but rather was sparked by lectures that he had attended by Rolf Nevanlinna on complex analysis. He moved to Berlin in April 1937 and habilitated there in March 1938. It is interesting to note that in his report on Teichmüller's thesis Bieberbach chose to put in a critical comment about his earlier algebraic papers influenced by Hasse. In Berlin, Teichmüller had someone in Bieberbach who shared his extreme Nazi views. The two, however, were also exceptional mathematicians and Teichmüller's two years in Berlin were golden years in terms of his research.

Teichmüller's main contribution is in the area of geometric function theory. He wrote 34 papers in the space of about 6 years, 21 being published in *Deutsche Mathematik*, the journal for German style mathematics founded by Bieberbach which we referred to above. He introduced quasi-conformal mappings and differential geometric methods into complex analysis. K Strebel, in a review of [2], gives the following summary of Teichmüller's contributions:-

In1936Teichmüller published five papers about various algebraic topics, and three more in1937. But it was already in that same year that two papers in function theory appeared, one on value distribution and the other on the type problem, using quasiconformal mappings. He was already an expert in the Nevanlinna theory and evidently greatly influenced by Ahlfors' contributions to it.

Teichmüller's Habilitationsschrift: "Untersuchungen über konforme und quasikonforme Abbildungen"Ⓣ, which appeared in 1938, and the next paper: "Ungleichungen zwischen den Koeffizienten schlichter Funktionen"Ⓣcan be considered as the beginning of his great contributions to function theory, which culminated in his masterpiece: "Extremale quasikonforme Abbildungen und quadratische Differentiale"Ⓣ, 1939. In this paper and its complement: "Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen"Ⓣ(1943), Teichmüller laid the basis of what is now known as the theory of Teichmüller spaces. He further developed the theme in one of his last papers: "Veränderliche Riemannsche Flächen"Ⓣ(1944).

There are other things, like the extremal mappings of the pentagon(1941)or the "Verschiebungssatz"Ⓣwhere he shows with great mastery how to deal with special problems.

Some other papers on pure function theory, like "Eine Verschärfung des Dreikreisesatzes"Ⓣ, and on algebraic functions, round out the picture.

Teichmüller was drafted on 18 July 1939 as Germany prepared for World War II. He was originally supposed to do eight weeks training, but before the eight weeks were up, World War II began on 1 September 1939. He remained in the army and, in April 1940, took part in the German invasion of Norway. After this he was transferred to Berlin to undertake cryptographic work. Bieberbach requested that Teichmüller be released from military duties to lecture at the university and indeed Teichmüller was able to teach at the university beginning at the start of session 1942-43 while continuing his cryptographic work. He still found time to continue his mathematical research with five papers being published in 1944: *Über die partielle Differentiation algebraischer Funktionen nach einem Parameter und die Invarianz einer gewissen Hauptteilsystemklasse* Ⓣ; *Beweis der analytischen Abhängigkeit des konformen Moduls einer analytischen Ringflächenschar von den Parametern* Ⓣ; *Ein Verschiebungssatz der quasikonformen Abbildung* Ⓣ; *Veränderliche Riemannsche Flächen* Ⓣ; and *Einfache Beispiele zur Wertverteilungslehre* Ⓣ. The first of these appeared in Crelle's *Journal*, the other four in *Deutsche Mathematik*.

The Battle of Stalingrad raged between July 1942 and February 1943. The Germans attempted to take the city but there was stubborn resistance from the Russians. Eventually the German 6^{th} Army was tricked and, after becoming trapped, was largely destroyed. It was the first major military defeat for the German armies and a new call to arms was made across Germany. Teichmüller answered this call and, giving up his cryptographic position in Berlin, joined the forces attempting to recover from the Stalingrad defeat. The German aim was to shorten their eastern line by taking the area around Kursk where the Russian forces held positions. Teichmüller's unit took part in the offensive which began on 5 July 1943. A Russian counterattack in early August saw the Germans forced to fall back. Teichmüller was given leave of absence to return home. His unit was, at that stage, in the vicinity of Kharkov and, after a battle lasting from 3 August to 23 August, Kharkov was recaptured by the Russians. Most of Teichmüller's unit was wiped out but in early September he tried to rejoin them. He seems to have reached Poltava, southwest of Kharkov, but was killed in the confused situation as the German forces retreated in disarray before the Russian advance.

**Article by:** *J J O'Connor* and *E F Robertson*