After the German defeat at Stalingrad in February 1943, he gave up his position in Berlin to volunteer for combat on the Eastern Front.

After his father's death, she took him out of his school in Sankt Andreasberg which "he had long outgrown" and sent him to live with his aunt in Nordhausen, where he attended the Gymnasium.

[5] Among Teichmüller's professors were Richard Courant, Gustav Herglotz, Edmund Landau, Otto Neugebauer and Hermann Weyl.

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.

In his letter he stated Teichmüller had "extraordinary mathematical gifts" and that his teaching style was "painfully exact, in high degree suggestive, and impressive sort."

Teichmüller received the position and began to devote himself more to mathematics at the expense of politics, which led fellow NSDAP members to describe him as "eccentric".

Teichmüller's habilitation thesis, Untersuchungen über konforme und quasikonforme Abbildungen, was not influenced by Hasse, but by the lectures of Rolf Nevanlinna, who was a visiting professor at the University of Göttingen.

In Berlin with Bieberbach, Teichmüller had someone who shared his political views and who was also an exceptional mathematician, which led to two years of great productivity.

Afterwards, he was recalled to Berlin where he became involved in cryptographic work along with other mathematicians such as Ernst Witt, Georg Aumann, Alexander Aigner and Wolfgang Franz in the Cipher Department of the High Command of the Wehrmacht.

[2] After the German defeat at Stalingrad in February 1943, however, Teichmüller left his position in Berlin and volunteered for combat on the Eastern Front, entering a unit which became involved in the Battle of Kursk.

His unit was surrounded by Soviet troops and largely wiped out by late August, but in early September he attempted to rejoin them.

After his habilitation in 1938, Teichmüller turned to questions in the variation of conformal structures on surfaces, raised earlier by Bernhard Riemann, Henri Poincaré, Felix Klein, and Robert Fricke.

His most important innovation was the introduction of quasiconformal mappings to the field, using ideas first developed by Herbert Grötzsch and Lars Ahlfors in different contexts.

Teichmüller's main conjecture stated that variation of conformal structure can be realised uniquely by extremal quasiconformal mappings.

Teichmüller's habilitation thesis: Untersuchungen über konforme und quasikonforme Abbildungen ("Studies of conformal and quasiconformal mappings"), which appeared in 1938, and the next paper: Ungleichungen zwischen den Koeffizienten schlichter Funktionen ("Inequalities between the coefficients of simple functions") can be considered as the beginning of his great contributions to function theory, which culminated in his masterpiece: Extremale quasikonforme Abbildungen und quadratische Differentiale ("Extremal quasiconformal mappings and quadratic differentials") (1939).

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