Robert's Mistakes

Oswald Teichmüller, the Nazi maths genius

Feb 15, 2014

John Huerta called for posts about mathematicians, like this:

The idea is to fill Google+ with math. Whoever +1's this post will receive a mathematician's name and will have to post a theorem (or lemma, proposition, etc) by that mathematician along with a short description.

At age 3, Oswald Teichmüller (1913-1943) surprised his mother when she realized he already could count. Showing signs of a genius very early, he later on became a passionate Nazi...

Studying in Göttingen it was him who forced Edmund Landau to stop teaching because he was jewish. Most of his 54(!) papers got published in 'Deutsche Mathematik', a journal directed by his friend Ludwig Bieberbach, further dedicated to publish racist ideology alongside and emphasize "the superiority of german mathematics".

So Ludwig Bieberbach was also very much a nazi. When he tried to keep Oswald away from the war, he couldn't. Oswald was too idealistic. In the end he got killed at age 30 when ending his war-free-time early to rejoin his unit, which was in unorderly retreat. You can find more in this biography:

Yuck. Now i'll have to tell you that his mathematical results are free of racism or any ideology, beautiful, and also very important. Let me try and give you and idea of what a Teichmüller space is. We'll dive vertically so you'll have to plow through some ideas, building upon one another:

A holomorphic function is an especially smooth, complex function that, based on knowledge of it's values on a small disc around some point, can uniquely be extended to the whole complex plane! The information on the disc amounts to knowing every derivative at that point and there's a simple technique to construct it.

But sometimes we'd like to have multiple values for each input. The square root of 4 is 2, but -2 could also do the job. When that happens we say \sqrt{} is a manifold .

For example, the complex logarithm is a complex manifold . Like the (real) square root has two branches , the complex logarithm has an infinitude of them. If you travel once around zero you'll get to "another sheet" of the manifold. You could think of it as a topological surface, an infinitely wide helix.

A (linear) complex structure is the name for when a map FF in some vector space is such that its square is the negative identity F2=IdF^2 = -Id. For example, the even-dimensional euclidean real vector spaces R2nR^{2n} trivially admit a complex structure, coming from \Cn\C^n.

Complex structures are the tool to define almost complex manifolds. Those are like complex manifolds, but may live on any topological surface. Given some real topological surface MM, it's Teichmüller space TMT_M is like a catalogue of all possible ways to construct complex structures on MM.

Teichmüller spaces are a natural extension of the Riemann sphere to general topological surfaces. They themselves have complex manifold structure, a whole lot of metrics work on them, and they left a deep mark on modern geometric function theory.