Category Archives: Math

Steiner faces the third degree…

…Is the most amusing Google translation I saw of “Steiner, Flächen dritten Graden”. This is of course the short form of “Steiner, J. – Über die Flächen dritten Grades”, the article I have been translating from fucking German for a presentation I’m doing this week.

Anyway, most of the work is done, and my German is now a little more fluent. That is, up from “barely at all”. Most of the language is translated, leaving mostly just notation, plus a couple of technical terms like Asymptotenpuncte (asymptotic point) and Ebenenbüscheln (sheaves). You can see the original and the results here:

Есть такой медиаплеер: mk908

Steiner, J. – Über die Flächen dritten Grades [1857] (PDF)
Steiner, J. – Über die Flächen dritten Grades (text, edited OCR, German)
Steiner, J. – On surfaces of the third degree (text, edited and Google-translated OCR, English)

It’s some cool stuff, and it kinda makes me look forward to learning to do Math in German. But holy shit was this a pain in the ass.

Diffeomorphisms

In an effort to blog a bit more, and to spur myself to work harder, I’m going to try to blog more about the work I’m doing this summer.

Briefly, I’m exploring properties of subsets of the modular group, defined as the set of 2×2 matrices with determinant 1. I work with an eye toward the problem of generating congruence subgroups, which is not adequately solved. It is possible to check whether a given group is a subgroup, but it has shown to be difficult to work in the opposite direction.

This week I’m delving into Topology. Any modular subgroup can be described as a polygon on the hyperbolic plane, which in turn can be turned into a Riemann surface by identifying corresponding edges (see for example Wohlfahrt). Of particular interest is the nature of the covering of that surface. Read beneath the fold for diffeomorphisms.

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Why Number Theory is Cool

I’ve started doing an intro to number theory, and one of the cool things about it is that it examines some simple questions with complicated answers. Here’s a quick example:

Show that $$n^2 – n$$ is even, for any given integer n.

And here’s the proof: we can easily see that $$n^2 – n = n(n-1)$$. Since either $$n$$ or $$n-1$$ must be even, and the product of an even number with any other number is even, their product must also be even.

This sort of problem has a property similar to many geometry problems, which is that the original statement is easily comprehensible to people without a mathematical background. In this case, the solution also is widely accessible. There are some more complex proofs, though, and I’ve put one below the fold if you’re interested.
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