Because of some turbulence in real life (™) over the last few months, it’s been somewhat hard to concentrate on research. Fortunately the obligation of teaching sometimes exerts the right sort of psychological pressure to keep my mind on mathematics in short bursts. Concerning teaching I believe it was Bott who said (roughly): the trouble with teaching is that when you’ve done it you feel like you’ve accomplished something. But I think this is exactly wrong, and especially absurd coming from a gifted teacher like Bott.

This quarter I’m teaching an introductory graduate class on Kleinian groups. It’s something I could teach standing on my head, and during a couple of the classes I half suspected that I was. But every time I teach something, no matter how “elementary” or “familiar”, I find that I get something new out of it. This time around I have been thinking about the Schläfli formula for the variation of volume in a smooth family of hyperbolic polyhedra, and the way in which it relates to some other well-known and important volume formulae relating to hyperbolic manifolds and geometry, especially in 3 dimensions. It turns out that there are some elegant and easy ways to derive many otherwise quite complicated statements directly from Schläfli; probably this is well known to experts, but it wasn’t to me, and I think it might make an interesting blog post.