In the previous episode, we asked: if you have a family of foliations on a surface, do they arise as the movie of characteristic foliations of a contact structure? In this episode, we ask how unique these contact structures are.
We’ve seen that convex surfaces have wondeful foliations. We’re now going to consider the relationship between these foliations on surfaces, and contact structures
Starting from a Liouville 1-form on a surface, we have been led to 3-dimensional contact geometry, and convex surfaces. We now go in the other direction.
(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.
(Technical) I’d like to show you some very nice geometry, involving some vector fields and differential forms.
In the spirit of previous work in abstract algebra, I have, erm, adapted another nursery rhyme.
To the tune of “Mary had a little lamb”, a discussion of Noether’s theorem.
In 1949, Marcel Golay was thinking about spectrometry. Here’s what happened next…
I don’t believe in any patriotic holidays. But a patriotic holiday on such a terrible date needs to be moved, rebuilt, or abolished.
Entropy means many different things in different contexts, but there is a wonderful notion of entropy which is purely topological. It only requires a space, and a map on it. It is independent of geometry, or any other arbitrary features — it is a purely intrinsic concept. This notion, not surprisingly, is known as topological entropy.
In the spirit of hilariously advanced baby books like Chris Ferrie’s Quantum Physics for Babies, I have taken to incorporating absurdly sophisticated concepts into nursery rhymes.