dan – Daniel Mathews

Uniqueness of contact structures and tomography

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.

Convex surfaces and tomography

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

Liouville structures and convex surfaces

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.

From Liouville geometry to contact geometry

(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.

Lovely Liouville geometry

(Technical) I’d like to show you some very nice geometry, involving some vector fields and differential forms.

Emmy had a theorem (mathematical nursery rhyme #2)

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.

Golay Golay Golay (Top of the autocorrelation world)

In 1949, Marcel Golay was thinking about spectrometry. Here’s what happened next…

The “Australia day” category error

I don’t believe in any patriotic holidays. But a patriotic holiday on such a terrible date needs to be moved, rebuilt, or abolished.

Topological entropy: information in the limit of perfect eyesight

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.

« Previous