Here’s a nice maths problem, which I thought it would be fun to discuss. The question doesn’t involve any advanced concepts, but it leads on to a very nice result called Hensel’s lemma.
A long long time ago, in a galaxy far away, I wrote up an account of the Euler-Fermat theorem for school students.
Very few professional mathematicians have been involved in the “math wars”, and when they have, they have not always inspired confidence. I wondered why.
Mathematics can be written in many ways. One approach, very popular with professional pure mathematicians, is to write as little as possible. But there should also be others.
Humans have known how to multiply natural numbers for a long time. In primary school you learn how to multiply numbers using an algorithm which is often called long multiplication, but it’s called “long” for a reason! Recently, a new paper purports to give an algorithm to multiply faster.
(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…
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 is known as topological entropy.