A fun fact from Euclidean geometry that I thought was a wonderful enough gem to share. It’s standard, but it’s nowhere near any curriculum. I’ll try not to get too snarky about the curriculum.
On 7 December 2020 I gave a (virtual) lecture at the Australian Mathematical Olympiad Committee’s School of Excellence on congruences.
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.
Very few professional mathematicians have been involved in the “math wars”, and when they have, they have not always inspired confidence. I wondered why.
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 which I recall, via neurologist Oliver Sacks, some musings of Sylvester from 1877 on the limitlessness of mathematics.
In which I attempt to explain some of the ideas behind the h-principle.
On some aspects of the research funding system in the UK and Australia.