On 7 December 2020 I gave a (virtual) lecture at the Australian Mathematical Olympiad Committee’s School of Excellence on congruences.
From Here to Hensel
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.
Sitting out the math wars
Very few professional mathematicians have been involved in the “math wars”, and when they have, they have not always inspired confidence. I wondered why.
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.
Limitless as that space too narrow for its inspirations
In which I recall, via neurologist Oliver Sacks, some musings of Sylvester from 1877 on the limitlessness of mathematics.
Holy h-principle, Batman!
In which I attempt to explain some of the ideas behind the h-principle.
The Impact of Impact
On some aspects of the research funding system in the UK and Australia.
The Lost Art of Integration Impossibility
Integration is less a science and more an art form. It high time we shed some light on this lost art.