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.
Return of the Euler-Fermat theorem
A long long time ago, in a galaxy far away, I wrote up an account of the Euler-Fermat theorem for school students.
Topology: The shape of space
Monash Open Day in 2020 was a purely online affair, thanks to COVID. I recorded a video talking about Topology: The Shape of Space.
University forever
I don’t have anything against people who want to stay at university as long as possible: this is, after all, my life. I think long term economic trends point in this direction though.
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.
Not human, but inhabited by humans: writing mathematics
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.
A-polynomials, Ptolemy varieties, and Dehn filling, Melbourne June 2020
On 15 June 2020 I gave a talk in the topology seminar at the University of Melbourne, entitled “A-polynomials, Ptolemy varieties, and Dehn filling.”
Monash topology talk on Circle packings, Lagrangian Grassmannians, and Scattering Diagrams, April 2020
On 1 April 2020 I gave a talk in the Monash topology seminar, entitled “Circle packings, Lagrangian Grassmannians, and scattering diagrams”.
A-polynomials, Ptolemy varieties and Dehn filling
The A-polynomial encodes hyperbolic geometric information on knots and related manifolds. Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots. Here, we show how to compute A-polynomials by starting with a triangulation of a manifold, similar to Champanerkar, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplicifation of the defining equations. Our methods are a refined version of Dimofte’s symplectic reduction, and we conjecture that the result is equivalent to equations arising from the enhanced Ptolemy variety of Zickert, which would connect these different approaches to the A-polynomial.
We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus. Thus the change in A-polynomial under Dehn filling is given by an explicit twisted cluster algebra. We compute the equations for Dehn fillings of the Whitehead link.