As part of an upcoming workshop participants were asked to introduce themselves with a one-page slide. I took it as an extreme form of concision: summarise your maths research in one slide, Dan.
On 9 December 2020 I gave a (virtual) talk in the Topology session of the 2020 meeting of the Australian Mathematical Society.
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.”
On 1 April 2020 I gave a talk in the Monash topology seminar, entitled “Circle packings, Lagrangian Grassmannians, and scattering diagrams”.
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.
On 4 December 2019 I gave a talk in the Topology session of the 2019 Australian Mathematical Society meeting, entitled “Geometry and physics of circle packings”.
We give another version of Huang’s proof that an induced subgraph of the n-dimensional cube graph containing over half the vertices has maximal degree at least , which implies the Sensitivity Conjecture. This argument uses Clifford algebras of positive definite signature in a natural way. We also prove a weighted version of the result.
On 16 September 2019 I gave a talk in the Monash discrete mathematics seminar. The talk was entitled “The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras”.
On 31 July 2019 I gave a talk at Monash University in the topology seminar, entitled “The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras”.
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.