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.
AustMS 2019 talk on geometry and physics of circle packings
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”.
The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras
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.
Talk in Monash discrete mathematics seminar, September 2019
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”.
Monash topology talk on sensitivity conjecture and Clifford algebras, July 2019
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”.
Uniqueness of contact structures and tomography
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.
Convex surfaces and tomography
We’ve seen that convex surfaces have wonderful foliations. We’re now going to consider the relationship between these foliations on surfaces, and contact structures
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.