On 12 December 2017 I gave a talk in the Topology session of the 2017 meeting of the Australian Mathematical Society, entitled “Knot invariants and cluster algebras”.
Some pure mathematics and consciousness
In November 2017 I gave a talk to the Monash Consciousness Research Laboratory (Tsuchiya Lab). I talked about some pure mathematical ideas that have appeared in the literature on the frontiers of neuroscience and the study of consciousness — gauge theory, and category theory.
Plane graphs, special alternating links, and contact geometry, Sydney Oct 2017
On Thursday October 5 2017 I gave a talk in the Geometry and Topology seminar at the University of Sydney.
Tight contact structures on Seifert surface complements
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda’s method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman’s formal knot theory.
The Tutte polynomial and knot theory, Monash Sep 2017
On September 25, 2017 I gave a talk as part of the Bill Tutte centenary celebration at Monash University.
Plane graphs and Grassmannian positivity, September 2017
On 20 September, 2017 I gave a talk in the Monash topology seminar. The talk was entitled “Plane graphs and Grassmannian positivity”.
Polytopes, dualities, and Floer homology
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.
Morse structures on partial open books with extendable monodromy
Joint with Joan Licata.
We extend the notion of Morse structure on an open book to extendable partial open books in order to study contact 3-manifolds with convex boundary.
AustMS talk, December 2016
On 6 December, 2016, I gave a talk at the Austrlaian Mathematical Society Annual Meeting at ANU, Canberra. The talk was entitled “Strand algebras and contact categories”. Slides from the talk are available.
Talks at Low-dimensional topology workshop, Oct-Nov 2016
In October-November 2016 I gave two talks at the MSI Workshop on Low-Dimensional Topology & Quantum Algebra at ANU, Canberra. Some slides are available.