(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.
From Liouville geometry to contact geometry
Lovely Liouville geometry
(Technical) I’d like to show you some very nice geometry, involving some vector fields and differential forms.
The beauty of mathematics shows itself to patient followers
In September 2018 I gave a talk on the life and mathematics of Maryam Mirzakhani in the School of Physical and Mathematical Sciences colloquium at NTU in Singapore.
Talk on Counting Curves on Surfaces, September 2018
On 19 September 2018 I gave a talk at the National University of Singapore (NUS) in the Topology and Geometry seminar. The talk was entitled “Counting Curves on Surfaces”.
The algebra and geometry of contact categories, Melbourne July 2018
On Monday July 23 2018 I gave a talk in the Geometry and Topology seminar at the University of Melbourne.
A-infinity algebras, strand algebras, and contact categories
In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.
Knot Invariants and Cluster Algebras, AustMS Dec 2017
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.