In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3-manifolds such as Heegaard Floer homology and embedded contact homology. These invariants are based on holomorphic curves and moduli spaces, but in the simplest cases, some of their structure reduces to some elementary combinatorics and algebra which may be of interest in its own right. In this note, which is essentially a light-hearted exposition of some previous work of the author, we give a brief introduction to some of the ideas of contact topology and holomorphic curves, discuss some of these elementary results, and indicate how they arise from holomorphic invariants.
Contact topology and holomorphic invariants via elementary combinatorics, Monash Dec 2012
On 7 December, 2012 I gave a talk at Monash University. Slides are available.
Field-theoretic ideas from contact geometry, ANZAMP 2012
On 5 December, 2012 I gave a talk at the Australian and New Zealand Association of Mathematical Physics (ANZAMP) Inaugural annual meeting. The talk was entitled “Some field-theoretic ideas out of contact geometry and elementary topology”. Slides are available.
Dimensionally-reduced sutured Floer homology as a string homology
Joint with Eric Schoenfeld.
We show that the sutured Floer homology of a sutured 3-manifold of the form [latex](D^2 \times S^1, F \times S^1)[/latex] can be expressed as the homology of a string-type complex, generated by certain sets of curves on [latex](D^2, F)[/latex] and with a differential given by resolving crossings. We also give some generalisations of this isomorphism, computing “hat” and “infinity” versions of this string homology. In addition to giving interesting elementary facts about the algebra of curves on surfaces, these isomorphisms are inspired by, and establish further, connections between invariants from Floer homology and string topology.
Itsy bitsy topological field theory, USC April 2012
On 30 April, 2012 I gave a talk at the University of Southern California, for the Geometry & Topology Seminar. Slides are available.
Itsy bitsy topological field theory, MIT April 2012
On 23 April, 2012 I gave a talk at MIT, for the Geometry and Topology Seminar. The talk was entitled “Itsy bitsy topological field theory”.
Itsy bitsy topological field theory, Monash Mar 2012
On 6 March, 2012 I gave a talk at Monash University, entitled “Itsy bitsy topological field theory”.
Itsy bitsy topological field theory
We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Matic. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”.
Hyperbolic cone-manifolds with prescribed holonomy, Maryland Nov 2011
On 28 November, 2011 I gave a talk at the University of Maryland, for the Geometry-Topology Seminar. The talk was entitled “Hyperbolic cone-manifolds with prescribed holonomy”.
Sutured Floer homology and TQFT, Harvard May 2011
On 13 May, 2011 I gave a talk at Harvard University, for the Gauge Theory and Topology seminar. The talk was entitled “Sutured Floer homology and TQFT”.