School of Mathematical Sciences ResearchI am interested in everything. In particular I am interested in mathematics. Most of my mathematical research has been in the broad field of geometry and topology. My fields of research include contact topology, symplectic topology, hyperbolic geometry, Heegaard Floer homology and topological quantum field theory. This page contains the following: Papers and PreprintsMorse structures on partial open books with extendable monodromy  with Joan Licata Abstract:The first author in recent work with D. Gay developed the notion of a Morse structure on an open book as a tool for studying closed contact 3manifolds. We extend the notion of Morse structure to extendable partial open books in order to study contact 3manifolds with convex boundary.
Polytopes, dualities, and Floer homology Abstract: 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, 3manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.
Strand algebras and contact categories Abstract: We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.
Counting curves on surfaces  with Norman Do and Musashi Koyama Abstract:In this paper we consider an elementary, and largely unexplored, combinatorial problem in lowdimensional topology. Consider a real 2dimensional compact surface S, and fix a number of points F on its boundary. We ask: how many configurations of disjoint arcs are there on S whose boundary is F? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasipolynomial behaviour. This "elementary curvecounting" is in fact related to a more advanced notion of "curvecounting" from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasipolynomials governing the enumerative problem encode intersection numbers on moduli spaces. Furthermore, among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions, topological recursion, and quantum curves.
Strings, fermions and the topology of curves on annuli Abstract:In previous work with Schoenfeld, we considered a stringtype chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the corresponding "string homology" of annuli. We find this homology has a rich algebraic structure which can be described, in various senses, as fermionic. While for discs we found an isomorphism between string homology and the sutured Floer homology of a related 3manifold, in the case of annuli we find the relationship is more complex, with string homology containing further higherorder structure.
Topological recursion and a quantum curve for monotone Hurwitz numbers  with Norman Do and Alastair Dyer Abstract: Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.
An explicit formula for the Apolynomial of twist knots Abstract: We extend HosteShanahan's calculations for the Apolynomial of twist knots, to give an explicit formula.
Twisty itsy bitsy topological field theory Abstract: We extend the topological field theory (``itsy bitsy topological field theory"') of our previous work from mod2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has informationtheoretic (``itsy'') and quantumfieldtheoretic (``bitsy'') aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.
Contact topology and holomorphic invariants via elementary combinatorics Abstract: In recent times a great amount of progress has been achieved in symplectic and contact geometry, leading to the development of powerful invariants of 3manifolds 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 lighthearted 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.
Itsy bitsy topological field theory Abstract: 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 HondaKazezMatic. This topological field theory stores information in binary format on a surface and has "digital" creation and annihilation operators, giving a toymodel embodiment of "it from bit".
Dimensionallyreduced sutured Floer homology as a string homology  with Eric Schoenfeld Abstract: We show that the sutured Floer homology of a sutured 3manifold of the form $(D^2 \times S^1, F \times S^1)$ can be expressed as the homology of a stringtype complex, generated by certain sets of curves on $(D^2, F)$ 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.
Sutured TQFT, torsion, and tori Abstract: We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with $\Z$ coefficients, of certain sutured manifolds of the form $(\Sigma \times S^1, F \times S^1)$ where $\Sigma$ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with $\Z$ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on $(\Sigma \times S^1, F \times S^1)$ described by an isolating dividing set.
Sutured Floer Homology, Sutured TQFT and NonCommutative QFT Abstract: We define a ``sutured topological quantum field theory'', motivated by the study of sutured Floer homology of product $3$manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $\Z$ of sutured manifolds $(D^2 \times S^1, F \times S^1)$ where $F$ is finite. This generalises previous results of the author over $\Z_2$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a noncommutative Fock space, together with a bilinear form deriving from a certain combinatorial partial order; we show that the sutured TQFT of discs is isomorphic to this Fock space.
Chord diagrams, contacttopological quantum field theory, and contact categories Abstract: We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)dimensional topological quantum field theory defined by HondaKazezMati\'{c} in \cite{HKM08}. The $\Z_2$ $SFH$ of these solid tori forms a ``categorification of Pascal's triangle'', and contact structures correspond bijectively to chord diagrams, or sets of disjoint properly embedded arcs in the disc. Their contact elements are distinct and form distinguished subsets of $SFH$ of order given by the Narayana numbers. We find natural ``creation and annihilation operators'' which allow us to define a QFTtype basis of each $SFH$ vector space, consisting of contact elements. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams. We prove that contact elements are in bijective correspondence with comparable pairs of basis elements with respect to a certain partial order, and in a natural and explicit way. The algebraic and combinatorial structures in this description have intrinsic contacttopological meaning. In particular, the QFTbasis of $SFH$ and its partial order have a natural interpretation in pure contact topology, related to the contact category of a disc: the partial order enables us to tell when the sutured solid cylinder obtained by ``stacking'' two chord diagrams has a tight contact structure. This leads us to extend Honda's notion of contact category to a ``bounded'' contact category, containing chord diagrams and contact structures which occur within a given contact solid cylinder. We compute this bounded contact category in certain cases. Moreover, the decomposition of a contact element into basis elements naturally gives a triple of contact structures on solid cylinders which we regard as a type of ``distinguished triangle'' in the contact category. We also use the algebraic structures arising among contact elements to extend the notion of contact category to a 2category.
Hyperbolic conemanifold structures with prescribed holonomy II: higher genus Abstract: We consider the relationship between hyperbolic conemanifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic conemanifold structure on a surface, with all interior cone angles being integer multiples of $2\pi$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic conemanifold structure. In this paper we build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct fundamental domains for hyperbolic conemanifold structures, from the geometry of a representation. Central to these techniques are the Euler class of a representation, the group $\widetilde{PSL_2\R}$, the twist of hyperbolic isometries, and character varieties. We consider the action of the outer automorphism and related groups on the character variety, which is measurepreserving with respect to a natural measure derived from its symplectic structure, and ergodic in certain regions. Under various hypotheses, we almost surely or surely obtain a hyperbolic conemanifold structure with prescribed holonomy.
Hyperbolic conemanifold structures with prescribed holonomy I: punctured tori Abstract: We consider the relationship between hyperbolic conemanifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic conemanifold structure on a surface, with all interior cone angles being integer multiples of $2\pi$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic conemanifold structure. In this paper we prove results for the punctured torus; in the sequel, for higher genus surfaces. We show that a representation of the fundamental group of a punctured torus is a holonomy representation of a hyperbolic conemanifold structure with no interior cone points and a single corner point if and only if it is not virtually abelian. We construct a pentagonal fundamental domain for hyperbolic structures, from the geometry of a representation. Our techniques involve the universal covering group $\widetilde{PSL_2\R}$ of the group of orientationpreserving isometries of $\hyp^2$ and Markoff moves arising from the action of the mapping class group on the character variety.
The hyperbolic meaning of the MilnorWood inequality Abstract: We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientationpreserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several wellknown results, including the MilnorWood inequality, using purely hyperbolicgeometric methods. Our methods express inequalities in Milnor's function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.
Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori This is a previous version of the article Chord diagrams, contacttopological quantum field theory, and contact categories above. It contains less content, in particular about contact categories, but is less terse (or more prolix!) and contains more background. It might be useful for some readers, and so I retain it here, even though it has been superseded by that article. TalksOn 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".
In OctoberNovember 2016 I gave two talks at the MSI Workshop on LowDimensional Topology & Quantum Algebraat ANU, Canberra.
On 8 April, 2016 I gave a talk at the University of Melbourne in the Knot Invariants seminar. The talk was entitled ``Hyperbolic volume and the Mahler measure of the Apolynomial". On 14 March, 2016 I gave a talk at Monash University in the Discrete Mathematics seminar. The talk was entitled ``Trinities, hypergraphs, and contact structures".
On 12 January, 2016 I gave a talk at the Workshop on GromovWitten theory, Gauge Theory, and Dualities at ANU Kioloa. The talk was entitled ``Trinities, sutured Floer homology, and contact structure".
On 30 September, 2015 I gave a talk at the Australian Mathematical Society Annual Meeting at Flinders University, in Adelaide. The talk was entitled ``Counting curves on surfaces".
On 12 June, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar. The talk was entitled ``Geometric quantisation and calculation of Apolynomials". On 15 May, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar. The talk was entitled ``The Apolynomial, symplectic geometry, and quantisation". On 18 February, 2015 I gave a talk at Tokyo Institute of Technology. The talk was entitled ``Contact topology and holomorphic invariants via elementary combinatorics".
On 11 December, 2014 I gave a talk at the 8th Australia New Zealand Mathematics Convention, at the University of Melbourne, as part of the Geometry and Topology session. The talk was entitled ``String, fermions and the topology of curves on surfaces".
On 24 October, 2014 I gave a talk at the University of Melbourne, for the AlgebraGeometryTopology Seminar. The talk was entitled ``Strings, fermions and the topology of curves on surfaces".
On 12 May, 2014 I gave a talk at the Monash University Discrete Mathematics Seminar. The talk was entitled ``Discrete Contact Geometry".
On 30 September, 2013 I gave a talk at the 57th Australian Mathematical Society Annual Meeting, at the University of Sydney, as part of the Geometry and Topology session. The talk was entitled ``A YangBaxter equation from sutured Floer homology".
On 24 May, 2013 I gave a talk at the University of Melbourne, for the AlgebraGeometryTopology Seminar. The talk was entitled ``Sutures, quantum groups and topological quantum field theory".
On 16 April, 2013 I gave a talk at ANU, for the Algebra and Topology seminar. The talk was entitled ``Contact topology and holomorphic invariants via elementary combinatorics".
On 7 December, 2012 I gave a talk at Monash University, entitled ``Contact topology and holomorphic invariants via elementary combinatorics".
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 fieldtheoretic ideas out of contact geometry and elementary topology".
On 30 April, 2012 I gave a talk at the University of Southern California, for the Geometry & Topology Seminar. The talk was entitled ``Itsy bitsy topological field theory".
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". On 6 March, 2012 I gave a talk at Monash University, entitled ``Itsy bitsy topological field theory". On 28 November, 2011 I gave a talk at the University of Maryland, for the GeometryTopology Seminar. The talk was entitled ``Hyperbolic conemanifolds with prescribed holonomy". 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". On 6 April, 2011 I gave a talk at Brown University, for the Geometry and Topology seminar. The talk was entitled ``Sutured topological quantum field theory". On 1 October, 2010 I gave two talks at Columbia University. The first was for an informal sutured Floer homology seminar and was entitled ``Sutured topological quantum field theory and contact elements in sutured Floer homology". The second talk was for the Geometric Topology Seminar and was entitled ``Hyperbolic conemanifolds with prescribed holonomy". On 23 September, 2010 I gave a talk at Boston College, for the Geometry/Topology seminar. The talk was entitled ``Sutured topological quantum field theory and contact elements in sutured Floer homology". On 22 July, 2010 I gave a talk at the workshop on Geometry, Topology and Dynamics of Character Varieties, part of a program at the Institute for Mathematical Sciences of the National University of Singapore. The talk was entitled ``Hyperbolic conemanifolds with prescribed holonomy".
On 11 May, 2010 I gave a talk for the Institut Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris 6 and Université Paris Diderot, Paris 7 at Chevaleret, for the Séminaire de Topologie. The talk was entitled ``Sutured Floer homology and contacttopological quantum field theory". On 7 May, 2010 I gave a talk at the Institut Camille Jordan in Lyon, France, for the Séminaire Géométries. The talk was entitled ``Sutured Floer homology and contacttopological quantum field theory". On 22 April, 2010 I gave a talk at the Université Libre de Bruxelles, Belgium, for the seminar on symplectic and contact geometry. The talk was entitled ``Sutured Floer homology and contacttopological quantum field theory". On 7 April, 2010 I gave a talk at Micihgan State University, USA, for the 3 and 4Manifold Seminar, entitled ``Sutured topological quantum field theory and contact elements in sutured Floer homology". On 3 February, 2010 I gave a talk at Uppsala Universitet, Sweden. The talk was entitled "Chord diagrams, contacttopological quantum field theory, and contact categories". On 5 January, 2010 I gave a talk at the University of Melbourne, Australia, for the Algebra/Geometry/Topology Seminar. The talk was entitled "Chord diagrams and contacttopological quantum field theory". On 18 December, 2009 I gave two talks at the Institut Fourier in Grenoble, France. The first talk was entitled ``Chord diagrams, contacttopological quantum field theory, and contact categories". The second talk was entitled ``Construction of hyperbolic conemanifolds with prescribed holonomy". On 10 December, 2009 I gave a talk at the Université de Nantes, for the Séminaire de Topologie, Géométrie et Algèbre, entitled ``Chord diagrams, contacttopological quantum field theory, and contact categories". On April 17, 2009 I gave a talk at Columbia University, for the Symplectic Geometry and Gauge Theory Seminar, entitled ``Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori".
On March 16, 2009 I gave a talk at Stanford, for the Symplectic Geometry seminar, entitled ``Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori". In January 2006 I gave a talk on my masters work at the conference ``Manifolds at Melbourne".
ThesesPhD ThesisIn 2009 I completed my PhD at Stanford. I submitted my thesis ``Chord diagrams, contacttopological quantum field theory, and contact categories" on August 21, 2009.
I defended my thesis on May 29, 2009. I gave a beamer presentation; here are the slides.
Masters thesisIn September 2005 I completed my masters at Melbourne, also under Craig Hodgson. I studied representations of surface groups and the existence of hyperbolic cone manifold structures with prescribed holonomy (1.8 MB pdf, 171 pages). Honours thesisAt the University of Melbourne I completed an honours thesis under Craig Hodgson, studying Apolynomials, representation varieties of 3manifolds and their relationship with hyperbolic geometry (753k pdf, 122 pages). TranslationsIt's plus facile for me to lire mathematics in English than in French.
