School of Mathematical Sciences
Monash University
Clayton VIC 3800 Australia
Email: dan.v.(my last name)@gmail.com

## Research

I 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.

## Papers and Preprints

Morse structures on partial open books with extendable monodromy - with Joan Licata
(19 pages) - on the arXiv.

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 3-manifolds. We extend the notion of Morse structure to extendable partial open books in order to study contact 3-manifolds with convex boundary.

Polytopes, dualities, and Floer homology
(41 pages) - on the arXiv.

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, 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.

Strand algebras and contact categories
(31 pages) - on the arXiv.

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
(105 pages) - on the arXiv.

Abstract:In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional 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 quasi-polynomial behaviour.

This "elementary curve-counting" is in fact related to a more advanced notion of "curve-counting" 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 quasi-polynomials 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
(55 pages) - on the arXiv.

Abstract:In previous work with Schoenfeld, we considered a string-type 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 3-manifold, in the case of annuli we find the relationship is more complex, with string homology containing further higher-order structure.

Topological recursion and a quantum curve for monotone Hurwitz numbers - with Norman Do and Alastair Dyer
(23 pages) - on the arXiv.

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 A-polynomial of twist knots
(4 pages) - on the arXiv.

Abstract: We extend Hoste-Shanahan's calculations for the A-polynomial of twist knots, to give an explicit formula.

Twisty itsy bitsy topological field theory
(52 pages) - on the arXiv.

Abstract: We extend the topological field theory (itsy bitsy topological field theory"') of our previous work from mod-2 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 information-theoretic (itsy'') and quantum-field-theoretic (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
(35 pages) - on the arXiv.

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 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.

Itsy bitsy topological field theory
(54 pages) - on the arXiv - published in Annales Henri Poincaré.

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 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".

Dimensionally-reduced sutured Floer homology as a string homology - with Eric Schoenfeld
(29 pages) - on the arXiv.

Abstract: We show that the sutured Floer homology of a sutured 3-manifold of the form $(D^2 \times S^1, F \times S^1)$ can be expressed as the homology of a string-type 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
(29 pages) - on the arXiv - published in International Journal of Mathematics.

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.

• pdf (359 kb)
• ps (1.7 Mb)

Sutured Floer Homology, Sutured TQFT and Non-Commutative QFT
(49 pages) - on the arXiv - published in Algebraic & Geometric Topology.

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 non-commutative 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.

• pdf (559 kb)
• ps (1.6 Mb)

Chord diagrams, contact-topological quantum field theory, and contact categories
(74 pages) - on the arXiv - published in Algebraic & Geometric Topology.

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 Honda--Kazez--Mati\'{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 QFT-type 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 contact-topological meaning. In particular, the QFT-basis 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 2-category.

• pdf (840 kb)
• ps (2.3 Mb)

Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus
(25 pages) - on the arXiv - published in Geometriae Dedicata.

Abstract: We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold 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 cone-manifold 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 cone-manifold 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 measure-preserving 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 cone-manifold structure with prescribed holonomy.

• pdf (385 kb)
• ps (757 kb)

Hyperbolic cone-manifold structures with prescribed holonomy I: punctured tori
(40 pages) - on the arXiv - published in Geometriae Dedicata.

Abstract: We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold 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 cone-manifold 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 cone-manifold 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 orientation-preserving 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 Milnor--Wood inequality
(21 pages) - on the arXiv - published in Expositiones Mathematicae.

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 orientation-preserving 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 well-known results, including the Milnor--Wood inequality, using purely hyperbolic-geometric 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.

• pdf (279 kb)
• ps (891 kb)

Chord diagrams, topological quantum field theory, and the sutured Floer homology of solid tori
(89 pages)

This is a previous version of the article Chord diagrams, contact-topological 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.

• pdf (847 kb)
• ps (1.4 MB)

## Talks

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 here.

In October-November 2016 I gave two talks at the MSI Workshop on Low-Dimensional Topology & Quantum Algebraat ANU, Canberra.

• The first, introductory, talk, on 31 October, was entitled An introduction to contact geometry and topology". Slides from that talk are available here.
• The second talk, on 2 November, was entitled Strand algebras and contact categories".

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 A-polynomial".

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".

• The slides from the talk are available here.

On 12 January, 2016 I gave a talk at the Workshop on Gromov-Witten theory, Gauge Theory, and Dualities at ANU Kioloa. The talk was entitled Trinities, sutured Floer homology, and contact structure".

• The slides from the talk are available here.

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".

• The slides from the talk are available here.

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 A-polynomials".

On 15 May, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar. The talk was entitled The A-polynomial, 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".

• The slides from the talk are available here.

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".

• The slides from the talk are available here.

On 24 October, 2014 I gave a talk at the University of Melbourne, for the Algebra-Geometry-Topology Seminar. The talk was entitled Strings, fermions and the topology of curves on surfaces".

• The slides from the talk are available here.

On 12 May, 2014 I gave a talk at the Monash University Discrete Mathematics Seminar. The talk was entitled Discrete Contact Geometry".

• The slides from the talk are available here.

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 Yang-Baxter equation from sutured Floer homology".

• The slides from the talk are available here.

On 24 May, 2013 I gave a talk at the University of Melbourne, for the Algebra-Geometry-Topology Seminar. The talk was entitled Sutures, quantum groups and topological quantum field theory".

• The slides from the talk are available here.

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".

• The slides from the talk are available here.

On 7 December, 2012 I gave a talk at Monash University, entitled Contact topology and holomorphic invariants via elementary combinatorics".

• The slides from the talk are available here.

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".

• The slides from the talk are available here.

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".

• The slides from the talk are available here.

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 Geometry-Topology Seminar. The talk was entitled Hyperbolic cone-manifolds 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 cone-manifolds 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 cone-manifolds with prescribed holonomy".

• Slides are available here (1.3 MB pdf) or at the workshop website.

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 contact-topological 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 contact-topological 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 contact-topological quantum field theory".

On 7 April, 2010 I gave a talk at Micihgan State University, USA, for the 3- and 4-Manifold 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, contact-topological 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 contact-topological 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, contact-topological quantum field theory, and contact categories". The second talk was entitled Construction of hyperbolic cone-manifolds 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, contact-topological 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".

• Here (2.9 MB pdf) are the slides from my talk, in the form of a beamer presentation.

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".

• Here are the slides from the talk.

## Theses

#### PhD Thesis

In 2009 I completed my PhD at Stanford. I submitted my thesis Chord diagrams, contact-topological quantum field theory, and contact categories" on August 21, 2009.

• pdf (1.3 MB, 229 pages).

I defended my thesis on May 29, 2009. I gave a beamer presentation; here are the slides.

#### Masters thesis

In 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 thesis

At the University of Melbourne I completed an honours thesis under Craig Hodgson, studying A-polynomials, representation varieties of 3-manifolds and their relationship with hyperbolic geometry (753k pdf, 122 pages).

## Translations

It's plus facile for me to lire mathematics in English than in French.

• Translation of Giroux's 1991 paper, Convexite en topologie de contact" (12/7/10) (tex, dvi, pdf). From the French. Now updated. Thanks to Patrick Massot for corrections. (35 pages)
• Translation of Martinet's 1971 paper, Formes de contact sur les varietes de dimension 3" (30/12/07) (tex, dvi, pdf). A classic and seminal paper in contact geometry. From the French. (20 pages)