Andy Wand

School of Mathematics and Statistics
University of Glasgow
15 University Gardens
Glasgow G12 8QW
Scotland

Email: andy dot wand at glasgow dot ac dot uk

Office: 437 Maths and Stats

About me:

I specialize mostly in low dimensional contact and symplectic topology. My complete CV can be accessed here.

Some publications/preprints:

• Mapping class group relations, Stein fillings, and planar open book decompositions , Journal of Topology 2011; doi: 10.1112/ jtopol/ jtr025 {arXiv:1006.2550}
  Abstract: The aim of this paper is to use mapping class group relations to approach the `geography' problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple obstruction to planarity, which we interpret in terms of existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non-planar structures which cannot be shown non-planar by previously existing methods.
• Tightness is preserved by Legendrian surgery, Annals of Mathematics (2) 182 (2015), no. 2, 723-738. arXiv:1404.1705
  Abstract: This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.
• Factorizations of diffeomorphisms of compact surfaces with boundary, Geometry & Topology 19 (2015) 2407-2464 arXiv:0910.5691
  Abstract: We study diffeomorphisms of compact, oriented surfaces, developing methods of distinguishing those which have positive factorizations into Dehn twists from those which satisfy the weaker condition of right veering. We use these to construct open book decompositions of Stein-fillable 3-manifolds whose monodromies have no positive factorization.
• Detecting tightness via open book decompositions Geometry & Topology Monographs 19 (2015), 291-317.
  Abstract: This article is an expository overview of work by the author characterizing tightness of a closed contact 3-manifold in terms of arbitrary open book decompositions thereof. The intent is to provide a `user's guide' of the theory.
• Surgery and tightness in contact 3-manifolds . Conference proceedings of the Gokova Geometry and Topology Conference, 2014
  Abstract: This article is an expository overview of the proof of the author that tightness of a closed contact 3-manifold is preserved under Legendrian surgery. The aim is to give a somewhat more leisurely, motivated, and illustrated version of the ideas and constructions involved in the original paper.
• Algebraic torsion via Heegaard Floer homology. (with C. Kutluhan, G. Matic, and J. Van-Horn Morris) to appear in Proceedings of Symposia in Pure Mathematics: Breadth of Contemporary Topology
  Abstract: We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.
• Filtering the Heegaard Floer contact invariant. (with C. Kutluhan, G. Matic, and J. Van-Horn Morris) to appear in Geometry & Topology
  Abstract: We define an invariant of contact structures in dimension $3$ from Heegaard Floer homology. This invariant takes values in $\mathbb{Z}_{\geq0}\cup{\infty}$, is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, and non-decreasing under Legendrian surgery. We also discuss computability of the invariant, which we illustrate with examples.
• Stein-fillable open books of genus one that do not admit positive factorisations. (with V. Brejevs) to appear in Mathematical Research Letters
  Abstract:We construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.

Some conferences I have (co-)organised:

• London Mathematical Society Durham Symposium: Pseudoholomorphic Curves and Gauge Theory in Low-Dimensional Topology, 18 to 24 August 2019
• Geometric structures in 3 and 4 manifolds, Dubrovnik 2018
• Developments in Contact and Symplectic Topology, Glasgow 2016

Past/present PhD students: