Publications

preprints   published or accepted   other contributions   expository / books / lecture notes   slides from talks

The links below marked "pdf" and "ps" allow you to download the same preprint versions that are also available in the arXiv. Links to online published versions are also given where available; in most cases these lead to pages with abstracts (publicly available) and downloadable PDF files (available only to subscribers). If you're not viewing from a university or library that has subscriptions to the relevant journals, you'll be offered the opportunity to pay an exhorbitant price for access to the published articles; please do not do that. Instead, feel free to write me a quick e-mail (my surname at math dot hu dash berlin dot de) and I'll be happy to send you PDF versions of any of the published articles. I was tempted to post them all on this page, but I have trouble understanding precisely whether that's legal, so I'm erring on the side of caution.

Preprints

• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification (joint with Sam Lisi and Jeremy Van Horn-Morris)
  Preprint arXiv:2010.16330 (October 2020)
  
  • 1 page per side (120 pages): pdf (1.3 MB)
  This is the second in a two-part series introducing spinal open book decompositions as a tool to study symplectic fillings of contact 3-manifolds. Having established some geometric foundations in Part 1, Part 2 uses holomorphic curves to prove that contact manifolds supported by planar spinal open books can have their fillings classified in terms of Lefschetz fibrations, a result that implies the vast majority of known results on classification of fillings, plus new classification results for fillings of contact circle bundles, and a "quasi-flexibility" result concerning Stein surfaces whose Stein homotopy type is determined purely by their symplectic structures.
• On symplectic fillings of spinal open book decompositions I: Geometric constructions (joint with Sam Lisi and Jeremy Van Horn-Morris)
  Preprint arXiv:1810.12017 (October 2018)
  
  • 1 page per side (68 pages): pdf (790 kB)
  This paper is the first in a two-part series introducing spinal open book decompositions as a tool to study symplectic fillings of contact 3-manifolds. Part 1 addresses most aspects of the subject that do not require holomorphic curve theory, e.g. the existence/uniqueness of contact structures or symplectic/Stein structures compatible with a given spinal open book or Lefschetz fibration respectively, plus the construction of non-exact symplectic cobordisms that realize a topological operation known as spine removal surgery. The latter is used to derive some new symplectic filling obstructions.
  (Note: This paper and its sequel completely subsume the manuscript known as Contact fiber sums, monodromy maps and symplectic fillings, which was occasionally cited in the past as "in preparation" but is now officially abandoned.)
  A pretty good synopsis of the ideas and results in these two papers may be found in this blog post by Laura Starkston, summarizing a 3-part minicourse I gave at a workshop in Minnesota in Summer 2013. It was also advertised in this talk by Jeremy Van Horn-Morris (video) at the 2012 Georgia Topology Conference.
  
  

Published or accepted

• Transversality and super-rigidity for multiply covered holomorphic curves
  Ann. of Math. 198 (2023), no. 1, 93-230
  Preprint arXiv:1609.09867 (September 2016, last revised November 2022)
  
  • 1 page per side (96 pages): pdf (1.2 MB)
  This paper answers the long-standing open question about super-rigidity of holomorphic curves in symplectic Calabi-Yau three-folds: namely, simple J-holomorphic curves in this setting are super-rigid for generic compatible J, meaning there are only finitely many of them for each genus and the Gromov-Witten invariants can be reduced entirely to a finite sum of obstruction bundle calculations. In fact, super-rigidity holds generically for all simple closed index zero curves in dimension at least six, and (by different arguments that are not so new) also in dimension four for curves of genus zero or one. This paper also addresses the problem of regularity for multiply covered curves, proving in particular that unbranched covers of closed curves are generically regular, and finding sharp criteria to prove the same about branched covers. The main technical theorem behind these results defines a smooth stratification of the space of branched covers such that kernels and cokernels have constant dimension on each stratum. A prerequisite for this is to understand the splitting of a Cauchy-Riemann operator for a multiple cover in terms of the irreducible representations of its (generalized) automorphism group; this idea is adapted from Taubes's work on the Gromov invariant, and in that spirit, we also include a brief informal discussion of wall-crossing phenomena for generic homotopies of almost complex structures.
  (Note: readers interested in a less formal overview of the ideas and results in this paper might enjoy the series of three blog posts I wrote about it, starting with "Transversality for multiple covers, super-rigidty, and all that".)
  
  
• Spine removal surgery and the geography of symplectic fillings (joint with Sam Lisi)
  Michigan Math. J. 70 (2021), 403-422
  Preprint arXiv:1902.01326 (February 2019)
  
  • 1 page per side (15 notes): pdf (264 kB)
  This note is a spin-off of the spinal open book project with Lisi and Van Horn-Morris: we use spine removal cobordisms to prove that there is a bound on the geography of minimal symplectic fillings for any contact 3-manifold supported by a planar spinal open book. Contrary to what many readers may expect, the argument does not involve Dehn twist factorizations or mapping class groups. It does require a small amount of holomorphic curve technology, but nothing fancier than what was current in 1996.
  
  
• Unknotted Reeb orbits and nicely embedded holomorphic curves (joint with Alexandru Cioba)
  J. Symplectic Geom. 18 (2020), no. 1, 57-122
  Preprint arXiv:1609.01660 (September 2016, last revised May 2019)
  
  • 1 page per side (50 pages): pdf (647 kB)
  We exhibit a distinctly low-dimensional dynamical obstruction to the existence of Liouville cobordisms: for any contact 3-manifold admitting an exact symplectic cobordism to the tight 3-sphere, every nondegenerate contact form admits an embedded Reeb orbit that is unknotted, meaning it is not only contractible but is also the embedded boundary of an embedded disk. The same holds for all contact structures on reducible 3-manifolds. The proof is a mixture of standard SFT-type techniques with the intersection theory of punctured holomorphic curves, including at least one new tool that we expect to have wider application: a "local" adjunction formula for sequences of holomorphic annuli breaking along a Reeb orbit.
  
  
• Generic transversality for unbranched covers of closed pseudoholomorphic curves (joint with Chris Gerig)
  Comm. Pure Appl. Math. 70 (2017), no. 3, 409-443
  Preprint arXiv:1407.0678 (December 2014, last revised November 2016)
  
  • 1 page per side (31 pages): pdf (375 kB)   ps (541 kB)
  • 2 pages per side (16 pages): pdf (464 kB)   ps (553 kB)
  This is the first installment of a larger project to establish transversality results for multiply covered holomorphic curves in all dimensions, without abstract or domain-dependent perturbations. In this paper, we use an analytic perturbation technique of Taubes to show that for generic tame almost complex structures J, transversality can be achieved for all unbranched covers of index 0 closed J-holomorphic curves. As a consequence, the Gromov-Witten invariants (without descendants) in dimension four can be computed as finite counts of honest J-holomorphic curves, including both simple curves and multiple covers (with rational weights).
  
  
• Subcritical contact surgeries and the topology of symplectic fillings (joint with Paolo Ghiggini and Klaus Niederkrüger)
  Journal de l'École polytechnique - Mathématiques, 3 (2016), pp. 163-208 (open access)
  Preprint arXiv:1408.1051 (August 2014, last revised February 2016)
  
  • 1 page per side (42 pages): pdf (703 kB)
  One of the main results of this paper is the fact that the contact prime decomposition theorem does not extend to higher dimensions, i.e. in any dimension greater than three, there exist tight contact structures on connected sums that do not decompose as contact connected sums. This result arises from a more general investigation of the possible higher-dimensional generalization of Eliashberg's theorem stating that a symplectic filling of a contact connected sum in dimension 3 is always obtained by attaching a 1-handle to another filling. The higher-dimensional version applies to subcritical contact surgeries, and we prove that at least up to dimension 5, a homotopy theoretic statement along these lines is true: the belt sphere of a subcritical surgery must be nullhomotopic in any symplectically aspherical filling.
  
  
• Contact hypersurfaces in uniruled symplectic manifolds always separate
  J. London Math. Soc. 89 (2014), no. 3, 832-852
  Preprint arXiv:1202.4685 (February 2012, last revised December 2013)
  
  • 1 page per side (24 pages): pdf (315 kB)   ps (457 kB)
  • 2 pages per side (12 pages): pdf (385 kB)   ps (467 kB)
This paper proves that nonseparating contact hypersurfaces can never exist in a closed uniruled symplectic manifold, hence (by a result of G. Lu) the Weinstein conjecture is known for all contact hypersurfaces in such settings. This is in some sense a higher-dimensional followup to my earlier paper with Albers and Bramham, published in AGT.
(The final version, which is three times the length of the original, implements the Cieliebak-Mohnke framework for achieving transversality in Gromov-Witten theory, thus it does not require unnatural assumptions such as semipositivity. It includes an appendix showing that the forgetful map in this framework is a pseudocycle, and also some discussion of a useful recent result of Mohsen involving the intersection of a Donaldson hyperplane section with a pseudoconvex hypersurface.)


• Non-exact symplectic cobordisms between contact 3-manifolds
  J. Differential Geom. 95 (2013), no. 1, 121-182
  Preprint arXiv:1008.2456 (August 2010, revised February 2013)
  
  • 1 page per side (50 pages): pdf (624 kB)   ps (1.1 MB)
  • 2 pages per side (25 pages): pdf (677 kB)   ps (1.1 MB)
We introduce a technique for symplectically attaching certain generalized 4-dimensional handles along transverse links and pre-Lagrangian tori in contact 3-manifolds. This produces non-exact symplectic cobordisms in many situations where exact cobordisms are known not to exist, e.g. we show that all contact manifolds with planar torsion admit symplectic cobordisms to all other contact manifolds, and we characterize a large class of contact manifolds that admit symplectic caps containing symplectically embedded 0-spheres.


• A hierarchy of local symplectic filling obstructions for contact 3-manifolds
  Duke Math. J. 162 (2013), no. 12, 2197-2283
  Preprint arXiv:1009.2746 (January 2010, last revised February 2013)
  
  • 1 page per side (65 pages): pdf (811 kB)   ps (1.5 MB)
  • 2 pages per side (33 pages): pdf (884 kB)   ps (1.5 MB)
This paper introduces an infinite hierarchy of new symplectic filling obstructions known as "planar torsion", which generalizes overtwistedness and Giroux torsion, and causes the vanishing of the ECH contact invariant. The proof of this makes use of the existence of a special stable Hamiltonian structure which admits non-generic holomorphic open books of arbitrary genus (cf. the paper on open book decompositions below).
(Note: This paper supersedes the preprint formerly known as Holomorphic curves in blown up open books. An old version by that name remains available as arXiv:1001.4109 since it has been cited a few times in papers that are published.)


• Weak and strong fillability of higher dimensional contact manifolds (joint with Patrick Massot and Klaus Niederkrüger)
  Invent. Math. 192 (2013), no. 2, 287-373
  Preprint arXiv:1111.6008 (November 2011, revised January and September 2012)
  
  • 1 page per side (69 pages): pdf (1.0 MB)
  We generalize to higher dimensions several constructions and results that are standard in 3-dimensional contact topology, including weak symplectic fillings, Giroux torsion, and the Lutz twist. In particular, we find the first examples in dimension 5 of contact manifolds that are weakly but not strongly fillable, as well as examples in all dimensions that have various characteristics of tightness (e.g. lack of contractible Reeb orbits, lack of flexibility) and yet are not weakly fillable. As an ingredient in these constructions, we also generalize to all even dimensions the existence of exact symplectic manifolds with disconnected contact type boundary.
  
  
• Algebraic torsion in contact manifolds (joint with Janko Latschev; with an appendix by Michael Hutchings)
  Geom. Funct. Anal. 21 (2011), no. 5, 1144-1195
  Update February 2012: the published version of this paper contains a minor error in the appendix (see the erratum posted by Michael Hutchings on his blog). We have corrected this in the most recent update to the arXiv version.
  Preprint arXiv:1009.3262 (September 2010, last revised March 2012)
  
  • 1 page per side (53 pages): pdf (615 kB)   ps (872 kB)
  • 2 pages per side (27 pages): pdf (654 kB)   ps (892 MB)
We extract a contact invariant from Symplectic Field Theory that defines a higher order generalization of "algebraic overtwistedness", and thus measures an infinite scale of "degrees of non-fillability" for contact manifolds. We discuss examples in dimension three and use these to derive some non-existence results for exact symplectic cobordisms, some of which are complementary to the existence results in the paper on non-exact cobordisms above. As far as we know, this is the first known contact topological application of the "full" SFT algebra.


• Weak symplectic fillings and holomorphic curves (joint with Klaus Niederkrüger)
  Ann. Sci. École Norm. Sup. (4) 44, fascicule 5 (2011), 801-853
  Preprint arXiv:1003.3923 (March 2010, revised May 2010)
  
  • 1 page per side (42 pages): pdf (799 kB)
We introduce a large class of contact 3-manifolds that are tight but not weakly fillable, or weakly but not strongly fillable, including many that have no Giroux torsion. We also show that weak fillings of planar contact manifolds are always deformable to blow-ups of Stein fillings.


• On non-separating contact hypersurfaces in symplectic 4-manifolds (joint with Peter Albers and Barney Bramham)
  Algebr. Geom. Topol. 10 (2010) 697-737
  Preprint arXiv:0901.0854 (January 2009, last revised July 2009)
  
  • 1 page per side (30 pages): pdf (450 kB)   ps (682 kB)
  • 2 pages per side (15 pages): pdf (486 kB)   ps (694 kB)
We prove obstructions to the existence of non-separating contact hypersurfaces in symplectic 4-manifolds, e.g. they do not exist whenever the contact manifold is planar or has Giroux torsion, or if the symplectic manifold is a ruled surface. This also introduces the concept of a partially planar contact manifold, which will be important in some forthcoming papers.


• Open book decompositions and stable Hamiltonian structures
  Expo. Math. 28 (2010), no. 2, 187-199
  Preprint arXiv:0808.3220 (August 2008, last revised June 2009)
  
  • 1 page per side (13 pages): pdf (214 kB)   ps (364 kB)
  • 2 pages per side (7 pages): pdf (276 kB)   ps (370 kB)
A brief note proving that every planar open book decomposition of a contact manifold can be made pseudoholomorphic. This proves an important special case of a result by C. Abbas that was announced several years ago but not available until recently. It also shows that every open book can be lifted to a family of pseudoholomorphic curves for nongeneric data (this will be used in some work in progress).


• Strongly fillable contact manifolds and J-holomorphic foliations
  Duke Math. J. 151 (2010), no. 3, 337-384
  Preprint arXiv:0806.3193 (June 2008, last revised July 2009)
  
  • 1 page per side (44 pages): pdf (555 kB)   ps (820 kB)
  • 2 pages per side (22 pages): pdf (577 kB)   ps (836 kB)
This uses punctured holomorphic spheres in symplectic cobordisms to prove several new results about symplectic fillings of contact manifolds, e.g. "strongly fillable" and "Stein fillable" are equivalent notions when the contact manifold is planar, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blow-up.


• Automatic transversality and orbifolds of punctured holomorphic curves in dimension four
  Comment. Math. Helv. 85 (2010), no. 2, 347-407
  Preprint arXiv:0802.3842 (February 2008, last revised August 2009)
  
  • 1 page per side (58 pages): pdf (585 kB)   ps (757 kB)
  • 2 pages per side (29 pages): pdf (563 kB)   ps (778 kB)
This paper generalizes several previously known transversality criteria for holomorphic curves in dimension 4 to the context of non-simple and non-immersed curves with cylindrical ends in a symplectic cobordism, and then uses the setup to exhibit a natural class of moduli spaces that are smooth for generic J despite containing (unbranched!) multiple covers.


• Compactness for embedded pseudoholomorphic curves in 3-manifolds
  J. Eur. Math. Soc. (JEMS) 12 (2010), no. 2, 313-342
  Preprint arXiv:SG/0703509 (March 2007, last revised March 2008)
  
  • 1 page per side (32 pages): pdf (407 kB)   ps (601 kB)
  • 2 pages per side (16 pages): pdf (464 kB)   ps (613 kB)
This is the first step in a large project to justify the statement that "nice holomorphic curves degenerate nicely" (of which the smooth moduli spaces studied in the transversality paper above are another example). It classifies all possible degenerations of holomorphic curves in the symplectization of a contact 3-manifold which also have embedded projections into the 3-manifold: in particular, multiple covers can never appear, thus the compactified moduli spaces are smooth for generic J.


• Finite energy foliations on overtwisted contact manifolds
  Geom. Topol. 12 (2008) 531-616
  Preprint arXiv:SG/0611516 (November 2006, last revised March 2008)
  
  • 1 page per side (76 pages): pdf (944 kB)   ps (4.5 MB)
  • 2 pages per side (38 pages): pdf (1.1 MB)   ps (4.5 MB)
A cleaner and more elegant presentation of the main result from my thesis, that every overtwisted contact manifold admits a finite energy foliation.


• My Ph.D. thesis: Finite energy foliations and surgery on transverse links (January 2005)
  
  • 1 page per side (279 pages): pdf (2.6 MB)   ps (7.6 MB)
  • 2 pages per side (140 pages): pdf (2.6 MB)   ps (7.7 MB)
  Note: this is not the original version, but rather a revision from July 14, 2005. The original had a gap in the main compactness proof, resulting from an erroneous statement in Appendix B about degenerating Riemann surfaces with boundary. This is why I generally try to use words like "obvious" and "clearly" as little as possible. Anyway, the problem has been fixed in the revision.

Other contributions

• Stein structures on Lefschetz fibrations and their contact boundaries (joint with Sam Lisi)
  An appendix to the article Families of contact 3-manifolds with arbitrarily large Stein fillings by R. İnanç Baykur and Jeremy Van Horn-Morris
  J. Differential Geom. 101 (2015), no. 3, 423-465
  Preprint arXiv:1208.0528 (August 2012)
  
In their article, Baykur and Van Horn-Morris find counterexamples to a conjecture of Stipsicz and Ozbağci stating that all the Stein fillings of a given contact 3-manifold should have "uniformly bounded topology" in certain senses. Their counterexamples rely on a special case of a basic symplectic topological result proved in the paper in progress on symplectic fillings of "spinal open books" by Van Horn-Morris, Lisi and myself: every allowable Lefschetz fibration over a compact oriented surface with boundary (not only the disk) admits a Stein structure that fills a contact structure uniquely determined by the spinal open book at the boundary. Lisi and I provided our proof of this fact in this appendix since the larger paper on spinal open books is not yet available. In the main body of their paper, Baykur and Van Horn-Morris also provide their own completely different proof of a very similar result, using symplectic handle attachments and convex surface theory.

Expository / books / lecture notes

Published books

• Contact 3-manifolds, holomorphic curves and intersection theory
  Cambridge Tracts in Mathematics, Cambridge University Press, 2020
  Preprint arXiv:1706.05540 (September 2013, last revised August 2019)
  
  • 1 page per side (170 pages): pdf (1892 kB)
  This book is based on some lecture notes written originally for a 5-part lecture series I gave at the LMS Short Course "Topology in Low Dimensions" at Durham University in August 2013. The notes were meant to serve as a well-motivated and topologically oriented introduction to the intersection theory of punctured pseudoholomorphic curves (developed originally by Richard Siefring) and its applications in 3-dimensional contact topology. They begin with a brief review of the closed case and a sample application (McDuff's characterisation of symplectic ruled surfaces), and then explain the essential ideas and results from Siefring's intersection theory, concluding with an application to the classification of symplectic fillings of planar contact manifolds. The necessary technical background on holomorphic curves is provided (without proofs) but relegated to the appendices, so that the main body of the text can focus on topological issues instead of analysis. The final revision of the appendices also includes self-contained proofs of the similarity principle, positivity of intersections and a weak version of the Micallef-White theorem (including a survey of the bits of elliptic regularity theory that are needed for all this), and a quick reference on Siefring's intersection theory aimed at researchers who would like to use it.
• Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds
  Springer Lecture Notes in Mathematics, 2018.
  This book developed out of some lecture notes orginally written for a minicourse on holomorphic curves that I gave at IRMA Strasbourg in October 2012. The stated goal of the notes was to review the appropriate background material on analysis of holomorphic curves and topology of Lefschetz pencils, and then explain (from a modern perspective) complete proofs of the main results in McDuff's paper on rational and ruled symplectic 4-manifolds. A second goal, which ended up becoming significantly more prominent in the notes than in the original minicourse, was to relate those results to subsequent progress on certain questions in 3-dimensional contact topology, notably the Weinstein conjecture and the classification of symplectic fillings. The final version also contains an extra chapter explaining the main result of McDuff's followup paper on immersed symplectic spheres, its connection to Gromov-Witten invariants and the characterization of uniruled symplectic 4-manifolds.

Lecture notes from advanced / special topics courses

• Lectures on symplectic field theory
  Preprint arXiv:1612.01009 (December 2016)
  
  • 1 page per side (343 pages): pdf (3.1 MB)
  Update (12.04.2021): There is a new expanded version of these notes that is not yet on the arXiv but is available (and being updated periodically) on the webpage of a recent course I taught on SFT. It is expected to go to the publisher in 2021 but has not yet done so, thus comments and corrections are still highly encouraged and welcome.
  This is the preliminary manuscript of a book on symplectic field theory based on a lecture course for PhD students given in 2015-16. It covers the essentials of the analytical theory of punctured pseudoholomorphic curves, taking the opportunity to fill in gaps in the existing literature where necessary, and then gives detailed explanations of a few of the standard applications in contact topology such as distinguishing contact structures up to contactomorphism and proving symplectic non-fillability. This electronic version is missing the final three chapters, which will be included in the printed version, to appear in the EMS Lectures in Mathematics series. Updates on the publication of the book will be posted here periodically. Comments and corrections from readers are welcome!
• Lectures on Holomorphic Curves in Symplectic and Contact Geometry (work in progress, last revised May 2015)
  
  • 1 page per side (230 pages): pdf (1.97 MB)   ps (3.04 MB)
  • 2 pages per side (115 pages): pdf (1.84 MB)   ps (3.12 MB)
  I have called this a "book-in-progress" for several years, but I'm not honestly sure whether I will ever try to finish it, as much of what I originally intended to write in this book has in the mean time appeared in the three other books listed above. This was my first attempt to write lecture notes on holomorphic curve theory, originally for a graduate course called Holomorphic Curves in Symplectic and Contact Geometry that I taught at ETH Zürich and HU Berlin in 2010. It includes a detailed development of the technical background on closed holomorphic curves (though only part of the compactness theory), leading to an exposition of Gromov's proof of the nonqueezing theorem. Contrary to what is indicated by the title, it does not contain any nontrivial applications to contact geometry (but there's more of that in the other books above).
  Note: This is Version 3.3, which contains slightly more material than the version (3.2) currently available on the arXiv at arXiv:1011.1690.

Lecture notes from standard Bachelor/Master level courses

• Functional Analysis, Winter Semester 2020-21
  I did not start out with an intention to write lecture notes for this course, but ended up doing so for more than half of it anyway because I was unhappy with what I found in my textbooks.
  • L^p-spaces, Fourier analysis, Sobolev spaces and distributions (118 pages)
  • Fredholm operators (14 pges)
• Topology 1 and 2, 2018-19 (404 pages)
  I wrote these notes for a 2-semester introduction to algebraic topology taught at the HU Berlin in 2018-19.
• Lecture notes on bundles and connections, June 2007, last revised September 2008
  This is a slightly outdated version of the first lecture notes I ever wrote, for an "undergraduate" differential geometry course at MIT in 2007. I also sometimes call it a "book-in-progress", but I have no idea whether I will ever attempt to finish it, and in truth, there are probably already enough good books out there that cover the same material. (Though to be honest, there are also a lot of bad ones.) Someday soon I may at least get around to replacing this with the more recent revision I made for a similar course at the HU Berlin in 2016.
  (Why, you may ask, did I put scare-quotes around the word "undergraduate" up there? Suffice it to say that when some people say "undergraduate differential geometry", they really mean "curves and surfaces", whereas I emphatically do not mean that. I maintain nonetheless that my notes are aimed mainly at advanced undergraduates, not graduate students.)

Other expository stuff

• A beginner's overview of symplectic homology (May 2010)
  
  • 1 page per side (28 pages): pdf (352 kB)
  Not even remotely intended for publication, this was the outcome of an obsessive weekend in May 2010 after I gave the first talk in the Leipzig/Berlin Symplectic Homology Learning Seminar, which was great fun.

Slides from talks

• Equivariant transversality, super-rigidity and all that, Western Hemisphere Virtual Symplectic Geometry Seminar, April 10, 2020
  full pdf version (1.1 MB)
• On symplectic manifolds with contact boundary, or "when is a Stein manifold merely symplectic?", mathematics colloquium at the Université Libre de Bruxelles, February 9, 2018
  full pdf version (2.7 MB)
• What can have a 3-sphere as its boundary, and why should you ask Isaac Newton?
  This is a general audience talk for the UCL AdM Maths Society, given March 3, 2014.
  full pdf version (1.3 MB)
  Here also are a couple of nice videos (not by me) illustrating the 3-body problem.
• Some Tight Contact Manifolds are Tighter Than Others, UK-Japan Mathematical Forum on Algebraic Geometry and Symplectic Geometry, Keio University, Tokyo, January 25, 2013.
  (Note: I also gave a very similar talk at the 27th British Topology Meeting, University of Cambridge, September 8, 2012.)
  full pdf version (953 kB)   basic pdf version (262 kB)
• On Contact Topology, Symplectic Field Theory and the PDE That Unites Them, Durham University pure maths colloquium, December 3, 2012
  full pdf version (858 kB)   basic pdf version (227 kB)
  
• On Symplectic Cobordisms Between Contact Manifolds, from the Istanbul Contact Geometry and Topology Workshop, June 2010.
  pdf (361 kB)  ps (1.1 MB)
• Open Books and Fiber Sums, SFT and ECH: A Plethora of Obstructions to Symplectic Filling, some slides that I used for illustration purposes in an otherwise mostly blackboard talk at the MSRI workshop Symplectic and Contact Topology and Dynamics: Puzzles and Horizons, March 2010.
  pdf (149 kB)  ps (357 kB)
  Here's a brief explanation of the slides:
  • Slide 1: a depiction of the holomorphic planes that arise from a Lutz twist and can be used to prove that overtwisted contact manifolds have vanishing contact homology.
  • Slide 2: the analogous picture of holomorphic cylinders arising from Giroux torsion, which imply vanishing of the ECH contact invariant, as well as algebraic 1-torsion in Symplectic Field Theory.
  • The rest: these are the individual frames of a very low-tech "animation" summarizing the holomorphic curve argument by which planar torsion in a strongly symplectically fillable contact 3-manifold leads to a contradiction.
  For full details on most of this, see the paper A hierarchy of local symplectic filling obstructions for contact 3-manifolds listed above.