The Symplectic Working Group Seminar is a largely informal gathering for symplectic topologists to explain things to each other. Topics for talks may vary between the current research of members of UCL's symplectic group or other interested participants, contents of a paper that one of us recently read, or some piece of "standard" knowledge that is not as well known as it should be. Lengths of talks can vary between 1 and 2 hours, and they also may span multiple weeks. There are occasional talks by outside speakers.
If you'd like to suggest topics for the seminar, especially if you're volunteering to give talks about them, e-mail c dot wendl at ucl dot ac dot uk.
During 2015-16, meetings of the seminar are relatively infrequent and limited to special occasions since there are many other interesting things going on, such as the Homological Mirror Symmetry Seminar at King's College and Chris Wendl's Symplectic Field Theory course at UCL. See the schedule below for details.
Note: the following schedule, like the archived schedules further down on the page, is now displayed in reverse chronological order. This is hopefully less confusing than the way it was previously displayed.
| Friday 15 January, 2016 16:00 UCL maths, Room 500 |
Speaker: Dan Cristofaro-Gardiner (Harvard) Topic: Some open questions about ECH cobordism maps, and some applications Abstract: One can define cobordism maps on ECH by using Taubes' isomorphism between ECH and Seiberg-Witten theory. Hutchings and Taubes showed that these maps satisfy a "holomorphic curve axiom", and this has led to many applications. For various reasons, however, one would like to give a direct definition of these maps in terms of holomorphic curves. I will explain some of the motivation for why one would want to do this, and the basic challenges; I will also give examples of some cobordisms of interest. To illustrate an application, I will briefly explain some joint work with Hind, in which we use certain very special cases of the cobordism map to learn new information about higher-dimensional symplectic embeddings. This talk is meant as a continuation of the previous one, and the material there will be assumed. |
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| Wednesday 13 January, 2016 16:00 UCL maths, Room 505 |
Speaker: Dan Cristofaro-Gardiner (Harvard) Topic: An introduction to embedded contact homology Abstract: Embedded contact homology (ECH) is a kind of Floer homology for contact three-manifolds. Although formally similar to symplectic field theory (SFT), Taubes has shown that ECH is canonically isomorphic to the Seiberg-Witten Floer cohomology of the manifold. Thus, ECH encodes the topology of three- and four-dimensional manifolds into information about their contact and symplectic geometry. I will give an introduction to ECH, and discuss some of the foundational challenges involved in setting up this theory. An emphasis of the talk will be explaining how Hutchings and Taubes used "obstruction bundle gluing" to prove that the ECH differential squares to zero. |
| Tuesday 20 October, 2015 16:00 + epsilon King's College London, Strand Building S6.06 |
Speaker: John Pardon (Stanford and IHES) Topic: Contact homology and virtual fundamental cycles (part 2) This is a sequel to the talk on 19 October, and will directly follow the HMS seminar at KCL. |
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| Monday 19 October, 2015 16:00 Drayton B06 |
Speaker: John Pardon (Stanford and IHES) Topic: Contact homology and virtual fundamental cycles (part 1) Abstract: I will discuss a rigorous definition of contact homology. More specifically, I will discuss the construction of virtual fundamental cycles on the relevant moduli spaces of holomorphic curves. This will involve constructing atlases of finite-dimensional reductions (an "implicit atlas") on the moduli spaces. The first talk will focus on the construction of these atlases (and on the definition of an "implicit atlas"). We will then give a definition of the relevant virtual fundamental cycles, admitting certain abstract algebro-topological machinery as a black box. The second will focus on setting up the algebro-topological machinery used to extract the virtual fundamental cycle of a space admitting an implicit atlas. Literature: Pardon, Contact homology and virtual fundamental cycles |
| Tuesday 28 July, 2015 15:00 UCL maths, Room 500 |
Speaker: Otto van Koert (Seoul National University and Universität Augsburg) Topic: Fractional twists, invariant contact structures, and Brieskorn manifolds Abstract: In this talk we construct fractional twists, a class of symplectomorphisms that generalize Dehn twists. We discuss their isotopy problem and describe how they can be used in the construction of contact structures that are invariant under a circle action. In the second part of this talk, we specialize to Brieskorn varieties, a class of symplectic manifolds where these fractional twists can be constructed. We explain their role in contact topology and mention some open problems. |
| Tuesday 30 June, 2015 | No symplectic working group on this day, but Yasha Eliashberg is speaking in an exceptional meeting of the King's/UCL Geometry Seminar at 11am in room 500. |
| Friday 29 May, 2015 15:00 UCL maths, Room D103 |
Speaker: Momchil Konstantinov Topic: Negatively stabilised contact manifolds are not symplectically fillable Abstract: A negative stabilisation is a particular alteration of an open book decomposition of an odd-dimentional manifold which does not change its diffeomorphism type. Under the Giroux correspondence however it produces a very different contact structure. In dimension 3 it has been known that the corresponding contact manifold is overtwisted and very recently the same has been shown to hold in higher dimensions by Casals, Murphy and Presas. In particular, one can expect that such manifolds have vanishing contact homology and do not admit symplectic fillings. This is indeed the case as established by Bourgeois and van Koert and (for the second fact) Massot, Niederkruger and Wendl. In this talk we will investigate the very hands-on argument from the paper by Bourgeois and van Koert, concentrating on showing existence and uniqueness of a holomorphic plane in the symplectisation, asymptotic to a degree 1 Reeb orbit. I will also give indications as to how such a construction implies the above facts. Literature:
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| Thursday 21 May, 2015 15:00 UCL maths, Room 500 |
Speaker: Jacqui Espina Topic: The Conley conjecture and local Floer homology (continuation from 26 February) |
| Thursday 14 May, 2015 15:00 UCL maths, Room 500 |
Speaker: Alex Cioba Topic: Gluing results in Morse and Floer theory (continuation from 10 March) |
| Tuesday 10 March, 2015 15:00 UCL maths, Room 607 |
Speaker: Alex Cioba Topic: Gluing results in Morse and Floer theory |
| Thursday 26 February, 2015 15:00 UCL maths, 706 |
Speaker: Jacqui Espina Topic: The Conley conjecture and local Floer homology Abstract: The existence of infinitely many simple periodic orbits of a Hamiltonian diffeomorphism on a symplectic manifold is referred to as the Conley conjecture. This has been observed to be true for many closed symplectic manifolds and the conjecture was formulated by Conley for tori in 1984. This was proved for certain Hamiltonian diffeomorphisms, surfaces, and tori. In 2009, Ginzburg proved the Conley conjecture for closed symplectically aspherical manifolds using local Floer theory. The class of manifolds for which the conjecture holds has been since then broadly extended based on his approach. I will give the basic definitions and elements of local Floer homology and outline Ginzburg's proof. If time permits, I will talk about further applications of local Floer theories for Reeb flows of contact manifolds such as local contact and symplectic homology. Literature:
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| Tuesday 10 February, 2015 15:00 UCL maths, Room 607 |
Speaker: Jonny Evans Topic: More on Fukaya categories (continuation from 18 November) Abstract: Last time I introduced the graded Floer complex and gave a hint about how to define its A-infinity structure. I also explained how to form a category of twisted complexes, but I will review this. Then I will explain how to prove that one object "generates" another (and what this means). I will explain why the thimbles of a Lefschetz fibration on a Liouville domain X generate the compact Lagrangians in X; since thimbles are noncompact this will involve a digression into wrapped Fukaya categories. Finally I will explain how all this technology can be used to prove uniqueness of special Lagrangian submanifolds of R2n with a particular asymptotic behaviour (following Imagi-Joyce-dos Santos). Literature:
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| Thursday 5 February, 2015 15:00 UCL maths, Room 706 |
Speaker: Diarmuid Crowley (Aberdeen) Topic: Bordism methods in contact topology in detail Abstract: In this talk I will give details behind the proofs of the surgery theorem for topological Stein cobordisms from my earlier talk (in the King's/UCL Geometry Seminar on 3 February). I will also present more interesting examples of relatively simple almost contact manifolds which are not Stein fillable and not known to be symplectically fillable (so far as I know). The examples might be interesting "boundary" cases for the symplectic filling problem. Literature:
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| Tuesday 3 February, 2015 | No symplectic working group on this day, but Diarmuid Crowley is speaking in the King's/UCL Geometry Seminar, and will be giving a followup talk in our seminar on Thursday. |
| Tuesday 27 January, 2015 15:00 UCL maths, Room 607 |
Speaker: Oldřich Spáčil Topic: Classification of overtwisted contact structures in higher dimensions (continuation from 3 December) Literature: Borman, Eliashberg and Murphy, Existence and classification of overtwisted contact structures in all dimensions |
| Thursday 22 January, 2015 15:00 UCL maths, Room 706 |
Speaker: Chris Wendl Topic: Introduction to spinal open books (This talk was originally scheduled for Tuesday the 20th of January, but it has been rescheduled.) Abstract: One way that open book decompositions arise naturally in topology is as boundaries of Lefschetz fibrations over the disk: in fact, every Stein domain admits a compatible Lefschetz fibration over the disk whose boundary is an open book supporting the induced contact structure at the convex boundary. A spinal open book is a more general topological decomposition of a contact manifold that can arise as the boundary of a Lefschetz fibration over an arbitrary surface. I will introduce the basic definitions and describe some general classes of contact manifolds in dimension 3 and higher dimensions that can profitably be understood in terms of spinal open books. There are many applications: e.g. given a contact 3-manifold with a spinal open book that has a genus 0 page, one can often use holomorphic curves to classify all the symplectic fillings (joint work with Sam Lisi and Jeremy Van Horn-Morris). Another application is a symplectic cobordism construction known as spine removal surgery, which generalizes Eliashberg's symplectic capping construction for contact 3-manifolds, and has also been used (in joint work with Patrick Massot and Klaus Niederkrüger) to detect fillability obstructions for tight contact manifolds in higher dimensions. I will explain as much of this as I can. Literature:
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| Tuesday 13 January, 2015 | No seminar this week, but Ailsa Keating is speaking as part of a double bill in the King's/UCL Geometry Seminar |
| Wednesday 3 December, 2014 15:00 16-18 Gordon Square, Room G09 |
Speaker: Oldřich Spáčil Topic: Classification of overtwisted contact structures in higher dimensions Literature: Borman, Eliashberg and Murphy, Existence and classification of overtwisted contact structures in all dimensions |
| Wednesday 26 November, 2014 15:00 16-18 Gordon Square, Room G09 |
Speaker: Agustín Moreno Topic: Classification of overtwisted contact structures on 3-manifolds Literature:: Eliashberg, Classification of overtwisted contact structures on 3-manifolds |
| Tuesday 18 November, 2014 15:00 Rockefeller 339 |
Speaker: Jonny Evans Topic: Fukaya categories with a view to special Lagrangians Abstract: It's a beautiful theorem of Thomas and Yau that two special Lagrangians in a Calabi-Yau manifold which intersect transversely cannot be isomorphic in the Fukaya category. This can be used to great effect when we have a way to determine when two Lagrangians are isomorphic in the Fukaya category. With this as the goal, I will review the definition of the exact Fukaya category and explain how one goes about computing things in the category (e.g. what it means for a given collection of Lagrangians to "generate" and how to prove generation in special cases). Time permitting (or in a later seminar) I will endeavour to explain the precise statement (due to Abouzaid-Smith) about the Fukaya categories of plumbings which is used by Imagi-Joyce-dos Santos to prove a uniqueness result for special Lagrangians in Cn with given asymptotic behaviour. Note: A followup to this talk will be given by Jason Lotay on 19 November in the King's/UCL Geometric Analysis Reading Seminar. Literature:
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| Monday 10 November, 2014 15:00 16 Taviton Street, room 534 |
Speaker: Chris Wendl Topic: Transversality for multiple covers and Gromov-Witten theory in dimension four Abstract: An oversimplified summary of Gromov-Witten theory is that it defines symplectic invariants by counting J-holomorphic curves for generic choices of a tame almost complex structure J. In general this approach is too naive, because generic perturbations of J do not achieve transversality for multiply covered curves. Nonetheless, the naive approach almost works in dimension four, because for simple numerical reasons, most multiply covered curves can be confined to subsets of codimension at least two in the moduli space and thus safely ignored. The only exceptions are higher genus curves with index 0, which can occur as unbranched multiple covers of simple index 0 curves. In this talk, I will describe how an analytic perturbation theory technique originally due to Taubes and recently worked out in joint work with Chris Gerig can be used to prove that these unbranched covers are in fact Fredholm regular for generic J. I will also sketch some interesting implications that this might have for the Gromov-Witten invariants in dimension four. |
| Monday 3 November, 2014 | No seminar this week, but there is the one day Complex and Symplectic Geometry workshop at Warwick |
| Tuesday 28 October, 2014 15:00 25 Gordon Street, D103 |
Speaker: Alex Cioba Topic: Nicely embedded curves in symplectic cobordisms Abstract: A pseudoholomorphic curve in a 4-manifold is called "nicely embedded" if it has the right intersection-theoretic properties to fit into a local foliation by holomorphic curves. We attempt to describe the structure of the moduli space of nicely embedded curves in certain 4-dimensional symplectic cobordisms; in particular, the intersection theory of punctured curves due to R. Siefring aids us in classifying the degeneracies that occur in the boundary of the moduli space. As an application, we explain why contact manifolds cobordant to the tight 3-sphere must always possess a contractible Reeb orbit that is embedded and unknotted. |
| Tuesday 14 October, 2014 15:00 26 Gordon Square, Room G09 |
Speaker: Chris Wendl Topic: A biased survey on symplectic fillings Abstract: Given a closed contact manifold, for which (if any) compact symplectic manifolds can it occur as convex boundary? This question is intimately related to a number of other fundamental questions in symplectic and contact geometry, as well as low-dimensional topology. I will give a biased sampling of the known results on this subject and discuss some open questions that I find interesting. |
| Monday 31 March, 2014 15:00 UCL maths, Room 500 |
Speaker: Jonny Evans Topic: Lagrangian tori (continuation) |
| Monday 24 March, 2014 15:00 UCL 14 Taviton Street, Room 128 |
Seminar cancelled |
| Monday 17 March, 2014 | No seminar (Simons Center workshop) |
| Monday 10 March, 2014 15:00 UCL maths, Room 706 |
Speaker: Jonny Evans Topic: Lagrangian tori (continuation) 5A |
| Monday 3 March, 2014 15:00 UCL maths, Room 706 |
Speaker: Jonny Evans Topic: Lagrangian tori Abstract: The eventual goal is to explain Vianna's exotic Lagrangian torus in CP2. This will be a preliminary talk on superpotentials (covering material in Auroux's T-duality paper). |
| Wednesday 26 February, 2014 16:00 24 Gordon Square, room 105 |
Speaker: Paolo Ghiggini (Nantes) Topic: An application of L2 Betti numbers to Floer theory Abstract: I will define the Von Neumann dimension, L2 homology and L2 Betti numbers and describe their basic properties. I will also show by an example how they can be used to deduce information on the fundamental group from Floer homological constructions. |
| Monday 24 February, 2014 | No seminar (due to Brussels-London geometry seminar on Tuesday and special seminar on Wednesday -- see below) |
| Monday 17 February, 2014 | No seminar (UCL reading week) |
| Monday 10 February, 2014 15:00 UCL maths, Room 706 |
Speaker: Chris Wendl Topic: Fast pseudoholomorphic planes and intersection theory with constraints (part 2) |
| Friday 7 February, 2014 16:00 UCL maths, Room 505 |
Speaker: Chris Wendl Topic: Fast pseudoholomorphic planes and intersection theory with constraints Abstract: The notion of "fast" finite energy planes was introduced by Hryniewicz in his thesis (based on ideas of Hofer-Wysocki-Zehnder) as a class of holomorphic curves in symplectizations that are especially well suited to forming foliations in a contact 3-manifold. In this talk, I will explain how many of the useful properties of these curves can be derived from Siefring's intersection theory for punctured holomorphic curves. I will begin by giving a general overview of Siefring's intersection theory and a variant that allows constraints on asymptotic decay rates. (This talk is intended as something of an epilogue to Alex's previous two talks, though I will not assume knowledge of those talks.) Literature:
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| Monday 3 February, 2014 | No seminar |
| Monday 27 January, 2014 15:00 UCL maths, Room 706 |
Speaker: Alexandru Cioba Topic: A dynamical characterization of universally tight lens spaces (continuation) |
| Monday 20 January, 2014 15:00 UCL maths, Room 706 |
Speaker: Alexandru Cioba Topic: A dynamical characterization of universally tight lens spaces Literature:: Hryniewicz, Licata and Salomao, A dynamical characterization of universally tight lens spaces |
| Monday 13 January, 2014 15:00 UCL maths, Room 706 |
Speaker: Jacqui Espina Topic: S^1-equivariant symplectic homology and an alternative definition of contact homology (conclusion) |
| Friday 13 December, 2013 | No seminar (UCL end of term) |
| Monday
9 December, 2013 16:00 UCL maths, Room 706 |
Speaker: Jacqui Espina Topic: S^1-equivariant symplectic homology and an alternative definition of contact homology (continuation) |
| Friday 6 December, 2013 | No seminar (Symplectix in Paris) |
| Monday
2 December, 2013 16:00 UCL maths, Room 706 |
Speaker: Marcelo Alves (UL Brussels) Topic: Contact homology and topological entropy |
| Friday 29 November, 2013 17:00 UCL maths, Room 500 |
Speaker: Jacqui Espina Topic: S^1-equivariant symplectic homology and an alternative definition of contact homology Literature:: Bourgeois and Oancea, S^1-equivariant symplectic homology and linearized contact homology |
| Friday 22 November, 2013 | No seminar (Nantes-Orsay seminar in Orsay) |
| Monday
18 November, 2013 16:00 UCL Drayton B05 |
Speaker: Oldřich Spáčil Topic: On the homotopy type of the space of tight contact structures on the three-sphere (conclusion) |
| Friday 15 November, 2013 | No seminar (Madrid conference) |
| Friday 8 November, 2013 | No seminar (UCL reading week, also Symplectix in Paris) |
| Friday 1 November, 2013 17:00 UCL maths, Room 500 |
Speaker: Oldřich Spáčil Topic: On the homotopy type of the space of tight contact structures on the three-sphere (continuation) |
| Friday 25 October, 2013 17:30 (note late starting time!) UCL maths, Room 500 |
Speaker: Oldřich Spáčil Topic: On the homotopy type of the space of tight contact structures on the three-sphere Abstract: Recall that by a theorem of Eliashberg, there exists up to isotopy a unique tight contact structure on S3, or in other words, the space T of tight contact structures on S3 is connected. A parametric version of this result states that the subspase T0 of T consisting of all the tight contact structures which agree with the standard one at some specific point p0 is contractible. As a warm up, we will show how this implies that the homotopy type of T is in fact that of the two-sphere, and that the homotopy type of the group Cont(S3, ξ0) of diffeomorphisms of S3 preserving the standard contact structure is that of the unitary group U(2). We will then try to reconstruct Eliashberg's arguments using taming functions to prove his theorem. Literature: |
| Friday 18 October, 2013 17:00 UCL maths, Room 500 |
Speaker: Chris Wendl Topic: Donaldson hypersurfaces and the Cieliebak-Mohnke approach to transversality in rational Gromov-Witten theory (continuation) |
| Friday 11 October, 2013 17:00 UCL maths, Room 500 |
Speaker: Chris Wendl Topic: Donaldson hypersurfaces and the Cieliebak-Mohnke approach to transversality in rational Gromov-Witten theory Abstract: The Gromov-Witten invariants on symplectic manifolds are defined morally by counting pseudoholomorphic curves for generic tame almost complex structures, and there are standard ways of making this notion precise under certain extra assumptions, e.g. on semipositive symplectic manifolds. The question of how to ensure transversality so that the definition can be extended to all symplectic manifolds is one of the most technically challenging (and politically awkward) issues in symplectic topology: several approaches have been proposed, but many of them are known to have fundamental errors or gaps and are thus not universally accepted. One of the less controversial approaches was introduced in a 2007 paper by Cieliebak and Mohnke, where transversality is achieved in the genus 0 case using largely classical methods involving generic domain-dependent almost complex structures, but with a Donaldson hypersurface (sometimes also called a stabilizing divisor) as additional auxiliary data. Recent papers by Ionel-Parker and Gerstenberger build upon the original idea of Cieliebak and Mohnke to give a fully general definition of the Gromov-Witten invariant for arbitrary genus. In preparation for taking a closer look at those papers in future talks, I will begin by reviewing the basic idea of the Gromov-Witten invariants, the technical problems that arise in the definition, and the Cieliebak-Mohnke approach to their solution. Literature:
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For more information about the seminar contact Chris Wendl by sending e-mail to c dot wendl at ucl dot ac dot uk