** Mini-workshop on**

** Symplectic Topology**

** June 7-8-9, 2023**

**Description**

This workshop on Symplectic Topology will take place on June 7-9, 2023 at Imperial College London.

** Schedule and abstracts**

**Confirmed speakers**

Denis Auroux (Harvard University)

Guillem Cazassus (Oxford University)

Merlin Christ (University of Hamburg)

Aleksander Doan (UCL)

Mohamed El Alami (Edinburgh)

Ailsa Keating (University of Cambridge)

Noémie Legout (Uppsala University)

Hayato Morimura (SISSA)

Agustin Moreno (Heidelberg University)

Ed Segal (UCL)

Gleb Smirnov (University of Geneva)

Ivan Smith (University of Cambridge)

Michael Wemyss (University of Glasgow)

Filip Živanović (University of Edinburgh)

**Schedule**

Time |
Wednesday 7/06 |
Thursday 8/06 |
Friday 9/06 |

9:00-10:00 | Živanović | Cazassus | El Alami |

10:00-10:30 | Break | Break | Break |

10:30-11:30 | I. Smith | Doan | Christ |

11:40-12:40 | Smirnov | Segal | Legout |

12:40-14:00 | Lunch | Lunch | Lunch |

14:00-15:00 | Keating | Auroux | Wemyss |

15:00-15:30 | Break | Break | Break |

15:30-16:30 | Moreno | Morimura | |

17:00 | Reception |

**Titles/Abstracts**

Denis Auroux

Title: Holomorphic discs of negative Maslov index and extended
deformations in mirror symmetry

Abstract: Given a Lagrangian torus fibration on the complement of an
anticanonical divisor in a Kahler manifold, one usually constructs a
mirror space by gluing local charts (moduli spaces of local systems on
generic torus fibers) via wall-crossing transformations determined by
counts of Maslov index 0 holomorphic discs; this mirror also comes
equipped with a regular function (the superpotential) which enumerates
Maslov index 2 holomorphic discs. However, holomorphic discs of negative
Maslov index deform this picture by introducing inconsistencies in the
wall-crossing transformations; the geometric features of the resulting
mirror can be understood in the language of extended deformations of
Landau-Ginzburg models. We illustrate this phenomenon (and show that it
actually occurs) by working through the construction for an explicit
example (a 4-fold obtained by blowing up a Calabi-Yau toric variety).

Guillem Cazassus

Title: Hamiltonian actions on Fukaya categories

Abstract: We will talk about algebraic structures arising in Lagrangian Floer homology in the presence of a Hamiltonian action of a compact Lie group. First, we will show how the Lagrangian Floer complex can be equipped with an A-infinity module structure over the Morse complex of the group, and how this action permits to define equivariant versions of Floer homology. We will then explain how this structure interacts with the structure of the Fukaya category: both can be packaged into (our version of) an A-infinity bialgebra action, giving an alternative answer to a conjecture of Teleman. This should enable one to build an extended topological field theory corresponding to Donaldson-Floer theory. If time permits, we will also explain how our A-infinity bialgebra structure can be upgraded to an A-infinity version of Hopf algebra that we define. This is based on two joint work in progress, one with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and Thibaut Mazuir.

Merlin Christ

Title: Complexes of Fukaya-Seidel categories

Abstract: Given iterated Lefschetz fibrations on exact symplectic manifolds, meaning that the total space of each fibration is the fibre of the next, the associated Fukaya-Seidel categories organize into a chain complex of categories. By this, we mean that the categories are equipped with a differential that squares to the zero functor. On the mirror side, other examples of such categorical complexes arise from derived categories of coherent sheaves on normal crossing divisors. We will sketch the construction of these complexes, mention some properties and state a conjecture about mirror symmetry in this context. This is based on joint work with Tobias Dyckerhoff and Tashi Walde.

Aleksander Doan

Title: An invitation to the Fueter equation

Abstract: This talk is motivated by the idea that there should be a category associated with a pair of complex Lagrangians in a hyperkahler manifold. The building blocks of this putative category are pseudo-holomorphic strips and solutions to the Fueter equation, a nonlinear generalization of the Dirac equation. I will discuss some questions and results in analysis inspired by this proposal, as well as the special case of cotangent bundles. This talk is based on joint work with Semon Rezchikov.

Mohamed El Alami

Title: An open GW-formula for Lagrangians in Fano varieties

Abstract: Suppose you have a (branched) cyclic covering of Fano varieties X -> Y, and that you have a Lagrangian torus L in Y. I will explain a formula relating the super-potentials of L and its pre-image.

Ailsa Keating

Title: An infinitely generated symplectic mapping class group

Abstract: This talk will present recent joint work with Ivan Smith: there are Stein 3-manifolds whose symplectic mapping class groups cannot be finitely generated. The key example is the so-called conifold smoothing, i.e. the Stein manifold which is the complement of a smooth conic in $T^\ast S^3$; we will carefully describe relevant symplectic features, along with ideas from mirror symmetry used in the proof. Time allowing, we will sketch some consequences on the mirror side.

Noémie Legout

Title: Calabi-Yau structure on the Chekanov-Eliashberg algebra

Abstract: We will describe a Floer complex, called the Rabinowitz complex, associated to a pair of exact Lagrangian cobordisms in the symplectization of a contact manifold. In the case where the pair of cobordisms is a 2-copy of a trivial cylinder over a Legendrian sphere, we show that the acyclicity of the Rabinowitz complex is equivalent to the existence of a (n+1)-Calabi-Yau structure (in the sense of Ginzburg) on the Chekanov-Eliashberg algebra of the Legendrian. If time permits we will briefly talk about a work in progress with Georgios Dimitroglou-Rizell for the case when the Legendrian is not necessarily a sphere. In this case, and under certain hypothesis, the Chekanov- Eliashberg algebra admits a relative Calabi-Yau structure.

Hayato Morimura

Title: Homological mirror symmetry for complete intersections in algebraic tori

Abstract: Recently, Gammage--Shende established homological mirror symmetry(HMS) for very affine hypersufaces under a certain assumption, which allowed them to construct a global skeleton by gluing skeleta of pairs of pants appearing in the pants decompositions. Along the global skeleton, they glued HMS for pairs of pants. In this talk, I will illustrate how to prove HMS for complete intersections of very affine hypersurfaces. Extending the above approach is unfeasible, as the current knowledge does not provide any method to construct a global skeleton. Our approach bypasses this issue by constructing only smaller pieces of skeleta near the gluing regions. It yields new results already in hypersurface case. This is based on joint work with Nicolò Sibilla(SISSA) and Peng Zhou(UC Berkeley).

Agustin Moreno

Title: Symplectic geometry of Anosov flows in dimension three

Abstract: In this talk, I will illustrate how symplectic geometry can be leveraged in the study of uniformly hyperbolic dynamics. Namely, every Anosov flow on a three-manifold M has an associated non-Weinstein Liouville domain V=[-1,1]xM, whose invariants (e.g. symplectic cohomology, Rabinowitz-Floer cohomology, wrapped Fukaya category) are homotopy invariants of the flow. Moreover, every orbit of the flow gives an exact Lagrangian cylinder, all of which are nontrivial and independent of each other in the Fukaya category, such that the A_infty category they generate does not satisfy Abouzaid’s generation criterion, in contrast to the Weinstein case. This leaves the open question on whether these Lagrangians split-generate, in which case the whole Fukaya category exhibits starkly contrasting behavior as that of the Weinstein case; or whether there are undiscovered Lagrangians. This is based on joint work with Kai Cieliebak (Augsburg), Oleg Lazarev (UMass Boston), Thomas Massoni (Princeton).

Ed Segal

Title: Fukaya categories at singular values of the moment map

Abstract: Given a Hamiltonian torus action on a symplectic manifold, Fukaya and Teleman tell us that we can relate the Fukaya category of each symplectic quotient to an equivariant Fukaya category on the original manifold. I'll present some conjectures that extend this story - in certain special examples - to singular values of the moment map. I'll also explain the mirror symmetry picture that we use to support our conjectures, and how we interpret our claims in Teleman's framework of `topological group actions' on categories. This is joint work with your esteemed host.

Gleb Smirnov

Title: Lagrangian tori in K3 surfaces

Abstract: Sheridan and Smith proved that every Maslov-zero Lagrangian torus in a K3 surface has a nontrivial homology class. In this talk, we will prove this result under a weaker assumption on the Maslov class. Further, we will see that every homologically nontrivial Lagrangian torus is Maslov-zero.

Ivan Smith

Title: Quantum (Morava) K-theory

Abstract: I will explain how to construct Gromov-Witten invariants in K-theory or Morava K-theory for any closed symplectic manifold. The splitting axiom for the invariants in general involves the full tautological ring of moduli spaces at lower degree, and is governed by the formal group law of the underlying cohomology theory. Specialised to genus zero this yields associative products on quantum K-theory and quantum Morava K-theory (at primes p>2). This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Michael Wemyss

Title: Derived Deformations of Crepant Curves

Abstract: Motivated by various contraction conjectures, I will describe the full A_infty structure associated to a general (-3,1)-curve inside a smooth CY 3-fold. From a mirror symmetry point of view, what this really does is construct a B-side geometric model to every "necklace" potential in the 2-loop quiver. This is joint work with Gavin Brown.

Filip Živanović

Title: Symplectic C*-manifolds

Abstract: I will define a broad family of open symplectic manifolds admitting pseudoholomorphic C*-actions, which contains many interesting spaces, such are equivariant resolutions of affine singularities, twisted cotangent bundles, semiprojective toric varieties and Higgs moduli. I will explain how one can construct symplectic cohomology and spectral sequences converging to it, although the symplectic structures on these spaces are typically highly non-exact at infinity. As a consequence, we get a filtration on their quantum cohomology rings by ideals, which should be thought of as a Floer-theoretic analogue of Atiyah-Bott filtration. This filtration is in practice computable using the aforementioned spectral sequences, which we do for twisted cotangent bundles, ADE resolutions, certain Slodowy varieties and certain parabolic Higgs moduli spaces. For the last ones, we compare it with the “P=W“ filtration. This is joint work with Alexander Ritter.

** Registration**

Everyone is welcome. To participate, please register your interest by sending an e-mail to * n.porcelli@imperial.ac.uk .*

** Practical information**

The mini-workshop will take place in Lecture Theatre 1 in the Blackett Laboratory at Imperial College London. Click here for directions.

For any query, get in touch with the organisers:

Yankı Lekili * y.lekili@imperial.ac.uk*

Noah Porcelli * n.porcelli@imperial.ac.uk*

Ilaria Di Dedda * ilaria.di_dedda@kcl.ac.uk*

Daniil Mamaev * d.mamaev22@imperial.ac.uk*