# Geometri- och topologiseminariet

Wednesdays at **14:15** on ZOOM.

*For more info contact the organiser: Georgios Dimitroglou Rizell.*

**22 September**** **(two short talks 14:15-15:00 and 15:15-16:00)**: Keita Nakagane **(Uppsala University)

Title:* **HOMFLY homology and its extreme parts*

Abstract: I will roughly review the theory of the HOMFLY polynomial and the Khovanov--Rozansky HOMFLY homology, and discuss what I know about them: results on certain extreme parts of the homology, one possible relation with Legendrian knot theory, and so forth. This talk will contain a joint work with E. Gorsky, M. Hogancamp, and A. Mellit, and one with T. Sano.

Zoom link:https://uu-se.zoom.us/j/68210068219

Zoom Meeting ID: 682 1006 8219

**Upcoming talks**

**29 September: Alice Hedenlund **(Uppsala University)

**6 October: Alice Hedenlund **(Uppsala University)

**Previous talks**

**16 June****: Baptiste Louf **(Uppsala University)

Title:* **Discrete hyperbolic geometry*

Abstract: I will talk about combinatorial maps, which are discrete surfaces built by gluing polygons together. They have been given a lot of attention in the last 60 years, and here we will focus on the geometric properties of large random maps, in a rather new regime where the genus of the underlying surface goes to infinity.

I will give an overview of the existing results and open problems, guaranteed without technical details.

**12 May****: Noémie Legout **(Uppsala University)

Title:* **Rabinowitz Floer complex for Lagrangian cobordisms*

Abstract: I will define a Floer complex associated to a pair of transverse Lagrangian cobordisms in the symplectization of a contact manifold, by a count of SFT pseudo-holomorphic discs. Then I will show that this complex is endowed with an A_\infty structure. Moreover, I will describe a continuation element in the complex associated to a cobordism L and a small transverse push-off of L.

**5 May****: Agustin Moreno **(Uppsala University)

Title:* **On the three-body problem, cone structures, entropy and open books.*

Abstract: In this talk, I will describe how cone structures naturally appear in the context of Reeb flows adapted to iterated planar open books on contact 5-folds. I will also discuss a notion of topological entropy for cone structures on an arbitrary manifold with a metric, and discuss possible applications. In particular, I will outline how we expect to use this to prove that the dynamics of the spatial circular restricted three-body problem, in low energies and near the primaries, can be arbitrarily C^\infty-approximated by flows with positive topological entropy, whenever the planar problem is dynamicaly convex

**14 Avril **at** 11:00 ***(note the unusual time!)***: Youngjin Bae **(Incheon National University)

Title:* **Legendrian graphs and their invariants*

Abstract: Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship.

This is a joint project with Byunghee An, and partially with Tamas Kalman and Tao Su

**7 Avril: Guillem Cazassus **(University of Oxford)

Title:* **Hopf algebras, equivariant Lagrangian Floer homology, and cornered instanton theory*

Abstract: Let G be a compact Lie group acting on a symplectic manifold M in a Hamiltonian way. If L, L' is a pair of Lagrangians in M, we show that the Floer complex CF(L,L') is an A-infinity module over the Morse complex CM(G) (which has an A-infinity algebra structure involving the group multiplication). This permits to define several versions of equivariant Floer homology.

It also implies that the Fukaya categoy Fuk(M), in addition to its own A_infinity structure, is an A-infinity module over CM(G). These two structures can be packaged into a single one: CM(G) is an A-infinity bialgebra, and Fuk(M) is a module over it. In fact, CM(G) should have more structure, it should be a Hopf-infinity algebra, a structure (still unclear to us) that should induce the Hopf algebra structure on H_*(G).

Applied to some subsets of Huebschmann-Jeffrey's extended moduli spaces introduced by Manolescu and Woodward, this construction should permit to define a cornered instanton theory analogous to Douglas-Lipshitz-Manolescu's construction in Heegaard-Floer theory.

This is work in pogress, joint with Paul Kirk, Artem Kotelskiy, Mike Miller and Wai-Kit Yeung.

**31 March: Russel Avdek **(University of Uppsala)

Title:* **Simplified SFT moduli spaces for Legendrian links*

Abstract: We study the problem of counting "full SFT" holomorphic curves with boundary on the Lagrangian cylinder RxL over a Legendrian link L in contact 3-space, allowing the domain S to have non-trivial H^1. Our counting problems are formally similar to computations of differentials in Heegaard-Floer and can be expressed as calculations of Euler numbers of H^1 bundles over combinatorial moduli spaces. When S is not a disk, these counts can not in general be computed combinatorially. However, we have a Sarkar-Wang type result which says that after an isotopy of L, any "full SFT" differential can be computed from the disks of rational SFT in the style of Chekanov.

**17 March: Alexandre Jannaud **(University of Neuchâtel)

Title: *Dehn-Seidel twist, C^0 symplectic geometry and barcodes*

Abstract: In this talk I will present my work initiating the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory.

Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to $C^0$ settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that the generalized Dehn twist is not in the connected component of the identity in the group of symplectic homeomorphisms. Doing so, we prove the non-triviality of the $C^0$ symplectic mapping class group of some Liouville domains.

**10 March: Paolo Ghiggini** (Uppsala University)

Title: *From compact Lagrangian submanifolds to representations of the Chekanov-Eliashberg dga.*

Abstract: I will describe an A_\infty functor from the compact Fukaya category of a Weinstein manifold to the category of finitely dimensional modules over the Chekanov-Eliashberg dga of the attaching spheres of the critical handles in a Weinstein handle decomposition. This is a joint work in progress with Baptiste Chantraine and Georgios Dimitroglou Rizell.

**17 February: Côme Dattin** (Uppsala University)

Title: *The Legendrian homology of a fiber in US^3, stopped by the conormal of an hyperbolic knot*

The goal of this talk is to compute the homology of a simple Legendrian in a sutured contact manifold. Such a manifold can be seen as either generalizing the contactisation of a Liouville domain, or as a presentation of a contact manifold with convex boundary. We will also give a stopped point of view of those objects. For the manifold U S^3 \ U_K S^3, where K is an hyperbolic knot, we can instead study U(S^3\K). In this case the Legendrian homology of a fiber, with its product structure, recovers the fundamental group of S^3\K, thus it is a complete invariant of the knot.

**10 February: Jian Qiu** (Uppsala University)

Title: *Rozansky Witten theory, localization and tilting*

Abstract: This talk is based on my recent paper of the same title. The Rozansky-Witten (RW) theory is a 3D topological field theory that can be used to produce 3 manifold invariants valued in the (equivariant) cohomology ring of a chosen hyperKahler variety. Physically, the theory itself arose from the low energy limit of some 3D supersymmetric gauge theory, mathematically, there is a 2-category construction with the target category constructed from D^b(X).

In this talk, I will first spend some time reviewing the RW theory especially its similarity to the Chern-Simons theory, which is perhaps more familiar to the audience. I will review how the Hilbert space is constructed, how concrete computations can be done. Over a restricted set of 3 manifolds, one can obtain exact results via the localisation technique, though I will be rather brief on this one. However, I would like to speak more about using tilting bundle as a tool to obtain the so called Verlinde formula for computing the dimension of the Hilbert space. The Verlinde formula Is something that appears often in the vertex algebra context, but its relevance in RW theory was proposed by Gukov and company last summer and was what started this paper.

**3 February **at **15:45*** (note the unusual time!)***: Johan Asplund** (Uppsala University)

Title: *Chekanov-Eliashberg dg-algebras for singular Legendrians: Applications and computations*

Abstract: In this talk we continue the discussion from last time about the Chekanov-Eliashberg dg-algebra for skeleta of Weinstein manifolds. We explain how our natural surgery pushout diagram leads to a proof of the stop removal formulas of Ganatra-Pardon-Shende. We then explicitly compute the Chekanov-Eliashberg dg-algebra in some examples, including links of some Lagrangian arboreal singularities. Finally we discuss exact singular Lagrangian cobordisms of singular Legendrians, and indicate in some examples that exact singular Lagrangian fillings need to be "sufficiently singular", depending on the singularities of the Legendrian. The talk is based on joint work with Tobias Ekholm.

**27 January: Tobias Ekholm** (Uppsala University)

Title: *Chekanov-Eliashberg dg-algebras for singular Legendrians*

Abstract: The Chekanov--Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of push-out diagrams for wrapped Fukaya categories and stop removal formulas from Ganatra-Pardon-Shende. It furthermore leads to a proof of the conjectured surgery formula relating partially wrapped Floer cohomology to Chekanov--Eliashberg dg-algebras with coefficients in chains on the based loop space. The talk reports on joint work with Johan Asplund.