Geometri- och topologiseminariet
Wednesdays at 14:15 on ZOOM.
For more info contact the organiser: Georgios Dimitroglou Rizell.
1 December: Benoit Joly (Ruhr University, Bochum)
Title: Barcodes for Hamiltonian homeomorphisms of surfaces
Abstract: In this talk, we will study the Floer Homology barcodes from a dynamical point of view. Our motivation comes from recent results in symplectic topology using barcodes to obtain dynamical results. We will give the ideas of new constructions of barcodes for Hamiltonian homeomorphisms of surfaces using Le Calvez's transverse foliation theory. The strategy consists in copying the construction of the Floer and Morse Homologies using dynamical tools like Le Calvez's foliations.
Zoom Meeting ID: 682 1006 8219
8 December: Irena Matkovic (Uppsala University)
15 December: Thomas Kragh (Uppsala University)
Title: Spaces of cobordisms and spaces of Legendrians in jet-1 bundles.
Abstract: I will construct a map from certain spaces of cobordisms into the space of Legendrians in cotangent bundles, I will then argue using generating functions why these are often highly non-trivially on homotopy groups. This is joint work with Yasha Eliashberg.
24 November: Georgios Dimitroglou Rizell (Uppsala University)
Title: Non-squeezing for Legendrian knots, and C^0 contactomorphisms
Abstract: In joint work with M. Sullivan where we show, that a contact isotopy cannot squeeze a Legendrian knot onto a non-Legendrian knot, meaning that the distance between the knots tend to zero while the two knots are smoothly isotopic in some tubular neighborhood. A similar mechanism allows us to show that any smooth knot which is the C0-limit of a Legendrian knot is itself contactomorphic to the original Legendrian. In particular, smooth C0-limits of Legendrian knots are Legendrian.
10 November: Lukas Nakamura (Uppsala University)
Title: Immediate displaceability of non-Lagrangian/non-Legendrian and C^0-limits of Lagrangian/Legendrian manifolds.
Abstract: Laudenbach and Sikorav proved that the immediate displaceability of a closed non-Lagrangian submanifold is only obstructed by the topology of the embedding. They used this to show that smooth C^0-limits of Lagrangians are again Lagrangian. In this talk, I will explain these results and some corresponding results for contact manifolds.
3 November: Johan Asplund (Uppsala University)
Title: Simplicial descent for Chekanov-Eliashberg dg-algebras
Abstract:The result of sectorial descent for wrapped Fukaya categories is a local-to-global principle due to Ganatra-Pardon-Shende which holds for sectorial covers of Weinstein sectors. In this talk we introduce simplicial decompositions of Weinstein manifolds, which is a surgery description of so-called good sectorial covers, and generalizes the operation of Weinstein connected sum. We prove that the Chekanov-Eliashberg dg-algebra of the attaching spheres satisfies a descent (cosheaf) property with respect to a simplicial decomposition.
20 October: Côme Dattin (Uppsala University)
Title: Sutured Legendrian homology, stops and the conormal of braids
Abstract: The unit conormal construction takes us from the smooth world to the contact world, hence Legendrian invariants of conormals yield invariants of smooth submanifolds. In this talk we will show that, if the conormals of two braids are Legendrian isotopic, then the braids are equivalent. The main tool will be the wrapped sutured homology, an invariant of Legendrians with boundary, and its associated exact sequence.On the way we will sketch the definition of a 2-sutured manifold, and present various descriptions of Liouville sectors. The result of sectorial descent for wrapped Fukaya categories is a local-to-global principle due to Ganatra-Pardon-Shende which holds for sectorial covers of Weinstein sectors. In this talk we introduce simplicial decompositions of Weinstein manifolds, which is a surgery description of so-called good sectorial covers, and generalizes the operation of Weinstein connected sum. We prove that the Chekanov-Eliashberg dg-algebra of the attaching spheres satisfies a descent (cosheaf) property with respect to a simplicial decomposition.
13 October: Adrian Petr (Nantes University)
Title: Invariant of the Legendrian lift in S^1 x P of an exact Lagrangian submanifold.
Abstract: Any exact Lagrangian submanifold L in a Liouville manifold P lifts to a Legendrian submanifold in S^1 x P. The goal of the talk is to relate the Chekanov-Eliashberg algebra of this Legendrian to the Floer A_infty algebra of L.
6 October: Alice Hedenlund (Uppsala University)
Title: The Tate spectral sequence II: The Tate construction and spectral sequences
Abstract: This is the second part of a two-part introduction to the Tate construction and the Tate spectral sequence. The Tate construction was first introduced by Greenlees and should be seen as a generalisation of Tate cohomology to the setting of homotopical algebra. We start by discussing Tate cohomology, to understand the classical context, and go on to define the Tate construction in the setting of spectra with group actions. Next, we explore the Tate spectral sequence, which is a spectral sequence that computes the homotopy groups of the Tate construction, and discuss what issues one might expect to pop up when studying this spectral sequence. Lastly, we sketch how these issues were solved in joint work with J. Rognes.
29 September: Alice Hedenlund (Uppsala University)
Title: The Tate spectral sequence I: Homotopical algebra and spectra
Abstract: This is the first part of a two-part introduction to the Tate construction and the Tate spectral sequence. In this talk, we give an brief introduction to the subject known as homotopical algebra. While an aspiring mathematician’s first exposure to “homotopy” often comes packaged in a topology course, one could argue that this is mostly a historical feature, and that there is nothing intrinsically topological about the concept, at all. In this way, homotopical phenomena is much more than just a subset of algebraic topology, with applications spanning through many diverse areas of mathematics. In particular, it is often useful to extend the world of algebra past homological algebra and into what is known as homotopical algebra. After discussing the basis ideas underpinning this field of mathematics, we introduce and discuss the mathematical objects called spectra, which could be viewed as the abelian groups of homotopical algebra. We will try to keep things conceptual, focusing on why rather than the more technical how. As such, the talk is meant to be accessible to a diverse audience.
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.
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.