# Previous geometry and topology seminars

## 2022

### 8 December: Thomas Kragh (Uppsala university)

Title: *Seiberg-Witten Floer Homotopy Types II: Family Floer theory and twisted spectra *

Abstract: I will start by discussing the example that was discussed at length last time during Alice's talk, and provide some variations and context (relating it to some notions of family Floer theory in cotangent bundles, background classes, partially wrapping, and micro-local support - hoping that each audience member will find at least one of these relations provides some insight into such examples and why they are interesting). I will then describe how to construct somewhat explicit twisted spectra out of finite dimensional approximations and Conley indices in the Seiberg-Witten (SWF) setting (this setting is actually much easier than the cotangent bundle). I will then talk about other twisted categories than that of twisted spectra. In particular I will describe how S^1-equivariant homotopy theory naturally arises in this context and how we actually construct a "twisted parametrized S1-equivariant spectrum" representing SWF (I will not assume much knowledge about any of these stable categories - and I will try and keep most of the general discussion at a heuristic level). This is joint work with A. Hedenlund and S. Behrens.

### 1 December: Alice Hedenlund (Uppsala university)

Title: *Seiberg-Witten Floer Homotopy Types I: Twisted Spectra *

Abstract: Seiberg–Witten theory has played a central role in the study of smooth low-dimensional manifolds since their introduction in the 90s. Parallel to this, Cohen, Jones, and Segal asked the question of whether various types of Floer homology could be upgraded to the homotopy level by constructing (stable) homotopy types encoding Floer data. In 2003, Manolescu constructed Seiberg-Witten Floer spectra for rational homology 3-spheres, and in particular used these to settle the triangulation conjecture. A precursor to Manolescu's Seiberg-Witten Floer spectra, the Bauer-Furuta invariant, was moreover used by Furuta to make significant progress on the "11/8 Conjecture" which deals with what quadratic forms are realisable as the intersection form on a smooth 4-manifold.

This is the first talk of a two-part series reporting on joint work in progress with S. Behrens and T. Kragh. In this talk I will give an expository introduction to twisted spectra, which are essential for constructing Floer homotopy types in the situation where our infinite-dimensional manifold is "non-trivially polarised". Roughly, one could think of twisted spectra as arising as global sections of a bundle whose fibre is the (infinity-)category of spectra. I will also explain how these mathematical objects naturally appear in Seiberg-Witten Floer theory. Next week, Thomas will go into the more concrete constructions of twisted spectra from Seiberg-Witten Floer data using finite-dimensional approximation and Conley index theory.

### 24 November: Tobias Ekholm (Uppsala university)

Title: *Counting bare curves III*

Abstract: This is a follow up of the talk "Counting bare curves II".

### 17 November: Tobias Ekholm (Uppsala university)

Title: *Counting bare curves II*

Abstract: This is a follow up of the talk "Counting bare curves I" which was held on October 20th.

### 10 November: Two talks

### Marco Golla (Nantes university)

Title:* **Alexander polynomials and symplectic curves in CP^2.*

Abstract: Libgober defined Alexander polynomials of (complex) plane projective curves and showed that it detects Zariski pairs of curves: these are curves with the same singularities but with non-diffeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is work in progress with Hanine Awada*.*

### Renato Vianna (Federal university of Rio de Janeiro)

Title:* **Lagrangian fibrations on smoothing of algebraic cone.*

Abstract: This is joint work with Santiago Achig-Andrango. Given a lattice polytope Q in R^n, we can consider its cone C(Q), in R^n+1, given by taking the rays from the origin passing through Qx{1} in R^nxR.

Being a rational polyhedral cone, we associate an affine toric algebraic variety Y, which is often singular. Altaman shows how to construct deformations of Y from a Minkowiski decomposition Q = M_1 + … + M_k of Q. Gross constructs a special Lagrangian fibration out of this data, but with general Lagrangian fibre equals to T^n x R (rather than T^n+1). Auroux describes in his work how to obtain a singular Lagrangian fibration with a generic fibre a torus T^n out of a auxiliary complex (symplectic) fibration with a fibrewise preserving T^n-1 action.

In his work, Achig-Andrango imposes conditions in the Minkowiski decomposition to ensure that Altman’s deformation gives a smoothing of Y and constructs a complex fibration in the generic fibre Y_e, with genral fibre (C^*)^n and singular fibres with local model associated to each term M_i of the Minkowiski decomposition. Using that as an auxiliary complex fibration, Achig-Angrango constructs a singula Lagrangian fibration, studies its monodromies and show it can be represented by a convex base diagram with cuts, whose image in R^n+1 is the dual of the cone C(Q). Moreover, the fibration contains a one parameter family of monotone Lagrangian fibres, whose superpotential can be obtained under wall-crossing formulas and described in terms of the Minkowiski decomposition of Q.

We will present Achig-Andrango’s work and time permiting, discuss some further developments.

**27 October: Georgios Dimitroglou-Rizell (Uppsala university)**

Title:* **Positive loops of non-Legendrians, and C^0 contactomorphisms.*

Abstract: We show that any n-dimensional non-Legendrian submanifold of a 2n+1-dimensional contact manifold admits a small positive loop that preserves parametrisations, which is generated by a non-negative global contact isotopy. The existence of the positive loop is then used to show that a homeomorphism that arises as the C0-limit of contactomorphisms has the property that the image of a Legendrian submanifold is again a Legendrian, under the assumption that the image is smooth. This is joint work in progress with M. Sullivan.

### 20 October: Two talks

### Tobias Ekholm (Uppsala university)

Title:* Counting bare curves I*

Abstract: In a series of talks we will discuss how to count bare holomorphic curves with Lagrangian boundary conditions in a Calabi-Yau 3-fold. Here bare means that the curves does not have any symplectic area zero components. The main application and motivation for this is the construction of invariant skein valued counts of open curves. In this first talk we will explain why the usual approaches to Gromov-Witten counts does not work to define skein valued counts and present one of the key technical results for bare counts: we finds a class of perturbations for the Cauchy-Riemann equations that has the property that if a ghost bubbles forms in the limit of sequence of bare curves then it leaves a trace (e.g. vanishing complex derivative) on the limiting positive area curve on which it is attached. In later talks we will construct a perturbation scheme using polyfold language with perturbations of this kind that gives rise to solutions of the perturbed Cauchy-Riemann equations that admit forgetful maps. The talks report on joint work with Vivek Shende.

**Yuan Yao (University of California, Berkeley)**

Title:* **Computing Embedded Contact Homology in the Morse-Bott Setting using Cascades *

Abstract: I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to compute ECH using cascades.

**13 October: **Matthew Habermann (Hamburg university)

Title:* **Homological Berglund--Hübsch--Henningson mirror symmetry for curve singularities*

Abstract: Invertible polynomials are a class of hypersurface singularities which are defined from square matrices with non-negative integer coefficients. Berglund—Hübsch mirror symmetry posits that the polynomials defined by a matrix and its transpose should be mirror as Landau—Ginzburg models, and an extension of this idea due to Berglund and Henningson postulates that this equivalence should respect equivariant structures. In this talk, I will begin by giving some background and context for the problem, and then explain my recent work on proving the conjecture in the first non-trivial dimension; that of curves. The key input, inspired by the derived McKay correspondence, is a model for the orbifold Fukaya—Seidel category in this context.

**29 September: Wenyuan Li (Northwestern university)**

Title:* **Lagrangian cobordism functor in sheaf theory*

Abstract: Given a Lagrangian cobordism between Legendrian submanifolds in the unit cotangent bundles, we construct a functor between the categories of sheaves on the base manifold with singular support on the Legendrians. This provides a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies enhanced with loop space coefficients. The construction works for singular exact Lagrangian cobordisms between singular Legendrians as well. I will then deduce some new obstructions to Lagrangian cobordisms between high dimensional Legendrians. If time permits, I will discuss immersed exact Lagrangian cobordisms based on future work in progress by constructing the cobordism functor from a different perspective.

### 22 September: Filip Strakoš (Uppsala university)

Title:* **DGA-homotopy criteria for augmentations of CE-algebras*

Abstract: In a suitably generic situation the differential of Chekanov-Eliashberg algebra can be augmented. The set of all DGA-homotopy classes of augmentations is then a Legendrian isotopy invariant, and so is its cardinality. We will use linearized and bilinearized Legendrian contact homology to derive DGA-homotopy criteria for connected and disconnected Legendrian submanifolds respectively. In particular, we will see that while in the connected case the DGA-homotopy criterion descends to the chain level, this is no longer true in the disconnected case. Then we will apply the results to the question of geography of the bilinearized Legendrian contact homology generalizing results of Bourgeois and Galant.

### 23 June: Yash Deshmukh (Columbia university)

Title:* **Homotopically extending actions of properads of Riemann surfaces*

Abstract: A result of G. Drummond-Cole relates the framed little disks operad to the genus zero Deligne-Mumford operad. In this talk, I will discuss the higher genus, multiple input-output analog of this result. I will explain that extending an action of the properad of Riemann surfaces with parameterized boundaries to an action of the properad of nodal Riemann surfaces is equivalent, homotopically, to providing the data of trivializations of certain operations coming from the moduli spaces of annuli. In the process, I will also introduce a new partial compactification of the moduli spaces of Riemann surfaces which is relevant to the study of symplectic cohomology, and describe a version of the above statement concerning extension of actions to these partial compactifications.

### 16 June: Luis Diogo (Uppsala university)

Title:* **The Floer cohomology of Lagrangian tori in the cotangent bundle of the 2-sphere*

Abstract: One of the simplest symplectic manifolds that one can consider is the cotangent bundle of the 2-sphere, and we will address the problem of classifying its monotone Lagrangian tori as objects in the Fukaya category. Specifically, we show that the Floer cohomology algebras of such Lagrangians are very similar to those of the classical Entov-Polterovich tori.

### 9 June: Two talks

### Cecilia Karlsson (University of Borås)

Title:* **Chekanov-Eliashberg DGA for Weinstein handle attachments in higher dimensions*

Abstract: The symplectic homology of a Weinstein manifold is encoded in the Chekanov-Eliashberg DGA of the attaching spheres of the top index handles. In this talk I describe a geometric set-up where we can calculate the Chekanov-Eliashberg DGA of the attaching spheres from Legendrians in 1-jet spaces. Since the latter is well-studied this simplifies the calculations. If time permits I will use these techniques to calculate the singular homology of the free loop space of CP^2.

This is a generalization of similar work by Ekholm-Ng to higher dimensions.

### Martin Bäcke (Uppsala university) *Master's thesis presentation*

Title:* **DG-algebra computations for singular Legendrians*

Abstract: We compute the homology of the Chekanov-Eliashberg algebra in the boundary of a Weinstein surface, also known as the internal algebra of Ekholm-Ng. This algebra appears as a subalgebra of the Chekanov-Eliashberg algebra of singular Legendrian skeleta of Weinstein hypersurfaces, introduced by Asplund-Ekholm. We show a result relating the algebras arising from different choices of handle decomposition of the surface, and obtain new examples of Legendrians with computable homology.

### 12 May: Sylvain Douteau (Stockholm university)

Title:* **Stratified homotopy theory and invariants of embeddings*

Abstract: The study of stratified spaces - and more recently, of their associated homotopy theory - was originally understood as a way of dealing with singularities. This is illustrated by Whitney's historical theorem, which states that all analytic varieties over R or C can be stratified in such a way that the strata are smooth manifolds that glue together properly. However, stratifications can also be used to encode structures on smooth objects, such as embeddings. In this context, stratified homotopy theory can be used to obtain relevant invariants. In this talk, I will present an overview of the homotopy theory of stratified spaces and of its applications to embeddings, by investigating the case of knots

### 5 May: Thibaut Mazuir (IMJ-PRG, Paris)

Title: *Higher algebra of A-infinity algebras in Morse theory*

Abstract: In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being a Kan complex. The combinatorics of n-morphisms are moreover encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.

Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory.

### 28 April: Johan Rydholm (Uppsala university)

Title:* Homological mirror symmetry for A_n resolutions and cyclic plumbings*

Abstract: We show a homological mirror symmetry equivalence between cyclic plumbings, that is, Milnor fibres of type A_n with a conic removed, and the resolutions of A_n singularities. This is done by symplectically realizing the Ginzburg algebra of the quiver with extended A_n form, using the surgery formula, and showing that this algebra is formal.

### 21 April: Georgios Dimitroglou-Rizell (Uppsala university)

Title: *Floer homology for monotone Weinstein skeleta*

Abstract: We discuss the connections between refined potentials, Floer homology of singular monotone Lagrangians, and deformations of the Hochschild cocomplex. This is joint work in progress with P. Ghiggini.

### 31 March: Daniel Kaplan (Hasselt University, Belgium)

Title: *DG algebras associated to plumbed cotangent bundles*

Abstract: Given a graph, one can build a 4-dimensional symplectic manifold by plumbing cotangent bundles of 2-spheres according to the graph. Etgu--Lekili established that the wrapped Fukaya category of such a manifold is equivalent to the category of dg-modules for the dg multiplicative preprojective algebra. This motivates the question of whether such dg algebras are formal (i.e., quasi-isomorphic to their homology as dg-algebras). If yes, then the wrapped Fukaya category is equivalent to the category of modules for the multiplicative preprojective algebra. In the first half of the talk, I will sketch this motivation but then leave the setting of geometry/topology to discuss a purely algebraic approach to formality developed in joint work with Travis Schedler (following work of Etingof--Ginzburg on non-commutative complete intersections, which itself utilizes older work of Anick). We prove the formality of these dg algebras if the graph is connected with a cycle. We conjecture that formality holds if the graph is connected and not ADE Dynkin, by analogy to the additive dg-preprojective algebra (also called the Ginzburg dg algebra with zero potential). In the second half of the talk, I will explain how to employ Bergman's Diamond Lemma for ring theory to establish this formality. By focusing on small examples, and side-stepping technical details, I hope to make the talk accessible to both geometers and algebraists.

### 24 March: Russell Avdek (Uppsala University)

Title: *Relative RSFT via planar diagrams*

Abstract: I’ll describe work in progress on a new version of Legendrian rational SFT which generalizes Legendrian contact homology (LCH). It reformulates Ekholm’s RSFT using ideas from Chas-Sullivan's chord diagram formalism for string topology and Hutchings’ ``q-variables only’’ version of closed-orbit RSFT. The result is a differential graded algebra (DGA) with a special filtration which is used to enhance the usual augmentation theory. Basic computations show that LCH and our RSFT contain very different qualitative information. If time permits, I’ll demo software which computes augmentations and bilinearizations of the new RSFT for links in R3.

### 17 March: Fabio Gironella (Humboldt University)

Title: *Liouville orbifold fillings of contact manifolds.*

Abstract: The topic of the talk will be Floer theories on Liouville orbifolds with smooth contact boundary. More precisely, I will describe the construction, which only uses classical transversality techniques, of a symplectic cohomology group on such symplectic orbifolds. Time permitting, I will then describe how to deduce the existence, in any odd dimension at least 5, of a pair of contact manifolds with no exact symplectic (smooth) cobordisms in either direction. This is joint work with Zhengyi Zhou.

### 10 March: Angela Wu at 3pm *(note the unusual time!)* (Louisiana State University)

Title: *Obstructing Lagrangian concordance for closures of 3-braids *

Abstract: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself. The proof uses a variety of techniques in contact and symplectic geometry, from open book decompositions and Weinstein handle diagrams, to symplectic homology and the Legendrian contact homology DGA.

### 3 March at 2pm *(note the unusual time!)*: Honghao Gao (Michigan State University)

Title: *Obstructions to exact Lagrangian fillings*

Abstract: We study Legendrian knots and their exact Lagrangian fillings. Viewed as cobordisms, exact Lagrangian fillings functorially induce morphisms between moduli spaces between Legendrian invariants. Algebraic structures over these moduli spaces can be used to define obstructions to exact Lagrangian fillings. In this talk, I will report a joint work with Dan Rutherford, where we construct an obstruction from the algebraic geometry of the augmentation variety, and compare it with previously known obstructions arising from A-infinity algebras.

### 17 February: Paolo Ghiggini (Nantes University)

Title: *Speculations on singular Lagrangian submanifolds and homological mirror symmetry of CP^3.*

### 10 February: Oliver Leigh (Uppsala University)

Title: *r-Spin Hurwitz numbers via Stable Maps with Divisible Ramification*

Abstract: Hurwitz numbers enumerate smooth covers of the projective plane. Classically, one also imposes a condition requiring all ramification to be of order 1. There are many beautiful and deep results related to classical Hurwitz numbers. This includes the ELSV formula, a link to Gromov-Witten theory, links to mathematical physics and links to topological recursion.

A natural question to ask is: How many of these results hold when the condition "all ramificaiton is order 1" is replaced with "all ramificaiton is order r"? In this talk we will answer this quesion using the theory of stable maps with divisible ramification. This will include links to r-spin theory via Zvonkine’s r-ELSV formula and a discussion of the subtleties arising in this situation.

### 3 February at 15:30 *(note the unusual time!)*: Orsola Capovilla-Searle (UC Davis)

Title:* Infinitely many planar exact Lagrangian fillings and symplectic Milnor fibers*

Abstract: We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality.

## 2021

### 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.

**8 December: Irena Matkovic **(Uppsala University)

Title: *On contact surgery*

Absract: I will give a hands-on presentation of contact surgery in dimension 3, and relate the contact properties of a knot complement to those of its Dehn fillings. A large part of the talk will be expository.

**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.

**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 April **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 April: 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.

## 2020

### 24 June *(***new date!**)**: Baptiste Chantraine **(Université de Nantes)

**new date!**)

*Title: Invariant of Legendrian submanifolds in non co-orientable contact structure*

Abstract: I want to report on some work in progress on invariants of Legendrian submanifolds in contact manifolds where the contact structure is not coorientable. We will see two structures arising when considering the Z_2 action on the lift of the Legendrian to its coorientation cover: one is a structure of involution on the augmentation category and the other is a Z_2 equivariant version of linearised Legendrian contact homology. I will begin by considering natural examples of such Legendrian submanifolds and end by explaining how these invariants relate to other equivariant theories and how this could be used to compute it in these natural situations.

### 10 June: Russell Avdek

Title: *A closed, tight contact 3-manifold with vanishing contact homology*

Abstract: Contact homology (CH) is an invariant which assigns a differential graded algebra to each closed contact manifold. While this invariant shares some formal properties with Heegaard-Floer homology, CH is comparatively not well-understood due to a current lack of computational techniques. In this talk, we will describe the first example of a closed, tight contact 3-manifold whose CH is the zero algebra. In proving that CH=0 for this contact manifold, we will summarize some computational tools which relate surgeries and cobordisms to dynamical systems and holomorphic curves in a combinatorial fashion.

### 3 June: Lione**l Lang **(Stockholm University).

Title: *Coamoebas, dimers and vanishing cycles*

Abstract: In this joint work in progress with J. Forsgård, we study the topology of maps P:(C*)^2 \to C given by Laurent polynomials P(z,w). For specific P, we observed that the topology of the corresponding map can be described in terms of the coamoeba of a generic fiber. Under these circumstances, the coamoeba contains a dimer (a particularly nice graph) together with a distinguished collection of cycles (the fundamental cycles) that turn out to be the vanishing cycles of the map P. For a given degree, the existence of such P is widely open: it relies on the existence of maximal coamoebas. In the meantime, we can bypass coamoebas by going directly to dimers using a construction of Goncharov-Kenyon. We obtain a virtual correspondence between the fundamental cycles of the dimer and the vanishing cycles. In this talk, we will discuss how this (virtual) correspondence can be used to compute the monodromy of the map P.

### 27 May: Tobias Ekholm (Uppsala Univerisity).

Title: *Alexander polynomial, Z-hat, and curve counts*

Abstract: We describe how to express the Alexander polynomial of a knot K via its augmentation polynomial. The formula allows for a natural deformation that connects to the physics inspired Z-hat invariant for 3-manifolds of the knot complement, F_K. In fact it leads to an enumerative geometry interpretation and an a-deformation of F_K that is annihilated by the quantum augmentation polynomial. We discuss possible geometric origins of non-uniqueness of solutions to the quantum augmentation polynomial which is related to branching in the augmentation variety.

### May 20: Agnès Gadbled (Université Paris-Saclay)

Title: *Weinstein handlebodies for complements of smoothed toric divisors.*

### May 13: Yang Huang (University of Southern Denmark)

Title: *IKEA in contact 5-manifolds.*

### April 29: Jian Qiu (Uppsala Univerisity).

Title: *Toric Hyper Kahler varieties and special functions*

Abstract: The talk will be about some work done together with Andreas, Nikos, Maxim. But I will be a bit more broad and try to give some context of why are we doing this. The plan of the talk is

1. some elements of equivariant cohomology algebra

2. special functions constructed out of H_{eq} and their properties, with simple examples.

3. HyperKahler varieties and 3-Sasaki structures

4. some combinatorial results on special functions constructed from 3.

5. why are we doing this, and outlook.

### April 22: Paolo Ghiggini (Université de Nantes).

Title: *The standard contact structure on RP5 is not Liouville fillable.*

### April 15: Tobias Ekholm (Uppsala University).

**April 8: Noémie Legout **(Uppsala University).

Title: *Lagrangian cobordisms not coming from immersed ones after surgery*

Abstract: I will give examples of Legendrian knots admitting Lagrangian fillings of genus g with p transverse double points, but no Lagrangian fillings of genus g-1 with p+1 double points (with some additional conditions). This follows from a count of equivalence classes of augmentations: for Legendrian links L^- and L^+, if there is a Lagrangian cobordism from L^- to L^+ we show that the number of equivalence classes of augmentations of L^- must be smaller than that of L^+. This is a joint work with Orsola Capovilla-Searle, Maÿlis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

### April 1: Georgios Dimitroglou Rizell (Uppsala University).

Title: *Refined Floer homology for singular Lagrangians*

Abstract: We explain a construction in joint work with T. Ekholm and D. Tonkonog where we use Symplectic Field Theory techniques to define Floer homology groups for pairs of singular exact Lagrangians. The examples that we consider are self-plumbings of Lagrangian surfaces, where these Floer homology groups become derived hom's of representations of "derived (higher genus) multiplicative preprojective algebras".

### March 11: Thomas Kragh (Uppsala University).

Title: *Using co-algebras, co-modules, injective resolutions and Serre fibrations to compute SWF-bar.*

### March 4: Olof Bergvall (Uppsala University).

Title: *Del Pezzo surfaces and rational curves*

Abstract: In this talk we will consider Del Pezzo surfaces, i.e. smooth algebraic surfaces with ample anticanonical bundles, and rational curves on such surfaces. More precisely, I will give very concrete descriptions of certain moduli spaces of Del Pezzo surfaces marked with rational curves and explain how to compute their cohomology (as representations of certain Weyl groups).

### February 26: Jérémy Guéré (Université Grenoble Alpes).

Title: *Hodge-Gromov-Witten theory*

Abstract: Hodge-Gromov-Witten theory of a smooth projective variety X deals with the cap product of the virtual fundamental cycle on the moduli space of stable maps to X with the Euler class of the Hodge vector bundle. I recently studied its deformation invariance to singular varieties, allowing explicit computations in many cases. An important application of my theorem is a calculation of genus-zero GW invariants for some hypersurfaces in weighted projective spaces which do not satisfy the so-called convexity property. It is a first step towards a mirror symmetry statement for these hypersurfaces.

In a second part of the talk, I will describe my plan towards a calculation of GW invariants of the quintic hypersurface in P^4. It is based on a theorem Costello proved in 2003 expressing genus-g GW invariants of a projective variety X in terms of genus-0 GW invariants of the (g+1)-st symmetric power of X.

### February 19: Mark Lawrence (Nazarbayev University).

Title: *Polynomial hulls, knots, and holomorphic motions*

Abstract: The study of polynomial hulls of compact sets in C^{n} is too challenging to have a useful general answer. Even for smooth manifolds, there is little known. One restriction is to look at tori or unions of tori in S^{1} × C which fiber over the circle. The knot class of the torus plays a crucial role, both in a positive and negative direction. The main positive result of the author and J. Duval is that a torus modeled on a square root has a polynomial hull which is fibered by varieties. On the negative side, it appears that the knot type of the torus can exclude any hull from appearing over the unit disc, but only preliminary results exist in this direction. Using the theorem of Lawrence and Duval, new types of holomorphic motions

can be constructed, which have a limited amount of branching.

### January 22: Wanmin Liu (Uppsala University).

Title: *Contractibility of space of stability conditions on projective plane via global dimension function.*

Abstract: The global dimension function $\gldim$ is a continuous function defined on Bridgeland stability manifold, and it maps a stability condition to a non-negative real number. We compute the global dimension function $\gldim$ on the space of stability conditions on projective plane. It has the minimal value $2$, and $\gldim^{-1}(2)$ is contractible. Moreover, $\gldim^{-1}(2)$ is contained in the closure of the subspace consisting of geometric stability conditions. We show that $\gldim^{-1}([2,x)$ contracts to $\gldim^{-1}(2)$ for any real number $x>=2$. The $\gldim$ sheds some lights on why a space of Bridgeland stability conditions should be contractible. This is a joint work with Yu-Wei Fan, Chunyi Li and Yu Qiu.

## 2019

### December 18: Johan Asplund (Uppsala University).

Title: Partially wrapped Floer cohomology of a fiber and conormal stops

Abstract: It is well-known that wrapped Floer cohomology of a cotangent fiber is quasi-isomorphic to chains of based loops of the zero section equipped with the Pontryagin product. In this talk we will compute the partially wrapped Floer cohomology of a fiber in T*S when a conormal stop over a submanifold K \subset S is present. By pseudoholomorpic curve counts we show that it is quasi-isomorphic to chains of based loops on S \ K equipped with the Pontryagin product. This quasi-isomorphism is induced by a chain isomorphism between the partially wrapped Floer cochains of the fiber and a Morse-theoretic model of the loop space. If time allows, we will also upgrade the isomorphism in cohomology to an isomorphism of Z[\pi_1(S \ K)]-modules, and see in a family of examples how the partially wrapped Fukaya category knows about the Alexander invariant of knots via the Leray--Serre spectral sequence.

### December 11: **Laurent Côté **(Stanford University).

Title: Invariants of codimension 2 contact submanifolds

Abstract: The general machinery of Symplectic Field Theory (SFT) gives invariants of contact manifolds and Legendrian submanifolds. I will discuss work in progress with Francois-Simon Fauteux-Chapleau to define SFT invariants of codimension 2 contact submanifolds, which are the natural generalization of transverse knots to all dimensions. I will sketch the construction and main properties of these invariants, and discuss applications (some of them speculative) to contact topology and dynamics.

### November 27, Oliver Leigh (Stockholm University).

Title: Donaldson-Thomas Theory and the Banana Threefold

Abstract: Gromov-Witten/Donaldson-Thomas invariants can provide fundamental information about Calabi-Yau threefolds. This is especially the case when the invariants are assembled into partition functions, which themselves have remarkable properties and key relations to other areas of mathematics. However, these invariants are notorious for being hard to compute. In this talk we focus our attention on expanding the novel Donaldson-Thomas computation methods of Bryan-Kool to the recently discovered "banana threefolds". These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces with the property that the singular locus of a singular fibre is a non-normal toric curve known as a “banana configuration”. The partition functions of banana threefolds have remarkable automorphic properties and an underlying relation to elliptic genera.

### November 20. Cheuk Yu Mak (University of Cambridge).

Abstract: One of the earliest fundamental applications of Lagrangian Floer theory is detecting the non-displaceablity of a Lagrangian submanifold. Many progress and generalisations have been made since then but little is known when the Lagrangian submanifold is disconnected. In this talk, we describe a new idea to address this problem. Subsequently, we explain how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for every S^2 \times S^2 with a non-monotone product symplectic form, there is a continuum of disconnected, non-displaceable Lagrangian submanifolds such that each connected component is displaceable. This is a joint work with Ivan Smith.

### November 13, Bahar Acu (Northwestern University and Oberwolfach Research Institute for Mathematics)

Title: Obstructions to planarity in higher-dimensional contact manifolds

Abstract: Planar contact manifolds, those that correspond to an open book decomposition with genus zero pages, have been intensively studied to understand several aspects of 3-dimensional contact topology. In this talk, we present a higher-dimensional notion of planarity, iterated planarity, and provide several generalizations of results for planar contact 3- manifolds, to higher dimensions. This is partly joint work with A. Moreno.

### October 23, Tobias Ekholm (Uppsala)

Title: Legendrian surgery formula and Chekanov-Eliashberg algebras for singular Legendrian IV.

### October 16, Tobias Ekholm (Uppsala)

Title: Legendrian surgery formula and Chekanov-Eliashberg algebras for singular Legendrian III.

### October 9, Thomas Kragh (Uppsala)

Title: Spectra.

### October 2, Tobias Ekholm (Uppsala)

Title: Legendrian surgery formula and Chekanov-Eliashberg algebras for singular Legendrian Part II.

### September 18, Tobias Ekholm (Uppsala)

Title: Legendrian surgery formula and Chekanov-Eliashberg algebras for singular Legendrian.

### September 11, Georgios Dimitroglou Rizell (Uppsala)

Title: DGAs CoDGAs and A-infinity algebras.

Abstract: We give a basic introduction to the algebraic objects in the title and the relationship between them.

### September 4, Thomas Kragh (Uppsala)

Title: Operads.

Abstract: I will introduce the notion of operads (in various categories) and give important and (as easy as possible) examples that are relevant for symplectic geometry. In a later talk I will discuss how to try and lift the so-called "curved A_\infty operad" to spectra and why this is relevant for anything.

### July 31, Yu Qiu (Tsinghua University)

Title: Cluster exchange groupoids and quadratic differentials. (Yu Qiu's slides from July 31)

Abstract: We introduce the cluster exchange groupoid associated to a quiver with potential. In the case of the decorated marked surface S, the universal cover of this groupoid is a skeleton for a space of suitably framed quadratic differentials on S, which in turn models the space Stab(S) of Bridgeland stability conditions for the 3-Calabi-Yau (Fukaya) category associated to S. Finally, we show that Stab(S) is simply connected. This is a joint work with Alastair King.

### July 10, Yu Qiu (Tsinghua University)

Title: Q-deformation of stability conditions and quadratic differentials. (Yu Qiu's slides from July 10)

Abstract: Categorically, we introduce Calabi-Yau-X categories, which are q-deformation of Haiden-Katzarkov-Kontsevich's topological Fukaya categories of flat surfaces. Their Calabi-Yau-N orbit categories are subcategories of derived Fukaya categories. Geometrically, we introduce and prove a q-deformation of Bridgeland-Smith theory, that realizes q-stability conditions on Calabi-Yau categories via multi-valued quadratic differentials on the corresponding surfaces. This is a joint work with Akishi Ikeda.

### May 29, Shamil Shakirov (Uppsala)

Title: Genus two analogue of double affine Hecke algebra.

Abstract: We review an algebraic approach to computation of quantum invariants of torus knots, which is partly motivated by Chern-Simons theory in physics. We suggest a possible generalization to higher genus surfaces, and discuss genus two case in detail.

### May 22, Marco Golla (Nantes)

Title: Singular symplectic curves: isotopy and symplectic fillings

Abstract: I will be talking about symplectic curves (mostly in the projective plane) whose singularity are modelled over complex singularities. I will discuss existence and uniqueness (up to isotopy) of these curves, phrasing it in terms of symplectic fillings; the focus will be on rational curves with irreducible singularities. This is joint work with Laura Starkston (in progress).

### May 13, Mohammed Abouzaid (Columbia University)

Title: Local Fukaya categories.

### April 24, Sofia Tirabassi (Stockholm University)

Title: Derived categories of Enriques and Bielliptic surfaces.

Abstract: I will show that over an algebraically closed field of characteristic greater than 5 Enriques surfaces and bielliptic surfaces do not have any non trivial Fourier-Mukai partners.

This is a joint work with M. Lieblich and K. Honigs.

### April 17, Thomas Rot (UV Amsterdam)

Title: Non-linear Fredholm mappings and homotopy theory.

Abstract: Non-linear existence problems can attacked with topological methods. For example a map between closed manifolds of the same dimension is surjective if the degree is non-zero. The degree is a homotopy invariant and two mappings into the sphere are homotopic if and only if the degrees are equal. Framed cobordism is a generalization for maps between closed manifolds of different dimensions. Again homotopy classes into the sphere are fully captured by the Framed cobordism class. In this talk I will discuss joint work with Alberto Abbondandolo in which we generalize the theory to an infinite dimensional setting.

### April 10, Agnès Gadbled (Uppsala)

Title: Categorical action of the braid group of the cylinder: symplectic aspect

Abstract: Khovanov and Seidel gave in 2000 an action of the classical braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In a recent article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder.

The work of Khovanov and Seidel also had a symplectic aspect that we now generalize. In this talk, I will explain the strategy and tools to get a symplectic monodromy in our case and prove its injectivity. If time permits, I will explain how this action lifts to a symplectic categorical representation on a Fukaya category that should be related to the algebraic categorical representation.

This is a joint work in progress with Anne-Laure Thiel and Emmanuel Wagner.

### March 13, Stefan Behrens (Bielefeld University)

Title: Froyshov-type invariants via homotopy theory.

Abstract: Froyshov's h-invariants are numerical invariants of rational homology 3-spheres that appear in generalizations of Donaldson's diagonalizability theorem to 4-manifolds with boundary. They are defined in terms of the algebraic structure of monopole Floer homology. I will explain their relation to the primary obstructions in a series of extension problems that naturally arise in Manolescu's approach to Seiberg-Witten theory on 3-manifolds. The higher obstructions lead to refined, potentially stronger Froyshov-type invariants. This is joint work with Tyrone Cutler.

**March** 6, Noemie Legout** **(Shanghai Tech)

Title: A product structure on the Floer homology of Lagrangian cobordisms.

Abstract: In 2015, Chantraine, Dimitroglou-Rizell, Ghiggini and Golovko [CDRGG] have defined a Floer complex associated to a pair of Lagrangian cobordisms between Legendrian submanifolds. In this talk, I will explain how to construct a product structure on it. This product recovers the cup product on singular cohomology when we consider a Lagrangian filling of a Legendrian. Moreover, we show that this product is mapped to the product on Legendrian contact cohomology by a quasi-isomorphism from the Floer complex to the Legendrian contact cohomology complex defined in [CDRGG]. Finally, we can define higher order maps to extend the product (resp. the quasi-isomorphism) to A-infinity composition maps (resp. A-infinity functor).

### February 13, Thomas Kargh (Uppsala)

Title: Bauer-Furuta invariants.

Abstract: I will sketch the overall ideas surrounding Bauer and Furutas

definition of a stable homotopy invariant - refining the usual

Seiberg-Witten invariant (knowledge of which is not assumed). I will

also sketch some of their re-proofs of famous theorems (many of which

where first proved by Donaldson). I will then ask the audience for

volunteers to fill in details of select parts of this (those parts

people are most interested in) - so that we can use these talks as

"fillers" for future Wednesdays where we do not have a speaker.

Alternatively, you can suggests other topics afterwards you would rather

learn (more) about an we can try and go there instead.

### February 6, Yang Huang (Uppsala)

Title: Applications of convex hypersurface theory.

Abstract: Assuming some fundamental results from convex hypersurface theory, I will explain an h-principle for contact submanifolds. The h-principle of Borman-Eliashberg-Murphy for the existence of contact structures will be an immediate corollary. For this talk, I will assume basic familiarity with contact topology but nothing from my previous talks. Joint work with Ko Honda.

### January 16 and January 23, Wanmin Liu (Uppsala)

January 16, Title: Bridgeland stability conditions and applications I

January 23, Title: Bridgeland stability conditions and applications II

These will be introductory talks on the notion of Bridgeland stability conditions, which originate from Douglas's work on Π-stability for B-branes. It associates a triangulated category a natural topological space, called stability manifolds.

In the first part, I will introduce Bridgeland's idea of stability conditions on triangulated category from mathematical side. I will give examples and motivations.

In the second part, I will give applications of Bridgeland stability conditions on birational geometry of moduli spaces, i.e. using wall-crossing of stability conditions to run the minimal model program of moduli space of sheaves.

There are lots of nice references on the Bridgeland stability conditions. I only list a few links for the first part:

https://arxiv.org/abs/math/0207021

https://arxiv.org/abs/math/0212237

https://arxiv.org/abs/1607.01262

and the following links for the second part:

https://arxiv.org/abs/1203.4613

https://arxiv.org/abs/1301.6968

https://arxiv.org/abs/1501.06397

## 2018, FALL

### November 28, **Y****ang Huang **(Uppsala)

Title: Convex hypersurface theory 2.

Abstract: I will explain how to make a generic hypersurface convex in any contact manifold. Based on joint work with Ko Honda.

### November 14, Tobias Ekholm (Uppsala)

Title: Skein valued Gromov-Witten invariants, large N duality, and refinement.

Abstract: We define open Gromov-Witten invariants with values in the framed skein module and use them to prove large N duality and give a geometric definition of refined HOMFLY polynomials.

### October 17, Jeff Hicks (Berkeley)

Title: Tropical Lagrangians and Mutations

Abstract: Mirror symmetry is a conjectured relation between the symplectic geometry of a space X and the complex geometry of a mirror space Y. A mechanism for this duality comes from the Strominger-Yau-Zaslow conjecture, which states that mirror spaces are dual Lagrangian torus fibrations over a common base Q. The connection between the symplectic geometry on X and complex geometry on Y is seen via a degeneration to tropical geometry on Q.

The recent work of Matessi and Mikhalkin show how to lift tropical curves in Q to Lagrangian submanifolds of X. I will provide a different construction of these tropical Lagrangians inspired by homological mirror symmetry, and explore how different Lagrangian lifts of these tropical curves may be related to each other by Lagrangian mutation.

### October 10, Yin Li (UCL)

Title: Calabi-Yau completions as Chekanov-Eliashberg algebras

Abstract: We identify the Chekanov-Eliashberg algebras of certain links of Legendrian 2-spheres with Calabi-Yau completions of Fukaya-Seidel directed A infinity algebras. Together with a generalized version of Eilenberg-Moore equivalence established by Ekholm-Lekili, we prove the formality of the Fukaya A infinity algebras of vanishing cycles in certain 6-dimensional Milnor fibers associated to non-quasi-homogeneous singularities, therefore showing that the compact and wrapped Fukaya categories of these manifolds provide symplectic geometric realizations of the well-known Koszul duality between cyclic and Calabi-Yau completions. The Koszul duality between Fukaya categories then enables us to interpret the generation result of Chantraine-Dimitroglou-Rizell-Ghiggini-Golovko for wrapped Fukaya categories as the split-generation of the compact Fukaya categories by vanishing cycles.

### September 26, Thomas Kragh (Uppsala)

Title: Twisted generating families.

Abstract: This will be a report on work in progress with Abouzaid, Courte, and Guillermou. In this lecture I will discus a short proof of the existence of generating families for A Lagrangian embedding in a cotangent bundle with stable trivial Gauss map. I will then define the notion of a twisted generating family, and sketch why we think these exists/how we want to construct them. I will also discus what consequences this existence has.

## 2018, SPRING

### May 3, Mohammed Abouzaid (Columbia)

Title: From flow categories to homotopy types

Abstract: A formal step in the construction of a Floer homotopy type is the passage from flow categories to stable homotopy types. Cohen-Jones-Segal gave a description of this via iterated Pontryagin-Thom constructions. I will explain a reformulation of this construction which is more amenable to considering general cohomology theories. This is joint work with A. Blumberg.

### April 25, Wanmin Liu (IBS Center for Geometry and Physics)

Title: Classification of full exceptional collections of line bundles on three blow-ups of P^3 and on some projective bundles.

Abstract: A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this talk, we show that this conjecture holds for X as the blow-up of P^3 at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X. The preprint is available at IBS-CGP preprint [CGP17025].

I will also introduce some recent progress on some projective bundles, including blow-up of P^n at a point. This is a joint work with Song Yang and Xun Yu at Tianjin University.

### March 28, Johannes Nordstrom (University of Bath)

Title: Disconnecting the G_2-moduli space

Abstract: A natural question in the study of Riemannian 7-manifolds with holonomy G_2 is whether the moduli space on a closed 7-manifold can be disconnected, i.e. whether there can exist different metrics of holonomy G_2 on the same closed manifold such that one cannot be deformed to the other through a path of G_2 metrics. Any holonomy G_2 metric has an associated G_2 structure, which satisfies a certain partial differential equation. I will present topological and analytic invariants of G_2-structures (largely ignoring the PDE), and examples where these invariants are able to distinguish components of the G_2 moduli space.

### February 28, Georgios Dimitroglou Rizell (Uppsala)

Title: Curve counts on Lagrangian immersions and multiplicative preprojective algebras

Abstract: We define open Gromov-Witten invariants with an immersed Lagrangian surface as boundary condition. It turns out that the count should be considered as an element inside a multiplicatively preprojective algebra. (In the embedded case this simply means the group ring of the fundamental group.) We explain these notions, and show how the result can be used to distinguish immersed Lagrangian spheres with one double point inside the projective planes up to Hamiltonian isotopy. This is joint work with T. Ekholm and D. Tonkonog.

### January 17, Thomas Kragh (Uppsala)

Title: Generating families for exact Lagrangians with trivial stable Gauss map.

Abstract: Giroux proved that any Lagrangian embedding has a generating family if and only if its stable Gauss map is trivial. However, the type of generating family he considered may not have well-defined Morse theory (in the sense that there is no control over the compactness of gradient trajectories). In this talk I will sketch a proof that all Lagrangians with stably trivial Gauss map actually has a generating family with fiberwise compact set of gradient trajectories (this is very similar to having a g.f. quadratic at infinity).

### January 10, Jean-François Barraud (Toulouse)

Title: A Novikov fundamental group.

Abstract: Just like there might be homotopical constraints on the critical points of Morse functions that are not detected by the homology of the ambient manifold, closed 1 forms may have necessary critical points that are not detected by the Novikov homology. I will present an agebraic invariant that captures some of this information, and is an analogue in Novikov theory of the fundamental group in Morse theory. In particular, this "Novikov fundamental group" leads to new lower bounds for the number of index 1 and 2 critical points of closed 1-forms, that are essentially different from the classical Morse-Novikov inequalities. (jw with A. Gadbled and H.V.Le).

### Thomas Kragh (Uppsala)

Title: On the space of Legendrians isotopic to the zero section in Jet-1 bundles.

Abstract: In this talk I will introduce the notion of pseudo-isotopies and use these to prove that the space of Legendrians in a Jet-1 bundle (with high dimensionally base) has highly non-trivial homotopy type. This is joint work with Yasha Eliashberg.

## 2017, FALL

Organisers: Georgios Dimitroglou Rizell and Maksim Maydanskiy.

### November 29, Luigi Tizzano (Uppsala)

Title: Physics conjectures about topological Fukaya category.

### November 22, Luis Diogo (Uppsala)

Title: Lifting Lagrangians from Donaldson-type divisors.

Abstract: Given a closed symplectic manifold with a symplectic submanifold of codimension 2, we can sometimes lift monotone Lagrangians from the submanifold to the ambient manifold. We show that under some assumptions, it is possible to write the superpotentials of the lifted Lagrangians in terms of the superpotentials of the original Lagrangians (we may also need additional information coming from relative Gromov-Witten invariants). The superpotential of a Lagrangian is a count of pseudoholomorphic disks (of Maslov index 2) with boundary on the Lagrangian, and it plays an important role in Floer theory and mirror symmetry.

We will discuss applications of this result, including how it can be used to distinguish infinitely many monotone Lagrangian tori in complex projective planes, quadrics and cubics of complex dimension at least 3. This is joint work with D. Tonkonog, R. Vianna and W. Wu.

### November 15, Honghao Gao (Grenoble)

Title: Augmentations and sheaves for knot conormals.

Abstract: Knot invariants can be defined using Legendrian isotopy invariants of the knot conormal. There are two types of invariants raised in this way: one is the knot contact differential graded algebra together with augmentations associated to this dga, and the other one is the category of simple sheaves microsupported along the knot conormal. Nadler-Zaslow correspondence suggests a connection between the two types of invariants. Moreover, augmentations specialized to “Q=1” have been understood through KCH representations.

I will present a classification result of simple sheaves, and relate it to KCH representations and two-variable augmentation polynomials. I will also present a Radon transform for sheaf categories, and explain how it corresponds to the specialization of Q on the sheaf side.

### November 8, Evgeny Volkov (Uppsala)

Title: SFT without holomorphic curves.

Abstract: We construct a version of SFT for the cotangent bundle of a simply connected odd dimensional manifold without using holomorphic curves. There is a map from this SFT into string topology of the manifold. We show that this map induces an isomorphism on homology, and intertwines the SFT product with (a version of) the Chas-Sullivan product.

### October 25, Paolo Ghiggini (Nantes)

Title: A dynamical regard on knot Floer homology.

Abstract: Knot Floer homology is a family of abelian groups \widehat{HFK}(Y, K, i) for a null-homologous knot K in a closed, oriented 3-manifold Y which is indexed by an integer i \in [-g, g], where g denotes the minimal genus of an embedded surface bounding K. This invariant was introduced by Ozsváth, Szabó and Rasmussen using a Lagrangian Floer homology construction. I will show that, when K is a fibred knot (i.e. Y-K fibres over S^{1} and K is the boundary of the closure of every fibre), the group \widehat{HFK}(Y, K, -g+1) is isomorphic to a version of the fix point Floer homology of any area-preserving representative of the monodromy of the fibration on Y-K. I will also discuss some potential applications of this isomorphism. This is a work in progress in collaboration with Gilberto Spano.

### October 25, Roman Golovko (ULB)

Title: The wrapped Fukaya category of a Weinstein manifold is generated by the cocores of the critical Weinstein handles.

Abstract: We decompose any object in the wrapped Fukaya category of a 2n-dimensional Weinstein manifold as a twisted complex built from the cocores of the n-dimensional handles in a Weinstein handle decomposition. This is joint work with Baptiste Chantraine, Georgios Dimitroglou Rizell and Paolo Ghiggini.

### October 18, Sara Angela Filippini (Cambridge University)

Title: Refined curve counting and the tropical vertex group

Abstract: The tropical vertex group of Kontsevich and Soibelman is generated by formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in many problems in algebraic geometry and mathematical physics. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov-Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts. I will describe a refinement or "q-deformation" of this expansion, motivated by wall-crossing ideas, using Block-Göttsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined. This is joint work with Jacopo Stoppa.

### October 11, Yang Huang (Uppsala)

Title: On convex hypersurface theory.

Abstract: In 3-dimensional contact topology, convex surface theory, developed by Giroux, plays a prominent role. In this talk, I will describe a ‘parallel’ story in higher dimensions. In particular I will introduce the notion of overtwisted orange and bypass as technical tools. This is based on joint work with K. Honda.

### October 4, Tobias Ekholm (Uppsala)

Title: Skein relation from holomorphic curves.

### September 27, Johan Asplund (Uppsala)

Title: Flexibility of Legendrian immersions.

Abstract: Following Eliashberg and Mishachevs book, I will give a brief

overview of Gromovs famous h-principle and will discuss the fact that

Legendrian immersions satsifies the h-principle. Using the h-principle

I will then show that a closed n-dimensional submanifold has a

Legendrian immersion into standard contact R^{2n+1} if and only if the

complexified tangent bundle of the submanifold is trivial. Using this

fact, we will then exhibit surfaces that do not admit a Legendrian

immersion into R^{5} by computing some Chern classes.

## 2017, SPRING

Organisers: Thomas Kragh and Maksim Maydanskiy.

### May 31, Davide Alboresi (Utrecht)

Title: Towards Fukaya categories of stable generalized complex manifolds.

Abstract: Stable generalized complex manifolds are a class of generalized complex manifolds which are symplectic almost everywhere, with a symplectic form that has a logarithmic singularity along a hypersurface. In this talk I will discuss an attempt to define a category of branes for such manifolds, mimicking the construction of the Wrapped Fukaya category.

### May 24, Peter Feller (MPI Bonn)

Title: Knots with a view toward embedding problems in complex geometry.

Abstract: We first discuss classical questions about polynomial embeddings of the complex line C into complex spaces such as Cm and affine algebraic groups. Next, we consider torus knots and discuss questions related to their non-sliceness motivated by singularity theory. Finally, we use a knot theory perspective to indicate proofs for the embedding questions discussed first.

### May 3, Momchil Konstantinov (UCL)

Title: Higher rank local systems for monotone Lagrangians.

Abstract: Lagrangian Floer homology is a tool for studying intersections of Lagrangian submanifolds of symplectic manifolds. It is defined by ``counting'' pseudoholomorphic discs with boundary on these submanifolds. One way to enrich this theory is to record some homotopy data about the paths that the boundaries of these discs trace on the Lagrangians. A simple algebraic way of accomplishing this is to use local coefficients. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not, one can always restrict to some natural unobstructed subcomplexes. I will showcase all of these constructions with some explicit calculations for the Chiang Lagrangian in CP3. Its Floer theory was computed by Evans and Likili, who also pointed out that standard Floer homology cannot tell us whether the Chiang Lagrangian and RP3 can be disjoined by a Hamiltonian isotopy. We will see how using a rank 2 local system in this example allows us to show that these two Lagrangians are in fact non-displaceable.

### April 26, Thomas Kragh (Uppsala)

Title: Waldhausen's K-theory of spaces and exact Lagrangian disc fillings of the Legendrian unknot.

Abstract: I will discus some basic notions around Waldhausen's definition of algebraic K-theory of spaces. I will then relate this to generating families quadratic at infinity for any Lagrangian in R^2n equal to the standard at infinity (previous talks was around the existence of such generating families - for this talk I will simply assume these). I will also discus an easier "Morse-type" proof of an important result by Bökstedt, which relates to the Lagrangian Gauss map of such Lagrangians. In fact, I will sketch a proof that the Gauss map relative to infinity is in fact trivial - i.e that any disc filling of the Legendrian unknot has trivial Gauss map.

### April 5, Dmitry Tonkonog (Uppsala)

Abstract: Refined curve counts for immersed Lagrangians, and stories in the neighbourhood

Title: The simplest open Gromov-Witten invariant is the count of holomorphic Maslov index 2 disks with boundary on a smooth Lagrangian submanifold. I will explain a "refined way" to count such disks on an immersed Lagrangian, focussing on dimension 4. I will talk about the surrounding context of local mirror symmetry, and as a different application, I will exhibit Lagrangian Whitney spheres in CP2 which are Hamiltonian non-displaceable from the complex line. This is joint work in progress with G. Dimitroglou Rizell and T. Ekholm.

### March 29, Jack Smith (Cambridge)

Title: Symmetries in monotone Lagrangian Floer theory.

Abstract: Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I'll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

Title: Spaces of Generating Families and the Hatcher-Waldhausen map.

Abstract: In a previous talk I outlined how to construct a generating family quadratic at infinity for any Lagrangian L in R2n, which is standard at infinity (equals Rn). In this talk I will talk about the consequences of having such a generating family, and about related classifying spaces. In particular I will show that the Langrangian Gauss map to U/O is homotopy trivial. This means that in all dimensions not equal 3 mod 4 the immersion class (relative infinity) is trivial (this is only news in dimension 4k+1).

### February 15, Hülya Argüz (Imperial College London)

Title: Log Geometric Techniques in Mirror Symmetry

Abstract: We will first discuss an algebraic geometric approach to the Fukaya category in symplectic geometry in terms of punctured log Gromov-Witten theory. For this our main object of study is a degeneration of elliptic curves, namely the Tate curve. This is the easiest non-trivial example of a toric degeneration in the Gross-Siebert program concerning mirror symmetry. We will also discuss more general toric degenerations as well as the topology of their real loci. For this we will look at Kato-Nakayama spaces associated to log schemes. This is mostly joint work with Bernd Siebert, some parts based on discussions with Mohammed Abouzaid.

### February 8, Tobias Ekholm (Uppsala)

Title: Legendrian surgery and partially wrapped Floer cohomology II.

January 11, Luis Diogo (Columbia)

Title: Monotone Lagrangians in cotangent bundles of spheres

Abstract: We show that there is a 1-parameter family of monotone Lagrangians in cotangent bundles of spheres with the following property: every (orientable spin) closed monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the Lagrangians in the family. The proof involves studying a version of the wrapped Fukaya category that includes monotone Lagrangians. This is joint work with Mohammed Abouzaid.

## 2016

### December 7, Tobias Ekholm (Uppsala)

Title: Legendrian surgery and partially wrapped Floer cohomology

Abstract: We generalize the surgery calculation of wrapped Floer cohomology to the partially wrappped setting using Chekanov-Eliashberg DG algebras with loop space coefficients, or in the simply connected case parallel copies. The talk reports on joint work with Y Lekili.

### November 30, Bruno Martelli (Università di Pisa)

Title: Hyperbolic cone-manifolds in dimension four

Abstract: A hyperbolic (or flat, spherical) cone-manifold is a kind of constant curvature manifold admitting some types of codimension-two "cone singularities". These objects have been studied a lot in dimension 2 and 3, but not much in higher dimensions. In this seminar we introduce the general theory of cone-manifolds (pioneered by Thurston and McMullen) and describe some new examples in dimension four. These examples are interesting because the structure can be deformed in an appropriate way, displaying some new flexibility phenomena in higher-dimensional hyperbolic geometry.

### November 23, Johannes Rau (Universität Tübingen)

Title: Real Hurwitz numbers -- a tropical approach

Abstract: The study of Hurwitz numbers, despite its long history, has been completely remodeled in the last twenty years with the discovery of deep connections for example to Gromov-Witten theory and matrix integrals, originating in string theory. While classical Hurwitz numbers count certain holomorphic maps, sometimes it is natural to look at the "real" version of the problem (counting holomorphic maps compatible with a given real structure). I will present a tropical approach (i.e. based on pair-of-pants decomposition) to calculate such real Hurwitz numbers. Another recent development is the construction of a signed invariant count of real maps by Itenberg and Zvonkine (in the spirit of Welschinger invariants). If time permits, I will discuss possible relations with the tropical approach.

### November 16, Lionel Lang (Uppsala)

Title: The vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)

Abstract: Take a generic curve C in a linear system L on a toric surface X. What are the simple closed curves in C that can be contracted along a degeneration to a nodal curve? This question can be rephrased in term of the image of the monodromy map given by the complement of the discriminant D ⊂ L into the mapping class group of C. Compared with degeneration in M̅g (g>1) where any cycles can be contracted, there are known obstructions to contract cycles in L, namely roots of the relative canonical bundle and hyperelliptic involution. We will review how we can detect them and show that there is no other obstructions. Indeed, we will show that the monodromy is surjective on MCG(C) when no such obstruction appears. If time permits, we will also discuss the image of the monodromy in some obstructed cases.

There will be two main ingredients along the proof. First we will use explicit degenerations on a well studied class of curves: simple Harnack curves. Then we will construct explicit element of the monodromy by applying Mihkalkin's approximation Theorem to well chosen loops in some tropical compactification of the linear system L.

### November 2, Georgios Dimitroglou Rizell (Uppsala)

Title: The classification of Lagrangians nearby the Whitney immersion

Abstract: We classify the Lagrangian tori and spheres in the four-dimensional vector space that are close (in the appropriate sense) to Whitney's immersed Lagrangian sphere with a single double point; up to Hamiltonian isotopy, they are either product tori, Chekanov tori, or rescalings of the Whitney sphere. While we believe that all Lagrangians in the vector space can be put in such a position, we also provide explicit and elementary examples showing that, for tori inside the unit ball, a significantly larger ball is sometimes needed by the Hamiltonian.

### October 26, Yankı Lekili (King's College London)

Title: Duality between Legendrian and Lagrangian invariants.

Abstract: We will discuss (derived) Koszul duality in the setting of pseudoholomorphic curve invariants associated to Legendrians and Lagrangians. This is mostly based on a joint work with Ekholm. Though, I will also allude to some previous results obtained jointly with Etgu.

### October 12, Evgeny Volkov (Uppsala)

Title: On one cyclic A∞ algebra.

Abstract: The de Rham algebra of a manifold is canonically a cyclic A infinity algebra with vanishing higher operations.

We pull this A infinity structure back to the subcomplex of harmonic forms and discuss the question of whether the pull back structure is also cyclic. This is work in progress joint with K. Cieliebak.

### October 12, Marco Golla (Uppsala)

Title: Signature defects, handles, and ribbon discs

Abstract: The homology groups of a manifold give a lower bound on the number of handles in a handle decomposition (or even on the cells of a CW decomposition). We use Casson-Gordon signatures to improve on this bound for rational homology 4-balls bounding a given rational homology 3-sphere. In turn, this gives information about slice and ribbon discs for knots in the 3-sphere.

This is joint work (in progress) with Paolo Aceto and Ana Lecuona.

### October 5, Alexander Berglund (Stockholm University)

Title: Automorphisms of manifolds and graph homology

Abstract: There is a classical programme for understanding automorphisms of high dimensional smooth manifolds whereby one studies, in turn, the monoid of homotopy automorphisms, the block diffeomorphism group, and finally the diffeomorphism group. The relative homotopy groups in each step are calculated by, respectively, the surgery exact sequence and, in a range, Waldhausen's algebraic K-theory of spaces.

I will talk about the calculation of the stable rational cohomology of the block diffeomorphism group of the g-fold connected sum #^g S^d x S^d relative to a disk (2d>4). Our result is expressed in terms of a certain graph complex, which, quite surprisingly, is related to Kontsevich's graph complex that calculates the cohomology of automorphism groups of free groups. I will also comment on the relation to the results of Galatius and Randal-Williams on the stable cohomology of the diffeomorphism group. This is joint work with Ib Madsen.

### September 28 Stéphane Guillermou (Université Grenoble Alpes)

Title: The three cusps conjecture.

Abstract: Arnol'd's three cusps conjecture is about the fronts of Legendrian curves in the projectivized cotangent bundle of the 2-sphere. It says that the front of a generic Hamiltonian deformation of the fiber over a point has at least three cusps.

We will recall some results of the microlocal theory of sheaves of Kashiwara and Schapira and see how we can use them to prove the conjecture.

### September 21, Stefan Behrens (Utrecht University)

Title: Surface diagrams and holomorphic quilts.

Abstract: I will report on work in progress with Katrin Wehrheim and Morgan Weiler. The rough idea is as follows. On the one hand, work of Donaldson-Smith and Perutz suggests that suitable counts of "holomorphic multi-sections" of generic maps from 4-manifolds to surfaces should contain information about the Seiberg-Witten invariants. On the other hand, a result of Williams shows the existence of particularly simple maps to the 2-sphere whose structure can be captured by curve configurations in a single fiber, so-called surface diagrams. Combining the two observations, our hope is to eventually find a description of the Seiberg-Witten invariants of 4-manifolds in terms of their surface diagrams.

### September 15 (Thursday, at 10:15): Tamas Kalman (Tokyo Institute of Technology)

Title: The Homfly polynomial and Floer homology

Abstract: I will report on a formula that expresses certain extremal coefficients in the Homfly polynomial of an alternating link from the Seifert graph G. This happens in a combinatorially novel way, using the so-called interior polynomial I(G). There is an intermediate step in the computation of I(G) where we consider a particular set of vectors called `hypertrees’. It turns out that hypertrees can be identified with spin-c structures that support a certain sutured Floer homology group. Hence in effect we are computing Homfly coefficients from Floer theory. If time permits, I will speculate on a possible generalization to the non-alternating case.

(I will mention joint results with A. Juhasz, H. Murakami, A. Postnikov, and J. Rasmussen.)

### September 7, Dmitry Tonkonog (Uppsala)

Title: Laurent phenomenon and symplectic cohomology

Abstract: The function x+y+1/xy stays Laurent under a sequence of certain birational changes of the coordinates x,y called cluster transformations. This phenomenon has been given algebro-geometric meaning through the work of Gross, Hacking and Keel; we will look at it from a mirror-symmetric point of view, through symplectic cohomology.

### May 18, Paolo Aceto (Rényi Institute)

Title: Knot concordance and homology sphere groups.

Abstract: We study two homomorphisms to the rational homology sphere group. One is the homomorphism from the knot concordance group C defined by taking double branched covers of knots. We show that the kernel of this map is infinitely generated by analyzing the Tristram-Levine signatures of a family of knots whose double branched covers all bound rational homology balls. We also study the inclusion homomorphism from the integral homology sphere group. Using work of Lisca we show that the image of this map intersects trivially with the subgroup generated by lens spaces. As corollaries this gives a new proof that the cokernel of this inclusion map is infinitely generated, and implies that a connected sum of 2-bridge knots is concordant to a knot with determinant 1 if and only if K is smoothly slice.

This is a joint work with Kyle Larson.

### May 18, Ana Garcia Lecuona (University of Marseille)

Title: Splice links and colored signatures.

Abstract: The splice of two links is an operation defined by Eisenbund and Neumann that generalizes several other operations on links, such as the connected sum, the cabling or the disjoint union. There has been much interest to understand the behavior of different link invariants under the splice operation (genus, fiberability, Conway polynomial, Heegaard Floer homology among others) and the goal of this talk is to present a formula relating the colored signature of the splice of two oriented links to the colored signatures of its two constituent links. As an immediate consequence, we have that the conventional univariate Levine-Tristram signature of a splice depends, in general, on the colored (or multivariate) signatures of the summands. If time permits we will discuss the intricacies of the non generic case.

This is a joint work with Alex Degtyarev and Vincent Florens.

### April 27, Thomas Kragh

Title: Generating families quadratic at infinity for exact Lagrangians in R^2n equal to R^n at infinity.

Abstract: In this talk I will describe how Laudenbach and Sikorav defined a generating family quadratic at infinity for any Lagrangian Hamiltonian isotopic to the zero section in a cotangent bundle (I will stay in R^2n for simplicity where the construction was originally due to Chaperon). I will then use this construction to create a generating family on contractible fibers with Well-defined fiber-wise Morse theory for any Lagrangian in R^2n equal to R^n at infinity. I will then explain how the "with well-defined Morse theory" can be turned into "quadratic at infinity" in some slightly weaker sense than the strictest sense imaginable. I will also explain how the strictest sense is different and how this defines an invariant of the Lagrangian.

April 20th is the day of the Twelth Uppsala Geometry and Physics Seminar.

### April 13, Johan Björklund

Title: Counting quadrisecants of smooth and real algebraic knots.

Abstract: A quadrisecant of a given knot K is a line intersecting K four times. A generic knot has a finite number of quadrisecants. Pannwitz, Morton and Mond has shown that any nontrivial knot in R^3 possess at least one quadrisecant. This result has been further refined by Kuperberg and Denne.

I will discuss how to use quadrisecants to obtain a smooth isotopy invariant for smooth knots in RP^3 (and show that Pannwitz, Morton and Monds statement does not hold true there) and how to use them to obtain a rigid isotopy invariant (utilizing the complex parts of a real algebraic knot). This is joint work (in progress) with Oleg Viro (Stony Brook).

### April 6, Anna Sakovich

Title: Mass and center of mass in mathematical general relativity.

Abstract: While the definition of mass and center of mass via the mass density is straightforward in Newton's theory of gravity, the situation in general relativity is more complicated. In this talk we will discuss how to define mass and center of mass of asymptotically Euclidean and asymptotically hyperbolic manifolds, which are important objects in mathematical general relativity arising as hypersurfaces of asymptotically Minkowskian spacetimes. We will focus on geometric aspects of these definitions and discuss some important related results.

### March 9, Evgeny Volkov

Title: Chain level string topology II

### March 2, Ralph Morrison (KTH)

Title: Bitangents of tropical plane quartic curves

Abstract: A smooth plane quartic curve over an algebraically closed field has 28 bitangent lines. In this talk, I will prove the corresponding result for the tropical world: a tropical smooth plane quartic curve has 7 tropical bitangent lines, up to a natural equivalence. The proof of this result will consider plane tropical curves both as embedded piecewise linear subsets of the Euclidean plane, and as abstract graphs with lengths on the edges. This is joint work with Matt Baker, Yoav Len, Nathan Pflueger, and Qingchun Ren.

### February 17, Marco Golla

Title: Symplectic hats

Abstract: A symplectic cap of a contact 3-manifold Y is a compact 4-manifold with concave boundary -Y; a symplectic hat of a transverse knot K in Y is a compact symplectic surface in a cap, whose boundary is -K.

We study the existence problem for hats and their topology, with a particular emphasis on the case of transverse knots in the standard 3-sphere; we will also discuss applications to fillings of contact 3-manifolds.

This is joint work (in progress) with John Etnyre.

### Febuary 10, Evgeny Volkov

Title: Chain level string topology I

## 2014

### October 8, Thomas Kragh

Title: Exact Lagrangians are simple-homotopy equivalent to the zero-section

Abstract: In this talk I will describe White-head torsion of finite CW complexes and explain how it is relevant for classification of smooth manifolds. Then I will describe an easy description of the Fukaya-Seidel-Smith spectral sequence and explain how it can be used to prove that any closed exact Lagrangian L in a cotangent bundle is simple homotopy equivalent to the zero section (the last part is joint with M. Abouzaid).

### September 24, Kai Cieliebak (Augsburg)

Title: A remark on the Euler equations of hydrodynamics

Abstract: The time evolution of an ideal incompressible fluid is described by the Euler equations. In this talk I will discuss a connection between stationary solutions of these equations and symplectic topology, as well as possible applications to questions of hydrodynamic instability.

### September 17, Tobias Ekholm

Title: Generalizations of knot contact homology and colored HOMFLY

### June 25, Emilia Lundberg

Title: A bar complex in Morse theory and what is it isomorphic to?

Abstract: I will define a Morse bar complex (or an A-infinity-structure on the Morse complex) originating from a Morse function on a closed Riemannian manifold M by perturbating vector fields. I will also say something about the homology of this complex being isomorphic to the homology of the free loop space of M (work in progress). If time permits I will calculate an example for the n-sphere.

### April 2, Cecilia Karlsson

Tilte: Orientations of Morse flow trees in Legendrian contact homology

I will briefly introduce the concept of Morse flow trees in Legendrian contact homology, defined for a Legendrian submanifold $L$ in the 1 jet space of a manifold $M$. Then I will discuss a way of put an orientation on the space of such trees, making it possible to calculate the homology with integer coefficients. This can be done provided $L$ is spin and $M$ is orientable. Using some geometric properties of Morse flow trees, i.e. the stable and unstable manifolds in $M$ corresponding to the gradient flow which define the trees, I will show a way to compute the orientation of rigid Morse flow trees explicitly. This is work in progress.

### March 26, Albin Eriksson Östman

Title: Fully noncommutative Legendrian contact homology with homotopy coefficients, linearizations, 2-copy links, and covers.

Abstract: We discuss the fully noncommutative Legendrian contact homology with homotopy coefficients of a Legendrian submanifold, L, in a contact manifold, P*\R, where P is an exact symplectic manifold. If the Legendrian contact homology algebra of L can be linearized, then the corresponding linearized complex of the 2-copy link of L contains a subcomplex which can be identified with the Morse complex of the universal cover of L. By reducing the coefficients in the Legendrian contact homology algebra, we get a subcomplex identified with the Morse comlpex of a normal cover of L.

As an example we prove an improved double point estimate for a displaceable exact Lagrangian immersion of the Poincaré homology sphere in an exact symplectic manifold P, provided the Legendrian contact homology algebra of L is linearizable.

### February 19, Till Brönnle

Title: Extremal Kähler metrics on projectivized vector bundles

Abstract*: *This talk shall be a sequel to my talk last year on the subject of extremal Kähler metrics on projectivised vector bundles. However, we shall focus more on the analytic aspects of the problem and will discuss how to deal with the fully non-linear fourth order PDE which has to be solved in order to construct such an extremal Kähler metric. The major technical difficulty is to control the adiabatic parameter involved in the construction and we shall explain how to deal with this.

## 2013

### March 6, Tobias Ekholm

Title: Constructing Lagrangian immersions with few double points

Abstract: We describe how Murphy's theory of loose Legendrian submanifolds leads to Lagrangian immersions with surprisingly few double points.

The talk reports on joint work with Elliashberg, Murphy, and Smith.

## 2012

### October 31, Tobias Ekholm

Title: Large N-dualities and knot contact homology III

Abstract: This is the third talk in a series where we intend to discuss recent observations relating Chern-Simons theory and topological strings to augmentation varieties arising in knot contact homology.

### October 17, Tobias Ekholm

Title: Large N-dualities and knot contact homology II

Abstract: This is the second talk in a series where we intend to discuss recent observations relating Chern-Simons theory and topological strings to augmentation varieties arising in knot contact homology.

### September 26, Tobias Ekholm

Title: Large N-dualities and knot contact homology I

Abstract: This is the first talk in a series where we intend to discuss recent observations relating Chern-Simons theory and topological strings to augmentation varieties arising in knot contact homology. During the first talk we discuss background material on both the physics a mathematics sides.

### September 19, Thomas Kragh (Uppsala)

Title: Stable homotopy types in Floer theory

Abstract: I will start by giving a crash course in Morse theory and describe how Morse homology is used to define invariants in symplectic topology. Then I will explain why spaces constructed from Morse theory are stronger invariants than Morse homology, but also why we can never hope to refine Floer homology to a space valued invariant. This naturally leads us to the definition of (pre-)CW-spectra, and I will explain how these are weaker invariants than spaces yet stronger than homology. Finally, I will discuss some cases where one can refine Floer homology into spectrum valued invariants and some results coming from this.

### September 5, Reza Rezazadegan (Uppsala)

Title: On a spectral sequence for Lagrangian Floer holomogy

Abstract: Fibered Dehn twists are certain symplectomorphisms associated to Lagrangian spheres and more generally spherically fibered coisotropic submanifolds of a symplectic manifold. They generalize the familiar two dimensional Dehn twists. In this talk I outline how the effect of a composition of such Dehn twists on Floer homology can be given as a hypercube of resolutions. An important tool here is quilted Floer homology. Time permitting, I outline how the "spectral sequence of branched double covers" is a special case of the one mentioned above. This spectral sequence implies that by adding extra terms to the Khovanov hypercube of a link L, coming from counting holomorphic polygons in symmetric products of surfaces, one obtains the Heegaard-Floer homology of the branched double cover of L.

### May 30, Johan Björklund

Title: Flexible isotopy classification of flexible knots

Abstract: In this talk we will define flexible knots, objects meant to capture the topological properties of real algebraic knots, and then use them to introduce flexible isotopy, that is, an isotopy which is at all times a flexible knot. We will also briefly present Viros encomplexed writhe using Ekholms interpretation in terms of the shade number. It will be shown that two genus 0 flexible knots of degree d are flexibly isotopic if, and only if, their real parts are smoothly isotopic and their encomplexed writhes coincide. If time allows we will also see that there are comparatively "many" flexible knots compared to real algebraic knots of a given degree (considered up to flexible and rigid isotopy respectively).

### March 7, Johan Källén (Dept. of Physics and Astronomy)

Title: Topological field theories and contact structures

Abstract: I will describe an interesting interplay between contact geometry and constructions of topological field theories. The focus is on a recently constructed five dimensional theory, and our formalism suggest a generalization of the instanton equations to five dimensional contact manifolds. For the special case of the five manifold being a circle bundle over a four dimensional symplectic manifold with integral symplectic form, I will describe how to calculate explicit expressions for the partition function of the theory using the technique of path integral localization, which is an infinite-dimensional generalization of the Atiyah-Bott-Berline-Vergne

localization theorem for finite dimensional integrals. These types of calculations have received a lot of interest in the case of three dimensional topological field theories, since it gives a new perspective on invariants arising from Chern-Simons theory on Seifert manifolds.

### February 22, Tobias Ekholm

Title: Legendrian knots and exact Lagrangian cobordisms

Abstract: An exact Lagrangian cobordism between Legendrian links induce a DGA-morphism from the Legendrian algebra of the link at the positive end to that of the link at the negative end. We give a flow tree description of such maps for cobordisms that are Lagrangian submanifolds in a class of symplectic manifolds with ends that are half-symplectization of standard contact 3-space. In particular, for elementary cobordisms, i.e. cobordisms that corresponds to certain modifications of Legendrian knots, this description leads to a purely combinatorial computation of the cobordism map. As an application we construct non-deformation equivalent exact Lagrangian surfaces that fill a fixed Legendrian link. Furthermore, we observe that an exact filling of a Legendrian knot that is composed of certain elementary cobordisms define an element in the Khovanov homology of the knot.

### January 26, Emilia Lundberg

Title: A bar construction in Morse-Witten homology (presentation of master thesis)

Abstract: We introduce an $A_\infty$-structure on the Morse-Witten complex of a smooth closed manifold $M$, where the operations are defined by counting perturbed Morse flow trees. Conjecturally, the corresponding Hochschild homology is closely related to the singular homology of the free loop space of $M$. We show, by direct calculation, that the two are isomorphic for products of spheres.

## 2011

### December 14, Vladimir Chernov (Dartmouth)

Title: Relations between contact and Lorentz geometry

Abstract: We show that for many spacetimes causal relation between two points is equivalent to the Legendrian linking of spheres of lights rays through these points in the contact manifold of all light rays. This gives solution to the Low Conjecture and the Legendrian Low conjecture formulated by Natario and Tod. We also discuss causal structure on the space of Legendrian submanifolds in a contact manifold. Finally we explain how globally hypebolicity of the Lorentz metric determines the smooth structure on the spacetime. This leads to the question whether the contact manifold of all light rays determines the spacetime and its Cauchy surface.

### November 23, Tobias Ekholm

Title: Exact Lagrangian immersions with one double point II

Abstract: We show the following: if $K$ is a $2k$-manifold, $k>2$, with Euler characteristic different from $-2$ that admits an exact Lagrangian immersion into ${\mathbb C}^{2k}$ with one transverse double point and no other self interscetions then $K$ is diffeomorphic to the standard $2k$-sphere. In particular, the result rules out Lagrangian immersions of exotic spheres with only one double point, even though such spheres admit Morse functions with only two critical points. The result discussed is joint work with Ivan Smith.

### November 16, Baptiste Chantraine (Université Libre de Bruxelles)

Title: Bilinearized Legendrian contact homology

Abstract: Linearisation of Legendrian contact homology is a tool to extract finite dimensional invariants out of the Chekanov algebra of a Legendrian submanifold. A drawback of the construction is that on the first order the theory becomes commutative. In this talk, we will introduce a generalisation of this tool called bilinearisation which keeps track of the non commutativity of the Chekanov algebra even at the first order. This construction also appears to be an effective tool to distinguish augmentations of a Legendrian submanifold. We will provide examples and geometrical interpretations of both of those aspects. This is a joint work with Frédéric Bourgeois.

### November 9, Tobias Ekholm

Title: Exact Lagrangian immersions with one double point

Abstract: We show the following: if $K$ is a $2k$-manifold, $k>2$, with Euler characteristic

different from $-2$ that admits an exact Lagrangian immersion into ${\mathbb C}^{2k}$ with

one transverse double point and no other self interscetions then $K$ is diffeomorphic to the standard $2k$-sphere. In particular, the result rules out Lagrangian immersions of exotic spheres with only one double point, even though such spheres admit Morse functions with only two critical points. The result discussed is joint work with Ivan Smith.

### November 2, Andreas Juhl

Title: Einstein metrics of negative curvature and conformally invariant differential operators

Abstract: I will first discuss the notion of Poincare-Einstein metrics in the sense of Fefferman and Graham. These are asymptotically hyperbolic metrics which can be considered as real analogs of the Kaehler-Einstein metrics of Cheng and Yau. In recent years, their study has been much stimulated by its interaction with the AdS/CFT-duality in physics. Poincare-Einstein metrics correspond to conformal classes on its boundary at infinity. This bulk-space/boundary correspondence is the basis of their usage in conformal differential geometry. Then I will describe how the correspondence can be used to understand the structure of conformally invariant powers of the Laplacian (which generalize the Yamabe operator). Among other things, these operators give rise to interesting action functionals, lead to the notion of Branson's Q-curvature and are related to functional determinants. In a sense, the results can be regarded as incarnations of the duality suggested by physics.

### October 26, Douglas LaFountain (Århus)

Title: The space of filtered screens and holomorphic curves

Abstract: For a genus g surface with s> 0 punctures and 2g+s> 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space.

In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we thus obtain a cell decomposition for a new compactification of moduli space; if time permits we indicate possible ways this technology may be used to study moduli spaces of holomorphic curves. This work is joint with R. Penner.

### September 14, Clement Hyvrier

Title: Hamiltonian fibrations and the Weinstein conjecture

Abstract: Using the relation between the non-vanishing of some Gromov-Witten invariants and the existence of a closed characteristics for any closed separating contact hypersurface due to Hofer and Viterbo, we will in show that the Weinstein conjecture holds for Hamiltonian fibrations over symplectically uniruled spaces, i.e. symplectic spaces for which there is a non vanishing genus zero Gromov-Witten invariant with one point constraint (or, roughly speaking, for which there is a rational curve through any point).

### April 20, Justin Pati (Uppsala)

Title: Convergence to Reeb Chords of Pseudoholomorphic Strips in Symplectizations

Abstract: We will discuss the assumptions and proofs of several results pertaining to asymptotic behavior of pseudoholomorphic strips.

These results are crucial for the analytic definition of the various versions of the differential in contact homology and Legendrian contact homology. The talk will be reasonably elementary and will hopefully bring out the contact geometric aspects of the pseudoholomorphic curve theory. We will begin with some rather general theory valid on the symplectization of any compact contact manifold. The main ingredients here are area and energy along with Hofer's bubbling lemma. Convergence to Reeb chords in the chord generic case will follow directly. Next we will try to understand the Morse-Bott case. This will involve a nice choice of coordinates which in fact cover the whole asymptotic story of a strip with a single coordinate chart.

### January 26, Licentiate seminar, Georgios Dimitroglou Rizell

Title: Knotted Legendrian surfaces with few Reeb chords

Abstract: For g > 0, we construct g + 1 Legendrian embeddings of a surface of genus g into J1(R2) = R5 which lie in pairwise distinct Legendrian isotopy classes and which all have g +1 transverse Reeb chords (g +1 is the conjecturally minimal number of chords). Furthermore, for g of the g + 1 embeddings the Legendrian contact homology DGA does not admit any augmentation over Z2, and hence cannot be linearized.

We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in J1(S2) from a similar perspective.

### January 19, Thesis presentation, Cecilia Karlsson

Title: Area preserving isotopies of self transverse immersions of S1 in R2

## 2010

### December 15, Ezra Getzler (Northwestern University)

Title: A Filtration of Open/Closed Topological Field Theory

### November 24, Ramon Horvath (Uppsala)

Title: On A∞ algebras II

### November 17, Ramon Horvath (Uppsala)

Title: On A∞ algebras

### November 3, Albin Eriksson Östman (Uppsala)

Title: A smooth isotopy invariant defined through relative contact homology

### October 20, Johan Björklund (Uppsala)

Title: Legendrian contact homology in the product of a punctured Riemann surface with the real line

### October 13, Georgios Dimitroglou Rizell (Uppsala)

Title: Knotted Legendrian surfaces with few Reeb Chords

### October 6, Tobias Ekholm (Uppsala)

Title: Legendrian surgery and the symplectic homology product II

### September 29, Tobias Ekholm (Uppsala)

Title: Legendrian surgery and the symplectic homology product

### April 21, Yacin Ameur

Title: *The Coulomb plasma*

### April 14, Dorin Cheptea (Romanian Academy)

Title: *The universal finite-type invariant of three-dimensional manifolds and of cobordisms, and applications.*

### April 7, Tobias Ekholm (Uppsala)

Title: Knot contact homology II

### March 31, Tobias Ekholm (Uppsala)

Title: Knot contact homology

### March 17, Jens Fjelstad (Karlstad)

Title: Some properties of quantum representations of mapping class groups.

### March 3, Johan Björklund (Uppsala)

Title: A new invariant for generic real algebraic surfaces in RP³

### February 17, Tobias Ekholm (Uppsala)

Title: Legendrian homology – S³ vs R³

### Wednesday February 3, Daniel Mathews

Title: Chord diagrams, contact-topological quantum field theory, and contact categories