The Friday Fish Seminar is a weekly seminar organized by the Poisson Geometry group in Utrecht. Our aim is to provide an opportunity for young mathematicians in the field of Poisson Geometry to present their research and to broaden their network. With this online edition, we hope to complement the Global Poisson Webinar.

**When:**Friday 16.30-17.30 (CET, i.e. Amsterdam time). Participants are very welcome to stay around afterwards for some informal discussion.**Where:**Zoom.**How:**Please send an e-mail to fridayfishseminar[at]gmail.com to receive weekly invitations to the Zoom meetings.**Junior Poisson Session:**Somewhat regularly we will have a “Junior Session”, consisting of two 30-minute talks given by junior members of the Poisson community. Any PhD student that wishes to give a talk is welcome to register here. Speakers should feel free to talk about work in progress.**The first talk will be at 4:30pm and the second one at 5:30pm**.**Additional information:**If you miss or want to revisit one of the talks, you can find a link to the slides/video below.**Note:**We have had some issues with our emails being sent to spam. Please take a look at your spam folder if you do not receive the weekly announcements.

#### Upcoming talks:

March 12, 2021

March 19, 2021

March 26, 2021

April 9, 2021

April 16, 2021

May 21, 2021

#### Previous talks:

February 26, 2021

**Title:** *Some aspects of $C^0$-symplectic geometry*

**Speaker:** Dušan Joksimović

A folklore theorem of Eliashberg and Gromov, which states that the group of symplectic diffeomorphisms is closed in the group of all diffeomorphisms w.r.t. C^0-topology, is considered the beginning of $C^0$-symplectic geometry which roughly investigates which symplectic phenomena persist under $C^0$-limits. It also allows us to define symplectic homeomorphisms (as $C^0-limits of symplectomorphisms) and $C^0$-symplectic manifolds. So far it is not known whether there are manifolds that admit a $C^0$-symplectic, but not smooth symplectic structures. One of the central question in the field, due to Hofer, is whether the spheres admit such structure in dimension greater than 2.

In this (mostly overview) talk we will state and prove Elisahberg-Gromov's theorem using another result from $C^0$-symplectic geometry, namely, the $C^0$-rigidity of Poisson brackets due to Cardin-Viterbo, Entov-Polterovich, and Buhovski. If time allows, we will discuss Hofer's question as well.

February 19, 2021

**Title:** *The deformation cohomology of a symplectic groupoid*

**Speaker:** João Nuno Mestre

Symplectic groupoids are Lie groupoids equipped with multiplicative symplectic forms, and serve as global counterparts to (integrable) Poisson manifolds, in the same way that Lie groups are global counterparts to Lie algebras.

In this talk I will present the construction of the deformation cohomology controlling deformations of symplectic groupoids, built out of the deformation complexes of a Lie groupoid and of a multiplicative form. We will see that it can be identified with the total complex of a sub double complex of the Bott-Shulman-Stasheff double complex.

I will then compute this cohomology in some particular cases (linear Poisson structures and proper groupoids), explain how to use it in a Moser path argument, and describe some relations to the deformation theory of the corresponding Poisson manifolds.

This is joint work with Cristian Cárdenas (UFF) and Ivan Struchiner (USP).

February 5, 2021

**Title:** *On the homotopy type of the contactomorphism group of a tight contact 3-manifold*

**Speaker:** Eduardo Fernandez ** (Junior talk, 5:30pm) **

One of the building blocks in the study of the homotopy type of the diffeomorphism group of a 3-manifold is the positive answer of A. Hatcher to the Smale conjecture. This result has its contact counterpart: Eliashberg's theorem about the contractibility of the contactomorphism group of the tight contact 3–ball.

In order to adapt the smooth techniques to the contact world, it is necessary to develop an understanding of the space of embeddings of surfaces into a given tight contact 3-manifold. In this talk we will see how to do this for simple surfaces such as disks and spheres. As a consequence, we will conclude that, with the exception of connected components, all the remaining homotopy groups of the group of contactomorphisms of a tight contact 3-manifold are controlled by topological invariants.

This talk is a continuation of the previous talk "Flexibility in contact 3-manifolds: from contactomorphisms to legendrian knots" by Javier Martínez-Aguinaga. Joint work with Javier Martínez-Aguinaga and Fran Presas.

February 5, 2021

**Title:** *Flexibility in contact 3-manifolds: From contactomorphisms to legendrian knots*

**Speaker:** Francisco Javier Martinez Aguinaga (**Junior talk, 4:30pm**)

The study of the homotopy type of the space of legendrian submanifolds in a contact manifold is a central problem in Contact Topology. In this talk, mimicking the techniques developed in Smooth Knot Theory (work of A. Hatcher and R. Budney), we will relate the homotopy type of such spaces with the homotopy type of the contactomorphism group of the complement.

In particular, we show that the inclusion of each connected component of the space of long legendrian embeddings into the space of smooth long embeddings is a homotopy equivalence. This result shows that, except for the number of connected components, the homotopy of the space of Legendrians is governed by the topology of the space of smooth knots.

This talk will be the first part of a 2-talk session, followed by "On the homotopy type of the contactomorphism group of a tight contact 3-manifold" by Eduardo Fernández. Joint work with Eduardo Fernández and Francisco Presas.

January 29, 2021

**Title:** *Real Forms of Holomorphic Hamiltonian Systems*

**Speaker:** Marine Fontaine

By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recovers the original system but also yields different real Hamiltonian systems which share the same complexification. In this talk I will explain the notion of real form for a symplectic manifold, more generally, for a Poisson manifold and will discuss their applications to integrable Hamiltonian systems. This is based on joint work with P. Arathoon.

January 15, 2021

**Title:** *A cohomological proof for the integrability of strict Lie 2-algebras*

**Speaker:** Camilo Angulo

A strict Lie 2-algebra is a groupoid object in the category of Lie algebras. These can naturally be seen as an infinitesimal version of strict Lie 2-groups which are groupoids in the category of Lie groups. Lie 2-algebras were proven to be integrable in this sense using the path strategy to integration, we present an alternative proof based on the van Est strategy.

December 18, 2020

**Title:** *Symmetry, Cartan connections and Rigidity*

**Speaker:** Katharina Neusser ** (Note: the talk will start an hour earlier, at 3:30pm) **

Many important geometric structures are geometrically rigid in the sense that their Lie algebra of infinitesimal automorphisms is finite-dimensional. Prominent examples of such structures are Riemannian and conformal manifolds, and in general all geometric structures admitting equivalent descriptions as so-called Cartan geometries. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified.

In this talk I will discuss several such classification results and explain how Cartan connections can be used to study local and global questions of geometric rigidity. The talk will also provide an introduction to Cartan geometries with a focus on the subclass of parabolic geometries.

December 11, 2020

**Title:** *Hyperkähler realizations of holomorphic Poisson surfaces*

**Speaker:** Maxence Mayrand

I will discuss the existence of hyperkähler structures on symplectic realizations of holomorphic Poisson manifolds, and show that they always exist when the Poisson manifold has complex dimension two. We obtain this structure by constructing a twistor space by lifting special deformations of the Poisson surface adapted from Hitchin's unobstructedness theorem. In the case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold. This talk is based on arXiv:2011.09282.

November 20, 2020

**Title:** *Strong homotopy structure of Poisson reduction*

**Speaker:** Jonas Schnitzer

Given a Hamiltonian symmetry on a Poisson manifold one can construct a Poisson structure on a reduced manifold. This can be achieved with the Poisson version of the Marsden-Weinstein reduction or equivalently with the BRST-method.

Fixing a Lie group action on a manifold, one can define a curved Lie algebra whose Maurer-Cartan elements are Poisson structures together

with momentum maps. Poisson structures on the reduced manifold are Maurer-Cartan elements of the usual DGLA of polyvector fields. Thus, reduction is just a map between these two sets of Maurer-Cartan elements.

In my talk I want to show that this map is actually the map on Maurer-Cartan elements induced by an $L_infty$-morphism.

November 13, 2020

**Title:** *Lie groups of Poisson diffeomorphisms*

**Speaker:** Wilmer Smilde **(Junior talk, 5:30pm)**

The group of symplectomorphisms of a symplectic manifold is a well-studied object. However, little is known about the Poisson diffeomorphism group of other Poisson structures.

In this talk, we present a first step in this direction. First, we discuss the intricate algebraic structure of the Poisson diffeomorphism group and its connection with coisotropic bisections of a Poisson groupoid. Then, we reformulate the problem of finding smooth structures on these objects to a linearization problem of Poisson structures around Lagrangian submanifolds.

In the second part, we dive deeper into linear Poisson structures and present a linearization result, generalizing the Lagrangian neighbourhood theorem to the setting of Lie algebroids and cosymplectic structures. This is applied to obtain smooth structures on Poisson diffeomorphism groups of several classes of Poisson manifolds.

This is part of my master thesis, supervised by Ioan Marcut.

November 13, 2020

**Title:** *A singular symplectic slice theorem*

**Speaker:** Anastasia Matveeva **(Junior talk, 4:30pm) **

Motivated by semilocal normal forms and invariants for (non)-commutative integrable systems on manifolds with boundary, we prove a singular symplectic slice theorem for $b^m$-symplectic manifolds and discuss some generalizations. This is joint work with Eva Miranda.

November 6, 2020

**Title:** *Vortex sheets in ideal fluids and coadjoint orbits*

**Speaker:** Cornelia Vizman

We describe the coadjoint orbits of the group of volume preserving diffeomorphisms of R3 associated to the motion of closed vortex sheets in ideal 3D fluids. We show that these coadjoint orbits can be identified with nonlinear Grassmannians of compact surfaces enclosing a given volume and endowed with a closed 1-form describing the vorticity density. Goldin, Menikoff, and Sharp present in [1] the case of open vortex sheets (tubes/ribbons).

If the vorticity density has a discrete period group and is nonvanishing, the vortex sheet is given by a surface of genus one fibered by its vortex lines over a circle. We determine the Hamilton equations for such vortex sheets relative to the Hamiltonian function suggested by Khesin in [2] and we prove that there are no stationary solutions having rotational symmetries. These coadjoint orbits are shown to be prequantizable if the period group of the 1-form and the volume enclosed by the surface satisfy an Onsager-Feynman relation.

Joint work with Francois Gay-Balmaz, ENS Paris.

[1] Goldin, G.A., Menikoff, R., Sharp, D.H., Quantum vortex configurations in three dimensions, Phys. Rev. Lett., 67 (1991), 3499-3502.

[2] Khesin, B., Symplectic structures and dynamics on vortex membranes, Moscow Math. Journal 12 (2012), 413-434.

October 23, 2020

**Title:** *On singular cotangent homotopies coming from the Poisson Sigma Model*

**Speaker:** Alejandro Cabrera

In this talk, the idea is to describe with some detail certain singular cotangent (algebroid) homotopies, associated to a Poisson manifold M, which appear as a semiclassical limit in the Poisson Sigma Model (PSM). To that end, first, we will review the heuristic motivation for the PDEs that define these homotopies and their role in quantization. We then proceed to describe the solutions and relate them to certain triangles in an integrating groupoid. The final aim is to show how these homotopies, when fed into a PSM action functional, yield a generating function for the structure of a (local) symplectic groupoid integrating the underlying Poisson M.

October 16, 2020

**Title:** *Deformations of Lagrangian submanifolds in log-symplectic manifolds*

**Speaker:** Stephane Geudens **(Junior talk, 5:30pm)**

Log-symplectic structures are a type of Poisson structures that are symplectic outside of a hypersurface. The aim of this talk is to discuss whether the deformation theory of Lagrangian submanifolds in this setting is as well-behaved as in symplectic geometry.

We will focus on deformations of a Lagrangian submanifold contained in the singular locus of a log-symplectic manifold. Using a normal form around the Lagrangian, we show that the deformation problem is governed by a DGLA. We discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give criteria for unobstructedness. If time permits, we also address equivalences of Lagrangian deformations under Hamiltonian and Poisson isotopies.

This is joint work with Marco Zambon.

October 16, 2020

**Title:** *On two notions of a gerbe over a stack*

**Speaker:** Praphulla Koushik ** (Junior Talk, 4:30pm) **

Let G be a Lie groupoid. The category BG of principal G-bundles defines a differentiable stack. On the other hand, given a differentiable stack D, there exists a Lie groupoid H such that BH is isomorphic to D. We define a gerbe over a stack as a morphism of stacks F : D -> C, such that F and the diagonal map Delta_F : D -> D times_C D are epimorphisms.

In this talk we explore the relationship between a gerbe, as defined above, and a Morita equivalence class of a Lie groupoid extension. This talk is based on our paper (j/w Saikat Chatterjee) titled "On two notions of a gerbe over a stack" (https://doi.org/10.1016/j.bulsci.2020.102886).

October 9, 2020

**Title:** *Stacky Lie algebroids*

**Speaker:** Miquel Cueca Ten

Differentiable stacks are useful models for singular spaces. Fifteen years ago, Tseng and Zhu defined stacky Lie groupoids and used them to study the integrability of Lie algebroids. In this talk we will introduce their infinitesimal counterpart, stacky Lie algebroids, and we shall give their basic properties.

If time permits, we will comment on the integrability of Courant algebroids. This is a joint work with Daniel Alvarez.

September 25, 2020

**Title:** *Moment maps in multisymplectic geometry*

**Speaker:** Leyli Mammadova

This talk will give an introduction to multisymplectic geometry and homotopy moment maps. I will start from the basics, defining notions like n-plectic manifolds, Hamiltonian vector fields, and Hamiltonian (n-1)-forms.

Then, before defining the Lie n-algebra of observables corresponding to an n-plectic manifold (definition due to C. Rogers), I will give a brief introduction to $L_{infty}$-algebras.

Finally, I will introduce two notions of moment maps, the first one due to M. Callies, Y. Fregier, C. L. Rogers, and M. Zambon, and the second one due to J. Herman. If time permits, we will also compare the two notions (this last part would be based on joint work with L. Ryvkin).

September 18, 2020

**Title:** *Weighted normal bundles*

**Speaker:** Yiannis Loizides

I will describe a variant of the normal bundle to a submanifold that is appropriate when one wants to treat normal directions to the submanifold as having different `weights'. The construction has applications to normal form theorems. This is joint work with Eckhard Meinrenken.

September 11, 2020

**Title:** *Universal centralizers and Poisson transversals*

**Speaker:** Ana Balibanu

Let G be a semisimple complex algebraic group of adjoint type. We consider a multiplicative analogue of the universal centralizer of G --- a family of centralizers parametrized by the regular conjugacy classes of the simply-connected cover of G. This multiplicative analogue has a natural symplectic structure and sits as a transversal in a variation of the quasi-Poisson double D(G). We show that D(G) extends to a smooth groupoid over the wonderful compactification of G, and we use this to construct a partial compactification of the multiplicative universal centralizer.

September 4, 2020

**Title:** *Haefliger's differentiable cohomology*

**Speaker:** Luca Accornero

The differentiable cohomology of a pseudogroup on a manifold M was defined by André Haefliger in the seventies, in the development of characteristic classes of foliations. In Haefliger's approach to the subject, foliations are seen as cocycles valued in the étale groupoid Gamma^q of germs of diffeomorphisms of R^q. From this point of view, the theory of characteristic classes for foliations is reminiscent of the theory of characteristic classes for flat principal g-bundles, which can be represented as cocycles valued in the discrete group G^delta.

For such bundles, a "geometric" characteristic map is defined. It is a map from the relative cohomology H*(g, K) of the Lie algebra g of G to the cohomology of M. The classical Van-Est isomorphism for Lie groups allows one to interpret it as a map from the differentiable cohomology H_d*(G) of G to H*(M). A similar "geometric" map is available for foliations and is defined from the cohomology GF^q of the Gelfand-Fuchs Lie algebra of formal vector fields on R^q relative to O(q). Haefliger defined a differentiable complex for the groupoid Gamma^q and proved a Van Est isomorphism between the differentiable cohomology H*_{diff}(Gamma^q) and the cohomology of the relative Gelfand-Fuchs Lie algebra, making the analogy with flat principal bundles complete.

His construction focuses on the groupoid Gamma^q and, probably also because of the seemingly ad-hoc approach, did not receive much attention. Furthermore, it is not clear which is the structure that makes the definition work. Our main driving question is to find such a structure. The outcome is the structure of "flat Cartan groupoid" on the infinite jet groupoid J^inftyGamma^q, which is well known for the role that plays in the geometric theory of PDE. After reviewing Haefliger's approach and introducing flat Cartan groupoids, we will define their "Haefliger cohomology", investigate its infinitesimal counterpart and relate the two by a Van Est map. Finally, we will construct a characteristic map for cocycles valued in a flat Cartan groupoid, which reduces to the "geometric" map for foliations when the groupoid is J^inftyGamma^q.

This is joint work with Marius Crainic.

August 28, 2020

**Title:** *Lie-Hamilton systems and their role in the current Covid pandemic*

**Speaker:** Cristina Sardón

We study Lie-(Hamilton) systems on the plane, i.e. systems of first-order (nonlinear) ODEs describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar (Hamiltonian) vector fields with respect to a geometric structure.

Lie-(Hamilton) systems enjoy a plethora of properties, e.g. they admit their general solution expressed as a (nonlinear) function through the so-called superposition rule of a finite set of particular solutions and some constants.

Lie-Hamilton systems are important because of their appearance in the physics, mathematics and biology literature. For example, they can be used to study Milne–Pinney, second-order Kummer–Schwarz, complex Riccati and Buchdahl equations, which occur in cosmology, relativity and classical mechanics. They also appear in the investigation of Lotka–Volterra, predator-prey or growth of a viral infection models. We are particulary interested in the latter.

In this talk, I present the geometrical properties of Lie-(Hamilton) systems and their application to SIS-pandemic models. We derive complete solutions for a SIS-pandemic model with fluctuations through the intrinsic properties of Lie systems and through the coalgebra method. We will present graphic representations of the solutions, and we will see how the number of infected individuals grows accordingly with a sigmoid-like function. This is precisely the expected behavior, and it is retrieved in two different geometric ways, from a symplectic point of view, and from a Poisson framework.

We finish this talk discussing whether this model is applicable to the current Covid pandemic.

August 21, 2020

**Title:** *Lie groupoids and logarithmic connections*

**Speaker:** Francis Bischoff

In this talk I'll describe a Lie groupoid based approach to the study of flat connections with logarithmic singularities on a hypersurface.

Flat connections on the affine line with logarithmic singularity at the origin are equivalent to representations of a groupoid associated to the exponentiated action of C. I'll describe a canonical Jordan-Chevalley decomposition for these representations, and show how this leads to a functorial classification.

Flat connections on a general complex manifold with logarithmic singularities along a hypersurface are equivalent to representations of a twisted fundamental groupoid. By using a Morita equivalence, the category of logarithmic flat connections can be localized to the normal bundle of the hypersurface. I'll explain how this can be used to prove a functorial Riemann-Hilbert correspondence for logarithmic flat connections.

August 14, 2020

**Title:** *The Atiyah class and ideal systems*

**Speaker:** Madeleine Jotz Lean

This talk describes the Atiyah class of a holomorphic vector bundle and explains how it can be understood as the Atiyah class of an infinitesimal ideal system. Then it explains how Molino's Atiyah class also fits in this context.

The Atiyah class of a Lie pair is an obstruction to the existence of an ideal system (à la Mackenzie and Higgins) on the Lie pair.

August 7, 2020

**Title:** *Poisson non-degeneracy of the Lie algebra so(3,1).*

**Speaker:** Florian Zeiser ** (Junior talk, 5pm)**

In this talk we take a look at the linearization question in Poisson geometry as first asked by Weinstein. We briefly recall what is

currently known for semisimple Lie algebras. Moreover, we outline a proof for the Poisson non-degeneracy of the Lie algebra so(3,1). Therefore so(3,1) is the first example of a non-compact semisimple Lie algebra which is Poisson non-degenerate. This is based on joint work with Ioan Marcut.

August 7, 2020

**Title:** *The singular Weinstein conjecture*

**Speaker:** Cédric Oms **(Junior talk, 4pm) **.

In this talk, we study contact structures that admit certain types of singularities, called $b^m$-contact structures. Those structures can be viewed as a particular case of Jacobi manifolds satisfying some transversality conditions. The motivation to study this generalization of contact structures arises from classical examples in celestial mechanics, as for example the restricted planar circular three body problem (RPC3BP), but also appears in the study of fluid dynamics on manifolds with cylindrical ends.

We will focus on understanding the dynamics of the associated Reeb vector field of $b^m$-contact forms. Due to singularities, the dynamics are fundamentally different to smooth Reeb dynamics and we will discuss a singular version of Weinstein conjecture on the existence of periodic orbits on those manifolds. Time permitting, we talk about generic existence of so called singular periodic orbits.

This is joint work with Eva Miranda and work in progress with Eva Miranda and Daniel Peralta-Salas.

July 31, 2020

**Title:** *Singular chains on Lie groups and the Cartan relations*

**Speaker:** Camilo Arias Abad

Let G be a simply connected Lie group. We denote by C(G) the differential graded Hopf algebra of smooth singular chains on G. We will

explain how the category of modules over C(G) can be described infinitesimally in terms of representations of the differential graded Lie

algebra Tg, which is universal for the Cartan relations. In case G is compact, this correspondence can be promoted to an A-infinity equivalence of differential graded categories. We will also explain how this equivalence is related to Chern-Weil theory and higher local systems on classifying spaces.

This talk is based on joint work with Alexander Quintero Vélez.

July 24, 2020

**Title:** *Characteristic classes for Lie groupoids: going to the basics*

**Speaker:** María Amelia Salazar

Characteristic classes are an important tool in algebra, geometry and topology as they encode invariants that can often be calculated explicitly. In this talk I will present some recent ongoing work about the definition of characteristic classes associated to representations of Lie groupoids. I will concentrate on the construction of such classes as this will shine some light on the underlying geometry and the relation with the classifying space.

July 17, 2020

**Title:** *Symplectic groupoids of elliptic Poisson manifolds*

**Speaker:** Ralph Klaasse

In this talk we describe an ongoing project to construct the adjoint (symplectic) groupoids associated to elliptic Poisson structures. These are a type of Poisson structure that is nondegenerate outside of a submanifold of codimension two. We will describe the geometry of these structures and see that their behavior depends on their so-called elliptic residue, and consequently so do the groupoids. Tools that are used in our constructions are the blow-up procedure for groupoids of Gualtieri-Li, a normal form result due to Witte, and the fact that elliptic Poisson manifolds can be blown-up to be log-Poisson, as noticed by Cavalcanti-Gualtieri.

July 10, 2020

**Title:** *Symplectic : Contact = Poisson : Jacobi = Affine : Projective.*

**Speaker:** Daniele Sepe

While contact geometry is often described as the odd-dimensional analogue of symplectic geometry, Arnol'd pointed out that the relation between these two geometries is analogous to that between affine and projective geometries. A natural (vague!) question is to investigate the extent to which this analogy extends to the degenerate versions of symplectic and contact structures, namely Poisson and Jacobi geometries.

The aim of this talk is to attempt to formalise the above question and to illustrate the intimate relation between these six geometries through a few specific problems, including isotropic realisations and Poisson/Jacobi manifolds of compact types. The talk is based on joint work with María Amelia Salazar, and on ongoing joint work with Camilo Angulo and María Amelia Salazar.

#### Organising committee:

- Marius Crainic (Utrecht University)
- Maarten Mol (Utrecht University)
- Álvaro del Pino (Utrecht University)