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 (CEST)**(note the change in schedule!)**. 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:

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

Speaker: Cornelia Vizman

Date: November 6, 2020

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.

Title: *TBA*

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

Date: November 13, 2020

TBA

Title: *The local nature of Poisson diffeomorphism groups*

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

Date: November 13, 2020

TBA

#### Previous talks:

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

Speaker: Alejandro Cabrera

Date: October 23, 2020

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.

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

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

Date: October 16, 2020

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.

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

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

Date: October 16, 2020

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).

Title: *Stacky Lie algebroids*

Speaker: Miquel Cueca Ten

Date: October 9, 2020

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.

Title: *Moment maps in multisymplectic geometry*

Speaker: Leyli Mammadova

Date: September 25, 2020

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).

Title: *Weighted normal bundles*

Speaker: Yiannis Loizides

Date: September 18, 2020

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.

Title: *Universal centralizers and Poisson transversals*

Speaker: Ana Balibanu

Date: September 11, 2020

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.

Title: *Haefliger's differentiable cohomology*

Speaker: Luca Accornero

Date: September 4, 2020

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.

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

Speaker: Cristina Sardón

Date: August 28, 2020

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.

Title: *Lie groupoids and logarithmic connections*

Speaker: Francis Bischoff

Date: August 21, 2020

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.

Title: *The Atiyah class and ideal systems*

Speaker: Madeleine Jotz Lean

Date: August 14, 2020

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.

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

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

Date: August 7, 2020

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.

Title: *The singular Weinstein conjecture*

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

Date: August 7, 2020

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.

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

Speaker: Camilo Arias Abad

Date: July 31, 2020

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.

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

Speaker: María Amelia Salazar

Date: July 24, 2020

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.

Title: *Symplectic groupoids of elliptic Poisson manifolds*

Speaker: Ralph Klaasse

Date: July 17, 2020

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.

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

Speaker: Daniele Sepe

Date: July 10, 2020

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)