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 16.00-17.00 (CEST) – 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:**The first Friday of each month 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 4pm and the second one at 5pm**.**Additional information:**If you miss or want to revisit one of the talks, you can find a link to the slides/video below.

**Upcoming talks:**

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: *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.

**Previous talks:**

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

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

Date: August 7, 2020

SlidesIn 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

Slides and videoLet 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

Slides and videoCharacteristic 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

Slides and videoIn 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

Slides and videoWhile 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)