This activity is part of the

Joint CRM-ICMAT Intensive Research Programme Poisson 2022

Poisson CRM Days 2022

Thematic day

From July 14, 2022
to July 15, 2022

Registration is free but mandatory. Limited seats available, the registration will be on a first come, first served basis.

Registration deadline 01 / 07 / 2022

PHOTO GALLERY

Description

Siméon-Denis Poisson, (born June 21, 1781, Pithiviers, France—died April 25, 1840, Sceaux)

As a part of the CRM-ICMAT Joint Intensive Research Programme POISSON, we are celebrating a special Poisson event at CRM during the French National Day in July 14 and 15.

PLENARY SPEAKERS & SHORT TALKS

PLENARY TALKS

On Poisson brackets and quantization

Lluís Santaló Lecturer: Alejandro Cabrera | Federal University of Rio de Janeiro

ABSTRACT

In this talk, we shall review the crucial role that Poisson brackets play in the idea of quantization and, thus, to help establish a connection between the formalisms behind classical and quantum mechanics. We shall first focus on highlighting some of the physical aspects behind these ideas and, finally, try to account for (at least part of) the corresponding wide range of ramifications in mathematics.

Formality and Reduction

Lecturer: Chiara Esposito | University of Salerno

ABSTRACT

In this seminar we discuss the compatibility between deformation quantization and phase space reduction. More explicitly, on the classical side one can reduce (formal) Poisson structures with formal momentum maps, and on the quantum side one can reduce equivariant star products. First, we describe these reductions by morphisms of L-infinity algebras. This allows us to rephrase the “reduction commutes with quantization” problem, originally formulated in the setting of geometric quantization, in terms of L-infinity morphisms.

This is a joint project with A. Kraft and J. Schnitzer.

Poisson geometry around Poisson submanifolds

Lecturer: Rui Loja | University of Illinois at Urbana-Champaign

ABSTRACT

The point of departure of this talk is the construction of local linear models around Poisson submanifolds. I will explain that this is not always possible, and that understanding this problem requires a theory of multiplicative Ehresmann connections, which extends the well-known theory of connections on principal bundles. This talk is based on joint work with Ioan Marcut.

Lie 2-groups actions, groupoid quotients and gerbes.

Lecturer: Alfonso Garmendia | Potsdam Uni

ABSTRACT

This talk reviews the notions of (strict) Lie 2-groups and bundle gerbes, actions of Lie 2-groups on a Lie Groupoid and its quotient. Moreover, it shows that for suitable quotients of groupoids there exists a bundle gerbe and viceversa.

Vortex filaments and membranes in the Euler hydrodynamics

Lecturer: Boris Khesin | University of Toronto

ABSTRACT

The binormal (or vortex filament) equation provides an approximation of the 3D incompressible Euler equation. We present a Hamiltonian framework for the binormal equation in higher-dimensions and its explicit solutions that collapse in finite time. By going to lower dimensions we observe a curious appearance of the golden ratio in the motion of point vortices in the half-plane. This is a joint work with C.Yang and H.Wang.

Existence and classification of $b$-contact structures

Lecturer: Cédric Oms | École normale supérieure de Lyon

ABSTRACT

$b$-contact structures are a generalization of contact structures and can be seen as a Jacobi structure which is not transitive but transversally vanishing. Using the $h$-principle, we will give existence results for those structures in dimension 3. Furthermore, the overtwisted/tight dichotomy can be prescribed in each contact leaf of the Jacobi structure. Time permitting, we will classify these structures for some manifolds with a distinguished hypersurface. This is based on joint work with Robert Cardona.

SHORT TALKS

A Floer complex for b-symplectic manifolds

Lecturer: Joaquim Brugués | University of Antwerp

ABSTRACT

Since the beginnings of the development of Floer theory there has been an interest to adapt this construction to non-compact manifolds. In our project, we aim at building one such complex for the open part of b-symplectic manifolds, using a cosymplectic foliation in a tubular neighbourhood near the singular hypersurface. In this talk we will review the concept of Floer complex, its relation to Morse theory, and the steps taken in our project to construct an adapted complex. This is a joint project with Eva Miranda and Cédric Oms.

Extension of the Large Color R-Matrix for Knot Complements to the so(8) case

Lecturer: Alberto Cavallar | UPC – Caltech

ABSTRACT

The Gukov-Manolescu series, denoted by FK(x,q), is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In a recent paper, Sunghyuk Park uses the large color R-matrix of sl(2) to study FK for some simple links, giving a definition of FK for positive braid knots and computing FK for various knots and links. This procedure has been extended by Angus Gruen to all other Lie algebras sl(n+1) beyond sl(2). In this work, we turn to the family so(2n) of complex semisimple Lie algebras in Cartan’s classification, mainly focusing on the so(8) case attracted by the three-fold symmetry in its Dynkin diagram D4.

Singular symplectic reduction

Lecturer: Anastasia Matveeva | Universitat Politècnica De Catalunya

ABSTRACT

The investigation of symmetries of $b$-symplectic manifold and folded-symplectic manifolds is well-understood when the group under consideration is a torus. However, reduction theory has not been set in this realm in full generality. We prove the Marsden-Weinstein reduction theory under general symmetries for $b^m$-symplectic manifolds and other singular symplectic manifolds, including certain folded symplectic manifolds. In this framework, the set of admissible Hamiltonian functions is larger than the category of smooth functions as it takes the singularities of the differential forms into account. We also show how different generalizations of the moment maps, including quasi-Hamiltonian can be used in this framework. This is a joint work with Eva Miranda.

Geometric quantization of b-symplectic toric manifolds

Lecturer: Pau Mir | Universitat Politècnica de Catalunya

ABSTRACT

We prove that Bohr-Sommerfeld quantization coincides with formal geometric quantization on symplectic toric and b-symplectic toric manifolds. Taking as starting point the construction of Guillemin-Sternberg, we prove the symplectic case. Taking into account the orientation of the connected components of the critical hypersurface Z, we introduce the Bohr-Sommerfeld quantization for b-symplectic toric manifolds and show that it is determined by its b-Delzant polytope. This is joint work with Eva Miranda and Jonathan Weitsman.

Hamiltonian mechanics and gauge theories on Lie algebroids

Lecturer: Pablo Nicolás | Universitat Politècnica de Catalunya

ABSTRACT

Many singular structures in smooth manifolds, like b-manifolds, b^k-manifolds, c-manifolds and, more generally, E-manifolds, are examples of Lie algebroids. With this motivation, we define a canonical Liouville form on the prolongation of the dual of any algebroid $A\rightarrow M$ . This provides us with a general framework for mechanics on Lie algebroids. After developing gauge theories in this setting, we generalize the hamiltonian and Poisson formulations of Wong’s equations introduced by A. Weinstein and R. Montgomery, respectively. We also present natural examples of Lie algebroid structures in general relativity and Yang-Mills theories. This is a joint work with E. Miranda and P. Mir.