Description

In the period May 13 - June 25, 2024, we plan to invite distinguished experts in Poisson Geometry and related theories to the INdAM Unit of the Department of Mathematics and Applications "Renato Caccioppoli" of the University of Napoli Federico II. The invited researchers will give seminars/mini-courses on their recent results. Below is a list of the experts who will visit us, including the dates of their visits and some details about the lectures that they will give.

Visitors

• Luca Accornero (MPIM Bonn)

• Kadri Berktav (Bilkent University)

• Nicola Ciccoli (University of Perugia)

• Peter Crooks (Utah State University)

• Miquel Cueca Ten (Universität Göttingen)

• Janusz Grabowski (IMPAN, Warsaw)

• Alberto Ibort (UC3M & ICMAT)

• Charlotte Kirchhoff-Lukat (KU Leuven)

• Niels Kowalzig (University of Rome Tor Vergata)

• Sylvain Lavau (Ruder Boskovic Institute)

• Antonio Maglio (University of Salerno)

• João Nuno Mestre (University of Coimbra)

• Antonio Michele Miti (Sapienza Università di Roma)

• Praful Rahangdale (Paderborn University)

• Leonid Ryvkin (University Claude Bernard Lyon  1)

• Matthias Schötz (IMPAN Warsaw)

• Wilmer Smilde (University of Illinois Urbana-Champaign)

• Salvatore Stella (University of L'Aquila)

• Mathieu Stiénon (Penn State University)

• Mikhail Vasilev (University of Glasgow)

• Patrizia Vitale (University of Napoli)

• Stefan Waldmann (University of Würzburg)

• Ping Xu (Penn State University)

Calendar

The seminars will take place at 15:00 in room SP1L, unless otherwise indicated.  

May 13

Kick-off Meeting: Poisson Geometry and Mathematical Physics

May 14

Seminar by Antonio Michele MITI (Sapienza Università di Roma) on

"Multisymplectic observables and higher Courant algebroids"

Abstract: Multisymplectic manifolds are a straightforward generalization of symplectic manifolds where one considers closed non-degenerate k-forms in place of 2-forms. Recent works by Rogers and Zambon showed how one could associate to such a geometric structure two higher algebraic structures: an L_infty-algebra of observables and an L_infty-algebra of sections of the higher Courant algebroid twisted by omega. Our main result is proving the existence of an L_infty-embedding between the above two L_infty-algebras generalizing a construction already found by Rogers around 2012 valid for multisymplectic 3-forms only. Moreover, we display explicit formulae for the sought morphism involving the Bernoulli numbers. Although this construction is essentially algebraic, it also admits a geometric interpretation when declined to the particular case of pre-quantizable symplectic forms. The latter case provides some evidence that this construction may be related to the higher analogue of geometric quantization for integral multisymplectic forms. This is joint work with Marco Zambon.

May 20-24

Mini-Workshop on Symmetry & Reduction in Poisson & Related Geometries 

May 29

Seminar by Patrizia VITALE  (University of Napoli) on

"Poisson Gauge Theory"

Abstract: I shall discuss Abelian gauge theories on Poisson manifolds. The gauge potential one forms   are described as sections of a symplectic realization of the spacetime manifold and infinitesimal gauge transformations as a representation of the associated Lie algebroid acting on the symplectic realization. Finite gauge transformations are obtained by integrating the sections of the Lie algebroid to bisections of a symplectic groupoid, which form a one-parameter group of transformations. A covariant electromagnetic two-form is obtained, together with a dual two-form, invariant under gauge transformations. The duality appearing in the picture originates from the existence of a pair of orthogonal foliations of the symplectic realization, which produce dual quotient manifolds, one related with space-time, the other with momenta.

June 4

Online Seminar by Charlotte KIRCHHOFF-LUKAT (KU Leuven) on

"The Fukaya category of a log symplectic surface"

Abstract: Floer theory and Fukaya categories constitute powerful invariants of symplectic manifolds. As a first step in the effort to extend these techniques to Poisson structures with degeneracies, I will present the construction of the Fukaya category for log symplectic Poisson structures on oriented surfaces, with a focus on the additional features of the theory arising from the degeneracy locus. 

June 11 - 14

Mini-Workshop on Lie Groupoids, Lie Algebroids & Differentiable Stacks

June 18

Seminar by Alberto IBORT (UC3M & ICMAT) on

"Symplectic groupoids, their representations and the structure of spacetime"

Abstract: The structure of spacetime will be analyzed from the perspective of the theory of groupoids. Any spacetime has associated a kinematical groupoid whose unitary irreducible representations provide Wigner’s elementary particles. A compatible Poisson structure on a given spacetime will provide a symplectic groupoid whose representations will describe some of its quantum characteristics. A few examples and implications will be presented.

June 25

Seminar by Salvatore STELLA (Università dell'Aquila) on 

"Deformation of Bases for Cluster Algebras"

Abstract: Since their introduction a major problem in the theory of cluster algebras has been to construct bases for them. Indeed, having a basis with some prescribed properties has deep structural consequences on a cluster algebra. Several different constructions were proposed over the years arising from a variety of perspectives (e.g. Lie theory, mirror symmetry, quiver representations, Teichmüller theory, and combinatorics). All these different constructions share a common feature: they consist of pointed elements. Recently Qin was able to parametrize all basses made out of pointed elements for a large class of cluster algebras. In particular, he proved that all of them are related by upper triangular linear maps. Unfortunately, his description of the deformation spaces these transformations span is not explicit limiting the scope of his result in practical applications. In this talk, after introducing the required notions, we will explain how to construct explicitly these deformation spaces for cluster algebras whose growth is at most linear. Time permitting we will relate this phenomenon to the geometry of certain subvarieties of Kac-Moody groups.