A oneday forum on multisymplectic geometry and related topics.
date: 10 January, 2024
time: 10:00–17:00
institute: Institut Camille Jordan, Université ClaudeBernard Lyon 1
room: Fokko de Cloux, Bâtiment Braconnier
organizers: Casey Blacker, Antonio Miti, Leonid Ryvkin
Francois Gieres (Université ClaudeBernard Lyon 1)
Covariant phase space and multisymplectic geometry We present the geometric formulation of explicitly timedependent Hamiltonian systems in classical mechanics in order to motivate the consideration of multisymplectic geometry for obtaining a relativistically covariant canonical formulation of classical field theories. To conclude, the approach of (multi‑)symplectic geometry is briefly compared with the socalled covariant phase space approach. 

Maxime Wagner (Université de Lorraine)
The 6sphere as a 2plectic manifold Firstly, we look at the 2plectic form on the sixsphere which is strongly linked to the exceptional Lie group G_2 and show that there exists no Darboux theorem for this structure. In a second part, we will be interested in the dynamics associated with this structure. 
Véronique Chloup and Phillippe Bonneau (Université de Lorraine)
An introduction to kPoisson structures Knowing what is Poisson geometry as a more general framework for symplectic geometry, what can be kPoisson geometry as a more general framework for k‑plectic geometry? We'll describe quickly but completely the answer given by H. Bursztyn et al. to this question. We'll generalise the question to Dirac/kDirac (Dirac geometry being a larger generalisation of Poisson geometry) as an easytoreach and more complete point of view, that includes prekplectic geometry. Seeing first a kPoisson (resp. kDirac) manifold as the infinitesimal object associated to a kplectic groupoid (resp. prekplectic groupoid) we'll then give equivalent and more easytouse definitions as structures on the manifold itself, without any more reference to a groupoid. Finally, we'll present results about foliations of these structures and illustrate all this with examples. Nothing in this talk will be original, all the material comes (mainly) from  Bursztyn, Cabrera, Iglesias : "Multisymplectic geometry and Lie groupoids"  Bursztyn, MartinezAlba, Rubio : "On higher Dirac structures". 

Casey Blacker (George Mason University)
Introduction to bundle gerbes In this exposition, we approach bundle gerbes as smooth assignments of groupoids enriched in U(1)torsors over a fixed manifold M. After first considering typical fibers, we present the general construction and proceed to define the Dixmier–Douady class, connective structures and surface holonomy. 
Casey Blacker  George Mason University 
Antonio Miti  Università Cattolica del Sacro Cuore 
Leonid Ryvkin  Université Lyon 1 
Olga Kravchenko  Université Lyon 1 
Claude Roger  Université Lyon 1 
Francois Gieres  Université ClaudeBernard Lyon 1 
Sucheta Majumdar  ENS de Lyon 
Marco Valerio d'Agostino  ENS de Lyon 
Maxime Wagner  Université de Lorraine 
Véronique Chloup  Université de Lorraine 
Phillippe Bonneau  Université de Lorraine 