Poisson CRM Days 2022

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.

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.

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.

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.

$b$-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.