In this talk, we shall review several formalisms involving the notion of quantization. These include deformation quantization of Poisson manifolds as well as Geometric quantization of symplectic manifolds. At the end, we shall sketch the relation between these notions of quantization and Lie theoretic constructions involving symplectic groupoids.

Despite their close relation with moduli spaces of Higgs bundles and flat connections through the non-abelian Hodge correspondence, the algebraic structure of moduli spaces of representations of surfaces groups, aka character varieties (our hard nut), remains being a mystery in many cases. To address this problem, in this talk we shall explain how to construct Topological Quantum Field Theories (our sophisticated and very powerful sledgehammer) to crack this problem and others in algebraic topology.Topological Quantum Field Theories were introduced by Witten to interpret the Jones polynomial, a knot invariant, in terms of Chern-Simmons theory. This construction, which also involves character varieties, was carried out through geometric quantization. In sharp contrast, our quantization will be inspired by Fourier-Mukai transformations of sheaves, which leads to new and exciting ways of computing motivic information of the character variety. Time permitting, we shall also discuss the emerging interference phenomena arising in parabolic character varieties.

In this talk, we present new results on the dynamics of three-dimensional Reeb flows defined by stable Hamiltonian structures. These flows generalize contact Reeb fields and arise as well on Hamiltonian systems restricted to a type of regular energy level set. We give a characterization of aperiodic stable Hamiltonian Reeb fields, obtaining a sharp refinement of the Weinstein conjecture in this context. Under a non-degeneracy assumption, we characterize Reeb fields with finitely many closed orbits, showing that in most manifolds they always admit infinitely many periodic orbits. Finally, building over recent results in contact Reeb dynamics, we give sufficient conditions for a Reeb field to be supported by some broken or a rational open book decomposition. This allows us to show that there is a dense set of stable Hamiltonian Reeb fields that admits a Birkhoff section: their dynamics can be studied via a first-return map on a surface with boundary. This talk is based on joint work with A. Rechtman.