  These are lecture notes for the course "Poisson geometry and deformation
quantization" given by the author during the fall semester 2020 at the
University of Zurich. The first chapter is an introduction to differential
geometry, where we cover manifolds, tensor fields, integration on manifolds,
Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter
covers the most important notions of symplectic geometry such as Lagrangian
submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics,
moment maps and symplectic reduction. The third chapter gives an introduction
to Poisson geometry where we also cover Courant structures, Dirac structures,
the local splitting theorem, symplectic foliations and Poisson maps. The fourth
chapter is about deformation quantization where we cover the Moyal product,
$L_\infty$-algebras, Kontsevich's formality theorem, Kontsevich's star product
construction through graphs, the globalization approach to Kontsevich's star
product and the operadic approach to formality. The fifth chapter is about the
quantum field theoretic approach to Kontsevich's deformation quantization where
we cover functional integral methods, the Moyal product as a path integral
quantization, the Faddeev-Popov and BRST method for gauge theories,
infinite-dimensional extensions, the Poisson sigma model, the construction of
Kontsevich's star product through a perturbative expansion of the functional
integral quantization for the Poisson sigma model for affine Poisson structures
and the general construction.
