Every branch of geometry (and not only) comes with its own problems regarding stability and rigidity phenomena (related to deformations and local forms of the structures involved). The aim of this mini-course is to explain some recent results of this type in Poisson geometry. If the time allows, I will also explain that the Poisson setting is actually a unifying one.

In the first lecture I will give an overview of some of the classical results:

- equivariant geometry: the stability theorem of orbits (Hirsch,
Stowe, Thurston) and the well-known slice theorem.

- foliation theory: the stability theorem of leaves (Thurston,
Langevin-Rosenberg) and the local Reeb stability theorem.

In the second lecture I will move on to Poisson geometry and I will explain the statements of the analogous results.

In the third lecture I will give an outline of the proofs. If the time allows I will explain how, due to the close relationship between Poisson manifolds and Lie algebroids, the classical results from the first lecture can be recovered from the Poisson ones.

Stokes variety arises in various disguises in several branches of mathematics. One can define it as a quotient of the space of meromorphic connections on a vicinity of the origin of the complex plane having a pole of given degree at zero by holomorphic gauge transformations. Another reincarnation of the same space (for the pole of order 2) is the group dual in the Poisson-Lie sense to a simple group. The main aim of the the mini-course is to give as explicit description of the Stokes variety as possible and discuss its properties. In particular we will construct explicit coordinate systems on it making the Poisson structure constant describe the action of the braid group on this space and, if time permits, discuss its relations to integrable systems and canonical basis.

Periods of the (pronilpotent completion of) fundamental group of a complex variety X can be calculated via iterated integrals. For example, if X is the complex projective line minus three points, the periods turn out to be precisely Euler's multiple zeta numbers. The simplest of the periods for the complex projective line minus four points are the classical polylogarithms. I will introduce a new way to calculate the periods, as correlators of certain Feynman integral related to the variety. In the above example it gives a completely new way to present the multiple zeta numbers.

In the 1980s low-dimensional topology encountered mathematical physics, and their interactions yielded many new invariants, called quantum invariants, of knots and 3-manifolds. In this mini-course, we will focus on two kinds of quantum invariants: the Reshetikhin-Turaev invariant (constructed via surgeries on links which are colored by means of a modular category) and the Turaev-Viro invariant (constructed by coloring triangulations of 3-manifolds with 6j-symbols of a fusion category). Then, if time permits, I will explain how to compare them (via the categorical center construction of Drinfeld).

I Gromov-Witten theory: an introduction

II Gauged Gromov-Witten theory (after Mundet, Salamon, etc.)

III Quantum Kirwan map (after Ziltener etc.)

IV Quantum non-abelian localization