Wall crossing is a relatively new phenomenon which appears in several mathematical frameworks: in Homological Algebra (namely, in Donaldson-Thomas invariants); in Combinatorics (in quiver mutations, cluster algebras, canonical bases); in Complex and Differential Geometry (explicit formulas for the hyperkähler metric on the Hitchin moduli space); in Analysis (Stokes phenomenon). Also the wall crossing phenomenon has «physical» origins in supersymmetric Gauge/String Theory, as marginal stability of BPS particles and supersymmetric black holes.

The moduli spaces of meromorphic quadratic differentials serves as the most basic example. In this case one can write the wall crossing formulae explicitly, and see directly its various algebraic and geometric interpretations. Moreover, the natural objects of study that the dynamics of the Teichmuller geodesic flow and interval exchange maps (i.e. separatrix intervals and closed geodesics) are, coincide with BPS states in Physics, stable objects in Algebra, cluster coordinates in Combinatorics.

The goal of these series of lectures is to introduce the community of specialists in flat surfaces and in Teichmuller flow (and, in fact, everybody) to the new exciting developments giving new viewpoints on the familiar objects, and proposing new paths for generalizations.