WALL CROSSING
                IHES, November 10-12, 2011
              Abstract of the minicourse of

                    Maxim Kontsevich

Wall crossing is a relatively new phenomenon which appears in several
mathematical frameworks:

1) (homological algbera): theory of Donaldson-Thomas invariants, i.e.
of  the  countiung  of  stable  objects in triangulated 3-dimensional
Calabi-Yau  categories  (by  Joyce  and  Song,  and by Kontsevich and
Soibelman).

2) (combinatorics): quiver mutations by Fomin and Zelevinsky, cluster
algebras, canonical bases of Fock and Goncharov.

3)  (complex  and  differential  geometry): explicit formulas for the
hyperkahler metric on the Hitchin moduli space (by Gaiotto, Moore and
Neitzke).

4)   (analysis)   Stokes   phenomenon  for  the  the  Bohr-Sommerfeld
asymptotics  of spectra and eigenfunctions of polynomial differential
operators depending on small parameters (works of Ecalle and Voros in
80-ies).

Also  the  wall  crossing  has  "physcial"  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 of
the  dynamics  of the Teichmuller geodesic flow and interval exchange
maps (i.e. separatrix intervals and closed geodesics) are exactly the
same as the BPS states in physics, stable objects in algebra, cluster
coordinates in combinatorics.

Our  goal in this series of lectures is to introduce the community of
specialists  in  flat surfaces and Teichmuller flow (and everybody in
fact!)  to  the new exiting developments giving new viewpoints on the
familiar objects, and proposing new venues for generalizations.