We illustrate a new method to induce stability conditions on
semiorthogonal decompositions and apply it to the Kuznetsov component
of the derived category of cubic fourfolds. We use this to generalize
results of Addington-Thomas about cubic fourfolds and to study the
rich hyperkähler geometry associated to these hypersurfaces. This is
the content of joint works with Arend Bayer, Howard Nuer, Martì
Lahoz, Emanuele Macrì and Alex Perry.