Derived categories and symmetry: spherical twists

I will review spherical twist symmetries of derived categories of coherent sheaves, and outline the mirror symmetry heuristics relating them to symplectic Dehn twists.
This is the first in a short series of talks. It will lead on to a discussion of recent developments, including my work on Grassmannian twists with Ed Segal, and recent advances by other authors in understanding derived categories of GIT quotients. The use of “grade restriction windows” is the central idea in both these areas. My aim is to give a survey, so I will keep technical prerequisites to a minimum.