Direct and Inverse Problems in Special Geometry

The inverse problem of special geometry (Seiberg-Witten geometry of 4d N=2 SCFT) asks for a recursive construction of all such geometries in rank $r$ by assembling together known lower-rank ``strata''. This leads to a program to understand/construct/classify all special geometries which looks surprising effective. After reviewing some advanced topics in special geometry, in this long note we define the inverse problem and introduce the basic tools of the trade. The program is essentially completed in rank 2, and we pave the way to proceed to higher ranks. A central role is played by various notions of geometric rigidity: in addition to the obvious one (triviality of the conformal manifold), Falting-Saito-Peters rigidity and Deligne-Simpson rigidity also enter in the story.