A Finiteness Criterion for Real Linear Subvarieties in the Moduli Space of Abelian Differentials

Real linear subvarieties in moduli spaces of holomorphic and meromorphic Abelian differentials arise in the study of flat surfaces and polygonal billiards. In this talk, I will discuss joint work in progress with Vincent Delecroix and Bradley Zykoski on a finiteness criterion for such subvarieties. This criterion gives a semi-decision procedure for testing whether a given germ defines a real linear subvariety. The proof uses o-minimal geometry on moduli spaces to relate this finiteness property to algebraicity.