Elena Mantovan of Caltech presents this Joint IAS/Princeton University Number Theory seminar talk on the Schottky problem, the classical question of characterizing which abelian varieties arise as Jacobians of curves. She focuses on positive characteristic settings, where the question becomes whether certain discrete invariants known to occur for general abelian varieties also occur for Jacobians specifically. Her chosen invariant is the Newton polygon, and she lays out an approach to the problem built on studying the geometry of Shimura varieties as they intersect the Torelli locus, the locus of Jacobians inside the moduli space of abelian varieties. The talk reports joint work with Wen-Ching Li and Rachel Pries (referred to as Singh in the source), delivered at the Institute for Advanced Study's Simonyi Hall. It assumes graduate-level familiarity with algebraic geometry and arithmetic geometry, and runs through the technical machinery connecting curve theory to the structure of moduli spaces.