Many algebro-geometric statements involve only finitely many data, and so counterexamples to such statements over infinite fields often come from the geometry of finitely generated algebras over Z, where residue fields are finite. We will explore a general motto: "If a complex-algebraic statement fails, there should be a 'finitary algebraic' obstruction which 'witnesses' this failure." This idea can be used to easily prove otherwise-hard-to-prove theorems by taking advantage of such basic concepts as cardinality in the context of finite fields. Speaker(s): Carsten Sprunger (UM)