  Although contemporary model theory has been called "algebraic geometry minus
fields", the formal methods of the two fields are radically different. This
dissertation aims to shrink that gap by presenting a theory of logical schemes,
geometric entities which relate to first-order logical theories in much the
same way that algebraic schemes relate to commutative rings.
  The construction relies on a Grothendieck-style representation theorem which
associates every coherent or classical first-order theory with an affine
scheme: a topological groupoid (the spectrum of the theory) together with a
sheaf of (local) syntactic categories. The groupoid is constructed from the
semantics of the theory (models and isomorphisms) and topologized using a
Stone-type construction. The sheaf of categories can be regarded as a logical
theory varying over the spectrum, and its global sections recover the theory up
to semantic equivalence. These affine pieces can be glued together to give more
general logical schemes and these are studied using methods from algebraic
geometry. The final chapter also presents some connections between schemes and
other areas of logic such as model theory, type theory and topos theory.
