Optical Tomography


Ehud Hrusovski, Hebrew University of Jerusalem

Abstract:

Model theory looks at p-adic, rigid or power series geometry through logical theories of valued fields. I will discuss only the simplest, the theory of algebraically closed valued fields. Model-theoretic analysis begins with the identification of the definable sets as semi-algebraic sets, defined by polynomial equalities and inequalities (Robinson.) The Ax-Kochen theorem on p-adic hypersurfaces of large degree, the extensive integration theories beginning with Denef and Kontsevich, as well as a new non-archimedean Euler characteristic, can be viewed as following from a description of the definable sets up to definable isomorphisms. Further model-theoretic developments are inspired by stability theory, pointing to the importance of generically stable types. This was first of use in work classifying imaginaries, or parameter spaces for semi-algebraic families. Unlike the real or complex cases, semi-algebraic families cannot be parameterized by semi-algebraic sets, but require the intervention of the a definable version of the affine Grassmanian. Glueing is notoriously difficult to define for disconnected spaces, with remedies offered by Krasner, Tate, Raynaud and Berkovich in the rigid analytic setting. Recent work with Loeser shows that the space of generically stable types forms a pro-definable set that can be viewed as a definable version of Berkovich spaces, and leads naturally to the introduction of a definable topology, with strong connections to o-minimality. Consequences for classical Berkovich spaces will be discussed.