The Globally Valued Fields (GVF) project is a joint effort with E. Hrushovski to understand (standard and) non-standard global fields - namely fields in which a certain abstraction of the product formula holds. One possible motivation is to give a model-theoretic framework
for various asymptotic distribution results in global fields.
Formally, a GVF is a field together with a "valuation" in the additive group of an L^1 space, such that the integral of v(a) vanishes for every non-zero a .
Natural model-theoretic questions regarding GVFs (existence of a model companion, stability, ...), translate to the development of some form of geometry over such fields. There are in fact two flavours for this - a "global" geometry, based on intersection theory, and a "local" geometry, based on the study of the fields at each (standard) valuation
separately, combined with a single local-global principle (or, at our level, axiom) called "fullness".