Definable Quotient Spaces for Unlikely Intersection Problems

Applications of o-minimality to unlikely intersection problems usually begin with the observation that the relevant analytic covering maps are definable.  However, this observation is almost never literally true in that the maps are definable on a suitable fundamental set, not on the full domain of definition, and this non-definabily is often leveraged in functional transcendence proofs.   Various formalisms to describe this situation have been introduced by, for example, Zilber, Bakker-Klingler-Tsimerman, and Eterović-Scanlon.  I will discuss a common refinement of these formalism and how it can be used to deduce automatic uniformity statements.