Complex cellular structures

In tame geometry, a cell (or cylinder) is defined as follows. A one dimensional cell is an interval; a two-dimensional cell is the domain bounded between the graphs of two functions on a one-dimensional cell; and so on. Cellular decomposition (covering or subdividing a set into cells) and preparation theorems (decomposing the domain of a function into cells where the function has a simple form) are two of the key technical tools in semialgebraic, subanalytic and o-minimal geometry.
Cells are normally seen as intrinsically real objects, defined in terms of the order relation on $\mathbb R$. We (joint with Novikov) introduce the notion of complex cells, a complexification of real cells where real intervals are replaced by complex annuli. Complex cells are naturally endowed with a notion of analytic extension to a neighborhood, called $\delta$-extension. It turns out that complex cells carry a rich hyperbolic-geometric structure, and the geometry of a complex cell embedded in its $\delta$-extension offers powerful new tools from geometric function theory that are inaccessible in the real setting. Using these tools we show that the real cells of the subanalytic cellular decomposition and preparation theorems can be analytically continued to complex cells.
Complex cells are closely related to uniformization and resolution of singularities constructions in local complex analytic geometry. In particular we will see that using complex cells, these constructions can be carried out uniformly over families (which is impossible in the classical setting). If time permits I will also discuss how this relates to the Yomdin–Gromov theorem on $C^k$-smooth resolutions and some modern variations.