Complex Cellular Structures
Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of cells homeomorphic to cubes. The cell decomposition can be built effectively, and is one of the most powerful tools in studying properties of real semialgebraic sets.
Another most useful tool, the Gromov-Yomdin Lemma, builds a uniform in parameters cover of real algebraic sets by images of $C^r$-smooth mappings of cubes.
There is a non-trivial obstruction to complexification of this result, related to inner hyperbolic metric properties of complex holomorphic sets.
We proved a new simple lemma about functions in one complex variables. This allowed us to construct a proper holomorphic version of the above results,
for complex (sub)analytic and semialgebraic sets, combining best properties of both. As a corollary, we prove an old Yomdin's conjecture on $\epsilon$-tail entropy for analytic maps.
This is a joint work with Gal Binyamini.