Abstract: In his work on Dulac's problem, Ilyashenko uses a quasianalytic class of functions that is a group under composition, but not closed under addition or multiplication. When trying to extend Ilyashenko's ideas to understand certain cases of Hilbert's 16th problem, it seems desirable to be able to define corresponding quasianalytic classes in several variables that are also closed under various algebraic operations, such as addition, multiplication, blow-ups, etc. One possible way to achieve this requires us to first extend the one-variable class into a quasianalytic algebra whose functions have unique asymptotic expansions based on monomials definable in $R_{an,exp}$. I will explain some of the difficulties that arise in constructing such an algebra and how far (or close) we are to obtaining it. (This is joint work with Tobias Kaiser.)