Abstract: It has been shown by Haskell, Hrushovski, and Macpherson that in an algebraically closed valued field, the residue field and the value group control the rest of the structure in the following sense: Suppose that L and M are valued fields containing a maximal field C, and suppose that the residue field of L (denoted k(L)) is algebraically independent from k(M) over C and likewise the value group of L (denoted $\Gamma(L)$) is independent from the $\Gamma(M)$ over C in the sense of vector spaces. Then tp(L/ M,k(L),$\Gamma(L)$ ) is implied by tp(L/ k(L),$\Gamma(L)$).
This behaviour is striking, because it is what typically occurs in a stable structure (where types over algebraically closed sets have unique nonforking extensions) but valued fields are far from stable, due to the order on the value group. Real closed valued fields are even farther from stable since the main sort is ordered, and one might expect the analogous theorem about real closed valued fields to be that tp(L/ M,k(L),$\Gamma(L)$) is implied by tp(L/ k(L)$\Gamma(L)$) together with the order type of $L/M$. In fact we show that the order type is unnecessary, that just as in the algebraically closed case one has that tp(L/ M,k(L),$\Gamma(L)$) is implied by tp(L/ k(L)$\Gamma(L)$). This is joint work with Haskell and Marikova.