A special case of Hironaka's QUESTION F, named F', asks about the strong factorization of birational maps between reduced nonsingular algebraic schemes, which is still open. Suppose that $\varphi : X\dashrightarrow Y$ is such a map, and let $U\subset X$ be the open subset where $\varphi$ is an isomorphism. This problem asks if there exists a diagram $$\xymatrix{ & Z \ar[dl]_{\varphi_{1}}\ar[dr]^{\varphi_{2}}\X \ar[rr]^{\varphi} & & Y}$$ where the morphisms $\varphi_{1}$ and $\varphi_{2}$ are sequences of blow-ups of non-singular centers disjoint from $U$. In this talk, we will discuss how strong factorization can be simplified by providing a complete answer to the problem of toroidalization of morphisms, while we introduce the strong Oda conjecture.

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