Title: Toroidalization of locally toroidal morphisms

Abstract: The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of locally toroidal morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. Aside from laying the foundation and discussing the motivation behind the problem, this talk will sketch a proof of an affirmative answer to the question when the morphism is between any arbitrary variety and a surface.