Torification and factorization of birational maps

This talk will be based on the paper of Dan Abramovich, Kalle Karu, Kenji Matsuki and Jaroslaw Wlodarczyk. We will discuss the proof of the weak factorization conjecture: every birational map between complete nonsingular varieties over an algebraically closed field of characteristic zero is a composite of blowings up and blowings down with smooth centers. Also we will see the difference between this proof and another proof, given by the fourth author.