Abstract: Bumpless pipe dreams were introduced by Lam, Lee, and Shimozono in the context of back stable Schubert calculus. Like ordinary pipe dreams, they compute Schubert and double Schubert polynomials. In this talk, I will give a bijective proof of Monk's rule for Schubert polynomials, and show that the proof extends easily to the proof of Monk's rule for double Schubert polynomials. As an application, I will explain how to biject bumpless pipe dreams and ordinary pipe dreams using the transition and co-transition formulas, which are specializations of Monk's rule. If time permits I will also briefly discuss and share some work in progress on how bumpless pipe dreams can be used to compute products of certain Schubert polynomials in generalizations of the Littlewood-Richardson rule for Schur polynomials. Please email Colleen at cer2 (at) illinois (dot) edu for the Zoom ID and password.