In this talk, we'll present our results about invariant Gibbs measures for the periodic cubic nonlinear wave equation (NLW) in 3D. The interest in this result stems from connections to several areas of mathematical research. At its core, the result concerns a refined understanding of how randomness gets transported by the flow of a nonlinear equation, which involves probability theory and partial differential equations. This is joint work with Bjoern Bringmann (Princeton), Yu Deng (UChicago) and Andrea Nahmod (UMass Amherst).