Motivated by robust mathematical finance, and also taking inspiration from the field of Optimal Transport, we ask:
What is the martingale, with prescribed initial and terminal marginal distributions, which is closest to Brownian motion?
Under suitable assumptions the answer to this question, in any dimension, is provided by the so-called Bass martingales. To imagine what these are, one pictures an underlying Brownian motion which is stretched in space in an order-preserving and martingale-preserving way. In this talk we discuss the properties of Bass martingales, their existence, and the duality theory required to study them. Talk based on joint work with Beiglböck, Schachermayer and Tschiderer.