Optimal transport provides a geometric way to compare probability measures and has become a common language in analysis, geometry, probability, and partial differential equations. In the classical two-marginal problem, the structure of optimal plans is closely related to convexity, cyclical monotonicity, and the geometry of the underlying space.

In this talk, I will start with a brief introduction to the two-marginal optimal transport problem and explain why many of its structural features become substantially richer in the multi-marginal setting. Then I will  discuss some joint work with Dengyu Liu and Zhuonan Zhu on multi-marginal optimal transport over general metric measure spaces, where one studies barycentric and curvature-type phenomena beyond the smooth or Euclidean framework.

The main part of the talk will focus on recent joint work with Zhuonan Zhu on a new structural viewpoint for multi-marginal optimal transport. The guiding principle is that, unlike the two-marginal case where finite cyclic monotonicity often captures global optimality, multi-marginal problems may exhibit genuinely higher-order obstructions. We formulate this phenomenon through symmetry, mode decomposition, and quantitative gap estimates. This perspective connects multi-marginal optimal transport with ergodic transport and some other problems in dynamical systems.