Surprising connections have recently emerged between two very active and previously independent research domains: random permutations and random geometry. This talk will uncover these connections, showing how random geometric objects can be directly used to reconstruct universal limits for random permutations.

We will illustrate this new general theory through concrete examples of Baxter permutations and monotone meanders, helping the audience build intuition. In the last part of the talk, we will explain how similar ideas led us to a new conjecture for the scaling limit of uniform random meanders and share progress on this long-standing open problem.