In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century.

In this talk, I will present the conjectural scaling limit of uniform meanders and some recent results on a related model called meandric systems.

A meandric system is a coupled collection of meanders. Also in this case, I will present (1) a conjecture which describes the large-scale geometry of a uniform meandric system and (2) several rigorous results which are consistent with this conjecture.

Based on joint works with Ewain Gwynne and Xin Sun, and Ewain Gwynne and Minjae Park.

Slides (PDF)