In 1912, Henri Poincaré asked the following simple question: “In how many different ways can a simple loop in the plane, called a meander, cross a line a specified number of times?”
Despite many efforts, this question remains open after over a century.
In this talk, I will focus on meandric systems, which are coupled collections of meanders.
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 work with Ewain Gwynne and Minjae Park.
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