Abstract: 

The moduli space of closed polygonal curves in \(\mathbb{R}^3\) with prescribed edge lengths, considered up to Euclidean isometries, is a rich and well-studied object. It has been investigated from a variety of perspectives by Deligne–Mostow, Thurston, Klyachko, and Kapovich–Millson. This space possesses a remarkable amount of structure: in particular, it carries a Kähler structure and is closely related to toric varieties.

In this talk, I will introduce the construction of this moduli space and explain, following joint work with Sasha Anan'in, how bending a polyhedral disk sweeps out Lagrangian subsets of the polygon space, and will use it to give a negative answer to a question of Kenyon about spanning domes. We will also discuss expected values of random variables on the space of polygons (like the length of the n-th diagonal), and related conjectures about its geometry.