The ACO Seminar (2023–2024)

A polyform is a planar shape which is constructed by gluing identical copies of a fixed polygon together along their edges. These objects are classically known as "lattice animals." Extremal numerical properties of lattice animals have been studied since the 70s, with growing interest in recent years in topological properties and generalizations to other metric spaces. In this talk, we will focus primarily on the question: what is the maximum number of holes that can be bounded by a set of n copies of a fixed polygon or polyhedron? We will discuss answers for this question and related work on corresponding extremal properties in a variety of settings, including the Euclidean plane, $\mathbb{R}^3$, and for (planar) hyperbolic animals. This is joint work with Fedor Manin, Érika Roldán, and Rosemberg Toalá-Enríquez.