Coming in Spring 2025!
Faculty mentor: Jianlong Liu
“Most of the interesting structures come from inverse limits.” The first rigorous example of an inverse limit you may have encountered is the Cantor set, constructed from iteratively removing the middle thirds of each interval. If you describe its construction more abstractly to non-mathematicians, they may recognize it as resulting in fractals, but inverse limits are much more ubiquitous. Reading off the date and time is a (finite) inverse limit. The car odometer is a (finite) inverse limit. The calculus/analysis limit is an inverse limit. In topology, they result in (generalized) Cantor sets. In groups, they result in p-adic groups (more generally profinite groups). Aperiodic tilings are inverse limits (of branched manifolds).
Explicitly computing topological invariants of a generic aperiodic tiling is difficult. The associated inverse limit generally involves infinitely many distinct branched manifolds, thus one needs to compute invariants for infinitely many such objects. The process becomesmuch easier if the tiling arises from a substitution rule, where a tile is expanded then sub divided into a finite collection of tiles. A famous theorem showed that any such tiling is homeomorphic to an inverse limit that uses a single branched manifold, but its construction requires collaring once.
Substitution tilings are a very restrictive class of aperiodic tilings (and their computations have been done by yours truly, among other people). We will explore a generalization, pseudosubstitution, or substitution-with-amalgamation. A similar theorem states that any pseudosubstitution tiling is also homeomorphic to an inverse limit using a single branched manifold, but the branched manifold possibly requires a higher number of collars. Our goal is to precisely determine the number of collars needed, and to write a script in Sage so that, given any pseudosubstitution tiling, it computes the resulting branched manifold and the corresponding induced self-map.