An n-dimensional rep-tile X is a compact codimension-zero submanifold of R^n which has the property that it be decomposed as the union of mutually isometric manifolds similar to X which have non-overlapping interiors. A trivial example is the unit n-cube, which can be written as the union of 2^n cubes the side length one half. More surprisingly, all knot exteriors in S^3 are homeomorphic to rep-tiles, by a 2021 result of Blair, Marley and Richards. First I will show how to construct new rep-tiles with interesting algebraic topology. Then I will give an isotopy classification of rep-tiles in all dimensions. Time permitting, I will either outline the main proof or explain the key idea, which is a technique we called an equivariant ball swap. This is based on joint work with Ryan Blair, Patricia Cahn and Hannah Schwartz (arxiv 2412.19986).

https://indico.math.cnrs.fr/event/13984/