A new mathematical dictionary links the way paths wander randomly in a plane to the shape of three-dimensional space.
April 29, 2026
Original Paper
CaTherine wheels
arXiv · 2604.24619
The Takeaway
CaTherine wheels are specific mathematical maps that bridge the gap between three wildly different fields of study. They create a formal connection between conformal dynamics, probability theory, and the geometry of hyperbolic 3-manifolds. Researchers discovered that these areas share a deep structural unity that was previously hidden. This means a problem that is impossible to solve in 3D geometry might have a simple solution in probability theory. This breakthrough provides a new set of tools for physicists and mathematicians to translate complex problems between different languages of math.
From the abstract
A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane ${\rm SLE}_\kappa$ for $\kappa \ge 8$, LQG metric trees) and elsewhere. We develop their t