CAT(0) geometry, robots, and society
This addresses path planning for discrete robots, though it appears incremental in applying known mathematical frameworks.
The paper tackles the problem of efficiently moving robots by leveraging CAT(0) cubical complexes in configuration spaces, resulting in optimal navigation methods using combinatorial and geometric group theory.
How do we move a robot efficiently from one position to another? To answer this question, we need to understand its configuration space, a 'map' where we can find every possible position of the robot. Unfortunately, these maps are very large, they live in high dimensions, and they are very difficult to visualize. Fortunately, for some discrete robots they are CAT(0) cubical complexes, a family of spaces with favorable properties. In this case, using ideas from combinatorics and geometric group theory, we can construct a 'remote control' to navigate these complicated maps, and move the robots optimally. Along the way, we face larger ethical questions that we cannot ignore.