Random Geometry via Volume Minimization

A single random path connecting a set of points—the shortest possible route—is being turned into a whole new way to generate random surfaces. The project generalises the classic Travelling Salesman Problem, which asks for the shortest tour visiting a collection of cities, to higher dimensions. Instead of a line, the solution becomes a surface, and by randomising the points, the resulting surface becomes a random geometric object. This matters because mathematicians have powerful tools for studying random surfaces like the Brownian map, but those models are often abstract and hard to connect to discrete, combinatorial problems. The new model offers a missing bridge: a way to build continuous random geometry from simple, combinatorial rules. It could unify two major strands of research—discrete random triangulations and Liouville Quantum Gravity—under a single framework. The project is fundamental mathematics. It has no immediate practical application. But similar work on random geometry has unexpectedly influenced network theory, image compression, and the design of communication algorithms. If this model succeeds, it could give physicists and computer scientists a new combinatorial handle on the geometry of random surfaces, with potential long-term impacts on how we model disordered materials or optimise spatial networks.