Finite dimension unravels the structural features at the glass transition

Abstract

Physics Nobel Prize winner P.W. Anderson famously wrote in 1995: “The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of the glass and the glass transition”. Despite much effort in the intervening years, the problem is still unsolved. We contribute a novel mathematical approach to this problem. The main new ingredient is finite dimension, a recently introduced “fractal” dimension defined only for finite sets. Our methods sharply distinguish the glass transition temperature and give hints as to the structural changes that occur in the transition.