Developments in Noncommutative Fractal Geometry

As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry.  At the quantum scale, the wave function of a particle, but not its path in space, can be studied.  Riemannian methods often rely on smooth paths to encode the geometry of a space.  Noncommutative geometry generalizes analysis on manifolds by replacing this requirement with operator algebraic data.  These same "point-free" techniques can also be used to study the geometry of classically pathological spaces like fractals.  The 2016 Nobel Prize in Physics was awarded for work on Hofstadter’s butterfly, which is a fractal that describes for theoretical condensed matter physicists the allowed energy levels for electrons confined to a crystalline atomic lattice as a function of the magnetic field applied to the system.  By expanding the formalism of fractal geometry to include the mathematical language of quantum theory, developments in noncommutative fractal geometry give both mathematicians and physicists the tools to gain insights about quantum behaviors in solids and any new materials made possible by these phenomena.