Noncommutative Geometry and Topology in Prague

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 spaces like fractals.  Michel Lapidus, Frédéric Latrémolière, and I identified conditions under which differential structures defined on fractal curves can be realized as a metric limit of differential structures on their approximating finite graphs.  Currently, I am using some of the same tools from that project to understand noncommutative discrete structures with Manuel Reyes.  Progress in noncommutative geometry has produced a rich dictionary of quantum analogues of classical spaces.  The addition of noncommutative discrete structure to this dictionary would enlarge its potential to yield insights about both noncommutative sets and classically pathological sets like fractals.  Time permitting, other works in progress, such as on classification of C*-algebras on fractals or the further development of noncommutative analogues of geometric notions on fractals, may be discussed.