Kellendonk has worked on the non-commutative topology of tilings, a
topic initiated by Alain Connes and Jean Bellissard which is important
for the gap-labelling of tight binding Hamilton-operators on quasicrystals.
He studied the K-theory of substitution tilings and, in joint work with
Alan Forrest and John Hunton, quasiperiodic tilings. He also gave a symbolic
description of tilings which is based on inverse semigroups. In joint work
with Tim Hoffmann, Nadja Kutz and Nicolai Reshetikhin, Johannes Kellendonk
investigated discrete nonlinear Toda lattices -- a discretation of Toda
lattices -- using the Poisson-geometry of (complex) Poisson-Lie-groups.
These lattices form integrable dynamical systems whose dynamics is given
by a factorization relation. Johannes Kellendonk is currently interested
in the integer quantum Hall-effect and here more precisely in the relation
between the formulation of the Hall-conductivity in the edge current picture
and the bulk-picture which expresses the conductivity as a Chern number.
Together with Thomas Richter and Hermann Schulz-Baldes he approaches the
subject using C*-algebras in order to include the disorder of the Hall-bar.