Noncommutative geometry of higher-dimensional phase spaces for solid state topologies

Prof. Dr. Duco van Straten
Prof. Dr. Yuriy Mokrousov

The principles and tools of noncommutative geometry (NCG), which is a modern branch of algebraic topology,
have been used with great success in such fields as nuclear physics or string theory. However, the prospects of NCG in the realm of solid state systems are still relatively unexplored. With our project, starting with
establishing the mathematical foundations of NCG built on the Moyal product between quantum operators, we
want to fill this gap, and develop necessary mathematical tools which could be later used in understanding the
geometrical properties of non-uniform non-periodic systems ranging from liquid crystals to Moiré lattices and
skyrmion fabrics, so as to ultimately formulate proper topological invariants and effective field theories which
govern their behavior and dynamics in multi-dimensional phases spaces of various parameters.