  The carbon monolayer band structure calculated in the approximation of weakly
interacting {\pi} electrons corresponds to massless electron excitations known
as Dirac fermions not previously observed in any other material. However, if
strong Coulomb and exchange interactions between {\pi} electrons are taken into
account, another picture of the {\pi} electron state emerges. These
interactions result in {\pi} electron localization and electron crystal
formation. The atomically flat layer can be regarded as a simplest
quasi-two-dimensional mesoscopic quantum system consisting of a carbon ion
plane and two {\pi} electron crystals on opposite sides of the plane. Such a
system must have dielectric and pronounced diamagnetic properties and a high
sensitivity to external factors distorting its electron crystals. The
instability manifests itself in a tendency of the monolayer to be transformed
into a more stable carbon modification with a rolled-up or wrapped-up carbon
skeleton which is observed as a monolayer corrugation (or ripples). The
corrugated monolayer is characterized by the presence of excited {\pi}
electrons which are responsible for its physical and chemical properties. The
approach can prove useful for investigation of carbon nanotubes, fullerenes,
graphite, and also topological insulators and complicated quantum systems with
a layered structure. Calculations of the quasi-two-dimensional quantum system
faces a many-body problem and are beyond the band-structure description, which
forces us to confine ourselves to purely qualitative analysis
