First steps of local contact algebra

V.I.Arnold

The belief that all simple (having no continuous moduli) objects in the
nature are controlled by the Coxeter groups is a kind of religion. The
corresponding theorem in singularity theory is due to A.B.Givental. It
classifies simple singularities of caustics and wave fronts, defined by the
projections of Lagrange and Legendre subvarieties of symplectic and contact
manifolds, in terms of the Coxeter Euclidean reflections groups, extending
to the case of singular varieties my previous $A-D-E$ -- classification
(corresponding to smooth submanifolds).

The present work is an attempt to start the classification of singular
simple curves in contact manifolds.