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First steps of local contact algebra V.I.Arnold

Summary: First steps of local contact algebra
Dedicated to H.S.M. Coxeter
1 Introduction
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 [1]. It classi es simple singularities of caustics and wave fronts,
de ned by the projections of Lagrange and Legendre subvarieties of symplectic and contact
manifolds, in terms of the Coxeter euclidean re ections groups, extending to the case of
singular varieties my previous A D E { classi cation [2] (corresponding to smooth
The present work is an attempt to start the classi cation of singular simple curves in
contact manifolds.
The idea that every reasonable mathematical theory should have a symplectic and con-
tact versions is also based on the success of Coxeter's extension of linear algebra (considered
as the theory of the root systems A) to other mirrors con gurations. The application of
this idea to calculus has led to the foundation of symplectic and contact topologies (see
In the present article the same idea is applied to a modest local problem. It is astonishing
that this problem { the classi cation of simple curves in a contact space { is rather diĘcult


Source: Arnold, Vladimir Igorevich - Steklov Mathematical Institute


Collections: Mathematics