Applications of flag contact singularities (Q2754584)

Discriminants of isolated hypersurface singularities arise in various problems of geometry, physics and differential equations. Mostly they occur as the projections (wavefronts) of Legendre submanifolds of projectivized cotangent bundles. Certain physical applications require the study of singularities on manifolds with corners and in general on the spaces of orbits of certain group actions. NEWLINENEWLINENEWLINEThe author shows that these constructions have natural counterparts in terms of singularities of the contact between a submanifold germ and a germ of a flag (a sequence of nested submanifolds) embedded in an ambient space. The author describes the generic singularities of the relative minimum arising in problems of parametric optimization, using flag contact classes. Among them those are distinguished which represent the generic singularities of the envelopes of moving wave-fronts in control systems. NEWLINENEWLINENEWLINEMoreover, the stability of the wave front of an exponential mapping in the simplest case of subriemannian geometry is proved by using flag contact group techniques.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].