Speciality:
01.01.04 (Geometry and topology)
Birth date:
23.09.1955
E-mail: ,
Keywords: singularities of convex hulls, caustics and wave fronts; global singularity theory; multidimensional generalizations of the $4$-vertex theorem
UDC: 515.16, 515.164.15, 514.75, 514.172 MSC: 58K, 57R45, 57R17, 53A04, 53A07, 53A20, 51L15, 32S20, 14E15
Subject:
Singularities of the convex hull of a smooth closed generic curve in $\mathbb{R}^3$ were classified with respect to diffeomorphisms of the ambient space. It was proved that singularities of convex hulls of curves in $\mathbb{R}^4$ and $k$-dimensional submanifolds in $\mathbb{R}^n$ for $k\geq 1, n\geq 5$ have functional moduli which can not be removed by small deformations of a submanifold. A three-dimensional generalization of the classical $4$-vertex theorem was obtained: “Any $C^3$-embedded closed curve in $\mathbb{R}^3$ which has nowhere vanishing curvature and lies on the boundary of its convex hull has at least $4$ geometrically different zero-torsion points”. New invariants of Arnold admissible homotopies of a closed curve in $\mathbb{R}P^3$ were found. We have constructed a correspondence between (Lagrangian) singularities of the envelope of the family of normals to a submanifold in $\mathbb{R}^n$ and (Legendrian) singularities of the set of tangent hyperplanes to its stereographic image in $\mathbb{R}^{n+1}$ (a symplectic generalization of Kneser lemma on flattening points of a spherical curve in $\mathbb{R}^3$). The adjacency indices of singularities of a generic wave front in a space of the dimension at most $6$ were calculated. It was proved that every connected component of the manifold of multisingularities of any given type for a germ of the image of a Lagrangian map with monosingularity of type $D_\mu$ is either contractible or homotopy equivalent to a circle; the number of components of each kind was calculated. We have developed a construction for resolving of stable corank $1$ multisingularities in the image of a smooth map of a closed manifold to a space of the same or higher dimension which generalizes Kleimans iteration principle: instead of cycles of multiple points, we consider cycles of arbitrary multisingularities. As a corollary, we obtained complete systems of universal linear relations with real coefficients between the Euler characteristics of the manifolds of multisingularities in the image of a stable map (arbitrary smooth, Lagrangian or Legendrian) having only corank $1$ singularities. We found many-dimensional generalizations of Bose formula which claims that the number of supporting curvature circles to a closed convex generic curve in $\mathbb{R}^2$ exceeds the number of supporting circles tangent to the curve at three points onto $4$. Namely, the numbers of supporting hyperspheres of various types having $(2n+1)$-th order of tangency with a closed convex generic curve in $\mathbb{R}^{2n}$ are related by the universal linear relation whose coefficients are defined by the Catalan numbers $c_k,k\leq n$.
Main publications:
V. D. Sedykh, “The four-vertex theorem of a convex space curve”, Funct. Anal. Appl., 26:1 (1992), 28–32.
V. D. Sedykh, “Relations between the Euler numbers of manifolds of corank 1 singularities of a generic front”, Dokl. Math., 65:2 (2002), 276–279.
V. D. Sedykh, “Resolution of corank 1 singularities of a generic front”, Funct. Anal. Appl., 37:2 (2003), 123–133.
V. D. Sedykh, “Resolution of corank 1 singularities in the image
of a stable smooth map to a space of higher dimension”, Izv. Math., 71:2 (2007), 391–437.
V. D. Sedykh,
“On the topology of wave fronts in spaces of low dimension”, Izv. Math., 76:2 (2012), 375–418.
