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Sedykh, Vyacheslav Dmitrievich

 Statistics Math-Net.Ru Total publications: 27 Scientific articles: 26

 Number of views: This page: 1768 Abstract pages: 7132 Full texts: 2171 References: 882
Doctor of physico-mathematical sciences (2006)
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 $c_k,k\leq n$.

Biography

Graduated (with honours) from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1978 (Department of differential equations). Ph.D. thesis was defended in MSU in 1982. Thesis title: Singularities of convex hulls. Supervisor: Professor V. I. Arnold.

D.Sci. thesis was defended in  V. A. Steklov Mathematical Institute of Russian Academy of Sciences in 2006. Thesis title: The global theory of real corank 1 singularities and its applications to the contact geometry of space curves. A list of my works contains more than 50 titles.

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.

http://www.mathnet.ru/eng/person14933
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/199578
http://elibrary.ru/author_items.asp?spin=7645-6757

Publications in Math-Net.Ru
 2018 1. V. D. Sedykh, “Topology of singularities of a stable real caustic germ of type $E_6$”, Izv. RAN. Ser. Mat., 82:3 (2018),  154–169    ; Izv. Math., 82:3 (2018), 596–611 2015 2. V. D. Sedykh, “On the topology of stable Lagrangian maps with singularities of types $A$ and $D$”, Izv. RAN. Ser. Mat., 79:3 (2015),  159–202        ; Izv. Math., 79:3 (2015), 581–622 2014 3. V. D. Sedykh, “The Topology of Adjacencies of Type $A$ and $D$ Lagrangian Singularities”, Funktsional. Anal. i Prilozhen., 48:4 (2014),  83–88        ; Funct. Anal. Appl., 48:4 (2014), 304–308 2012 4. V. D. Sedykh, “On Euler Characteristics of Manifolds of Singularities of Wave Fronts”, Funktsional. Anal. i Prilozhen., 46:1 (2012),  92–96        ; Funct. Anal. Appl., 46:1 (2012), 77–80 5. V. D. Sedykh, “On the topology of wave fronts in spaces of low dimension”, Izv. RAN. Ser. Mat., 76:2 (2012),  171–214        ; Izv. Math., 76:2 (2012), 375–418 2011 6. V. D. Sedykh, “On the topology of cooriented wave fronts in spaces of small dimensions”, Mosc. Math. J., 11:3 (2011),  583–598 2010 7. V. D. Sedykh, “Adjacency Indices for Singularities of Wave Fronts in Low Dimensional Spaces”, Funktsional. Anal. i Prilozhen., 44:3 (2010),  88–91      ; Funct. Anal. Appl., 44:3 (2010), 234–236 2007 8. V. D. Sedykh, “Resolution of corank 1 singularities in the image of a stable smooth map to a space of higher dimension”, Izv. RAN. Ser. Mat., 71:2 (2007),  173–222        ; Izv. Math., 71:2 (2007), 391–437 9. V. D. Sedykh, “On the Coexistence of Corank 1 Multisingularities of a Stable Smooth Mapping of Equidimensional Manifolds”, Tr. Mat. Inst. Steklova, 258 (2007),  201–226        ; Proc. Steklov Inst. Math., 258 (2007), 194–217 2005 10. V. D. Sedykh, “Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve”, Mat. Zametki, 78:3 (2005),  413–427        ; Math. Notes, 78:3 (2005), 378–390 2004 11. V. D. Sedykh, “A Complete System of Linear Relations between the Euler Characteristics of Manifolds of Corank $1$ Singularities of a Generic Front”, Funktsional. Anal. i Prilozhen., 38:4 (2004),  73–78      ; Funct. Anal. Appl., 38:4 (2004), 298–301 12. V. D. Sedykh, “On the topology of stable corank 1 singularities on the boundary of a connected component of the complement to a front”, Mat. Sb., 195:8 (2004),  91–130        ; Sb. Math., 195:8 (2004), 1165–1203 2003 13. V. D. Sedykh, “Resolution of Corank $1$ Singularities of a Generic Front”, Funktsional. Anal. i Prilozhen., 37:2 (2003),  52–64        ; Funct. Anal. Appl., 37:2 (2003), 123–133 14. V. D. Sedykh, “On the topology of singularities of Maxwell sets”, Mosc. Math. J., 3:3 (2003),  1097–1112 2001 15. V. D. Sedykh, “Some Invariants of Admissible Homotopies of Space Curves”, Funktsional. Anal. i Prilozhen., 35:4 (2001),  54–66      ; Funct. Anal. Appl., 35:4 (2001), 284–293 1996 16. V. D. Sedykh, “Theorem on Four Support Vertices of a Polygonal Line”, Funktsional. Anal. i Prilozhen., 30:3 (1996),  88–90      ; Funct. Anal. Appl., 30:3 (1996), 216–218 1995 17. V. D. Sedykh, “Invariants of Nonflat Manifolds”, Funktsional. Anal. i Prilozhen., 29:3 (1995),  41–50      ; Funct. Anal. Appl., 29:3 (1995), 180–187 18. V. D. Sedykh, “Strict convexity of a generic convex manifold”, Trudy Mat. Inst. Steklov., 209 (1995),  200–219      ; Proc. Steklov Inst. Math., 209 (1995), 174–190 1994 19. V. D. Sedykh, “Connection of Lagrangian singularities with Legendrian singularities under stereographic projection”, Mat. Sb., 185:12 (1994),  123–130      ; Russian Acad. Sci. Sb. Math., 83:2 (1995), 535–540 1993 20. V. D. Sedykh, “Invariants of Strictly Convex Manifolds”, Funktsional. Anal. i Prilozhen., 27:3 (1993),  67–75      ; Funct. Anal. Appl., 27:3 (1993), 205–210 1992 21. V. D. Sedykh, “A theorem on four vertices of a convex space curve”, Funktsional. Anal. i Prilozhen., 26:1 (1992),  35–41      ; Funct. Anal. Appl., 26:1 (1992), 28–32 1989 22. V. D. Sedykh, “An infinitely smooth compact convex hypersurface with a shadow whose boundary is not twice-differentiable”, Funktsional. Anal. i Prilozhen., 23:3 (1989),  86–87      ; Funct. Anal. Appl., 23:3 (1989), 246–248 1988 23. V. D. Sedykh, “Stabilization of singularities of convex hulls”, Mat. Sb. (N.S.), 135(177):4 (1988),  514–519      ; Math. USSR-Sb., 63:2 (1989), 499–505 1982 24. V. D. Sedykh, “Functional moduli of singularities of convex hulls of manifolds of codimensions 1 and 2”, Mat. Sb. (N.S.), 119(161):2(10) (1982),  233–247      ; Math. USSR-Sb., 47:1 (1984), 223–236 1981 25. V. D. Sedykh, “Moduli of singularities of convex hulls”, Uspekhi Mat. Nauk, 36:5(221) (1981),  191–192      ; Russian Math. Surveys, 36:5 (1981), 175–176 1977 26. V. D. Sedykh, “Singularities of the convex hull of a curve in $\mathbb{R}^3$”, Funktsional. Anal. i Prilozhen., 11:1 (1977),  81–82      ; Funct. Anal. Appl., 11:1 (1977), 72–73 2012 27. A. A. Agrachev, D. V. Anosov, I. A. Bogaevskii, A. S. Bortakovskii, A. B. Budak, V. A. Vassiliev, V. V. Goryunov, S. M. Gusein-Zade, A. A. Davydov, V. R. Zarodov, V. D. Sedykh, D. V. Treshchev, V. N. Chubarikov, “Vladimir Mikhailovich Zakalyukin (obituary)”, Uspekhi Mat. Nauk, 67:2(404) (2012),  187–190        ; Russian Math. Surveys, 67:2 (2012), 375–379

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