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Sedykh, Vyacheslav Dmitrievich

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Total publications: 27
Scientific articles: 26

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Sedykh, Vyacheslav Dmitrievich
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) in1978 (Department of differential equations). Ph.D. thesis was defended in MSU in1982. 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 in2006. Thesis title: The global theory of real corank 1 singularities and its applications to the contact geometry of space curves. Alist of my works contains more than 50titles.

   
Main publications:
  • V. D. Sedykh, The four-vertex theorem of a convex space curve, Funct. Anal. Appl., 26:1 (1992), 2832.
  • V. D. Sedykh, Relations between the Euler numbers of manifolds of corank 1 singularities of a generic front, Dokl. Math., 65:2 (2002), 276279.
  • V. D. Sedykh, Resolution of corank 1 singularities of a generic front, Funct. Anal. Appl., 37:2 (2003), 123133.
  • 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), 391437.
  • V. D. Sedykh, On the topology of wave fronts in spaces of low dimension, Izv. Math., 76:2 (2012), 375418.

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  mathnet  elib; Izv. Math., 82:3 (2018), 596–611  isi  scopus
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  mathnet  mathscinet  zmath  elib; Izv. Math., 79:3 (2015), 581–622  isi  scopus
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  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 48:4 (2014), 304–308  isi  scopus
2012
4. V. D. Sedykh, “On Euler Characteristics of Manifolds of Singularities of Wave Fronts”, Funktsional. Anal. i Prilozhen., 46:1 (2012),  92–96  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 46:1 (2012), 77–80  isi  elib  scopus
5. V. D. Sedykh, “On the topology of wave fronts in spaces of low dimension”, Izv. RAN. Ser. Mat., 76:2 (2012),  171–214  mathnet  mathscinet  zmath  elib; Izv. Math., 76:2 (2012), 375–418  isi  elib  scopus
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  mathnet  mathscinet  isi
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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 44:3 (2010), 234–236  isi  scopus
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  mathnet  mathscinet  zmath  elib; Izv. Math., 71:2 (2007), 391–437  isi  elib  scopus
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  mathnet  mathscinet  zmath  elib; Proc. Steklov Inst. Math., 258 (2007), 194–217  elib  scopus
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  mathnet  mathscinet  zmath  elib; Math. Notes, 78:3 (2005), 378–390  isi  elib  scopus
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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 38:4 (2004), 298–301  isi  scopus
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  mathnet  mathscinet  zmath  elib; Sb. Math., 195:8 (2004), 1165–1203  isi  scopus
2003
13. V. D. Sedykh, “Resolution of Corank $1$ Singularities of a Generic Front”, Funktsional. Anal. i Prilozhen., 37:2 (2003),  52–64  mathnet  mathscinet  zmath  elib; Funct. Anal. Appl., 37:2 (2003), 123–133  isi  scopus
14. V. D. Sedykh, “On the topology of singularities of Maxwell sets”, Mosc. Math. J., 3:3 (2003),  1097–1112  mathnet  mathscinet  zmath  isi
2001
15. V. D. Sedykh, “Some Invariants of Admissible Homotopies of Space Curves”, Funktsional. Anal. i Prilozhen., 35:4 (2001),  54–66  mathnet  mathscinet  zmath; Funct. Anal. Appl., 35:4 (2001), 284–293  isi  scopus
1996
16. V. D. Sedykh, “Theorem on Four Support Vertices of a Polygonal Line”, Funktsional. Anal. i Prilozhen., 30:3 (1996),  88–90  mathnet  mathscinet  zmath; Funct. Anal. Appl., 30:3 (1996), 216–218  isi
1995
17. V. D. Sedykh, “Invariants of Nonflat Manifolds”, Funktsional. Anal. i Prilozhen., 29:3 (1995),  41–50  mathnet  mathscinet  zmath; Funct. Anal. Appl., 29:3 (1995), 180–187  isi
18. V. D. Sedykh, “Strict convexity of a generic convex manifold”, Trudy Mat. Inst. Steklov., 209 (1995),  200–219  mathnet  mathscinet  zmath; 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  mathnet  mathscinet  zmath; 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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 27:3 (1993), 205–210  isi
1992
21. V. D. Sedykh, “A theorem on four vertices of a convex space curve”, Funktsional. Anal. i Prilozhen., 26:1 (1992),  35–41  mathnet  mathscinet  zmath; Funct. Anal. Appl., 26:1 (1992), 28–32  isi
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  mathnet  mathscinet  zmath; Funct. Anal. Appl., 23:3 (1989), 246–248  isi
1988
23. V. D. Sedykh, “Stabilization of singularities of convex hulls”, Mat. Sb. (N.S.), 135(177):4 (1988),  514–519  mathnet  mathscinet  zmath; 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  mathnet  mathscinet  zmath; 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  mathnet  mathscinet  zmath; Russian Math. Surveys, 36:5 (1981), 175–176  isi
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  mathnet  mathscinet  zmath; 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  mathnet  mathscinet  zmath  elib; Russian Math. Surveys, 67:2 (2012), 375–379  isi

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