Advertisement

Monge–Ampère Grassmannians, Characteristic Classes and All That

  • Valentin V. Lychagin
  • Volodya RoubtsovEmail author
Chapter
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

Abstract

We study the topology of integral Monge–Ampère grassmannians and related characteristic classes.

Notes

Acknowledgements

V. L. and V. R. express their deep thanks to the organisers of the Summer School ‘Wisla 2018’ and personally to Jerzy Szmit for a very stimulating summer school and for excellent working conditions. We are grateful to all participants for useful and fruitful discussions and inspiring atmosphere during the School.1 During preparation of the material, V. R. was partly supported by the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010, and by the Russian Foundation for Basic Research under the Grants RFBR 18-01-00461 and 19-51-53014 GFEN. The research of V. L. was partly supported by the Russian Foundation for Basic Research under RFBR Grant 18-29-10013.

References

  1. 1.
    V. Guillemin, S. Sternberg: Geometric Asymptotics, AMS, Series: Mathematical Surveys and Monographs Number 14 , revised ed. 1977, ISBN-13: 978-0821816332.Google Scholar
  2. 2.
    B. Banos, Opérateurs Monge-Ampère Symplectiques en dimensions 3 et 4, Thèse de Doctorat, Université d’Angers, (Novembre, 2002) 1–128.Google Scholar
  3. 3.
    V. V. Lychagin, Geometric theory of singularities of solutions of Nonlinear Differential Equations, J. Soviet Math., vol 51, 1990, n. 6, 2735–2357 (English transl. of russian ed. “Itogi Nauki VINITI”, 1988).Google Scholar
  4. 4.
    L. V. Zilbergleit, Characteristic classes of Monge-Ampère equations, “The interplay between differential geometry and differential equation”, AMS Transl. Ser.2, 167, 279–294, Providence, RI 1995.Google Scholar
  5. 5.
    A. Borel: La cohomologie mod 2 de certains espaces homogènes, Comment. Math. Helv., vol. 27(1953), p 165–197.MathSciNetCrossRefGoogle Scholar
  6. 6.
    D. B. Fuchs: About Maslov-Arnold characterstic classes, Soviet. Math. Dokl., vol. 178 (1968), n. 2, 301–306 (russian), English transl. Soviet. Math. Dokl., 9 (1968).Google Scholar
  7. 7.
    V. Lychagin, V. Rubtsov and I. Chekalov : A classification of Monge-Ampère equations, Ann. scient. Ec. Norm. Sup., 4 ème série, t.26, 1993, 281–308.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Kushner, V. Lychagin and V. Rubtsov, Contact geometry and Non-linear Differential Equations, Cambridge University Press, 2007.Google Scholar
  9. 9.
    B. Banos, Non-degenerate Monge-Ampère Structures in dimension 3, Letters in Mathematical Physics, 62 (2002) 1–15.MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. Rubtsov: Geometry of Monge-Ampère structures, Lectures delivered on the Summer School “Nonlinear PDEs, their Geometry and Applications -Wisla 2018”, this vol. to appear, ed. Birkhäuser, 2019.Google Scholar
  11. 11.
    R. Harvey and H. B. Lawson, Calibrated geometries, Acta. Math. 148, p. 47–157, 1982.MathSciNetCrossRefGoogle Scholar
  12. 12.
    L.Chekhov, M. Mazzocco, V. Rubtsov : Painlevé monodromy manifolds, decorated character varieties, and cluster algebras. Int. Math. Res. Not. IMRN., 2017, no. 24, 7639–7691.Google Scholar
  13. 13.
    M. Tibar, Polynomials and vanishing cycles., Cambridge Tracts in Mathematics, 170. Cambridge University Press, Cambridge, 2007. xii+253 pp.Google Scholar
  14. 14.
    S. Szilárd: Perversity equals weight for Painlevé spaces, arXiv:1802.03798, 2018.
  15. 15.
    D. D. Joyce, Singularities of special Lagrangian submanifolds, math.DG/0310460, 2003.Google Scholar
  16. 16.
    R. Bocklandt, T. Schedler, M. Wemyss, Superpotentials and higher order derivations., J. Pure Appl. Algebra , 214 (2010), no. 9, 1501–1522.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Whittaker, E. T.; Watson, G. N.: A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions , Reprint of the fourth (1927) edition, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. vi+608 pp. ISBN: 0-521-58807-3.Google Scholar
  18. 18.
    Ch. Conley, V. Ovsienko Lagrangian configurations and symplectic cross-ratios, arXiv:1812.04271, 2018.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.V. A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  2. 2.UiT Norges Arktiske UniversitetLangnes, TromsoNorway
  3. 3.LAREMA UMR 6093 du CNRS, Département de MathématiquesUniversité d’Angers AngersFrance
  4. 4.Theory Division of ITEPMoscowRussia

Personalised recommendations