Advertisement

Abstract

I review topics of my talk in Alcalá, inspired by the paper [1]. An isomonodromic system with irregular singularity at \(z=\infty \) (and Fuchsian at \(z=0\)) is considered, such that \(z=\infty \) becomes resonant for some values of the deformation parameters. Namely, the eigenvalues of the leading matrix at \(z=\infty \) coalesce along a locus in the space of deformation parameters. I give a complete extension of the isomonodromy deformation theory in this case.

Keywords

Isomonodromy deformation Stokes matrices Coalescing Eigenvalues Painlevé equations Frobenius manifolds 

MSC

Primary 34M56 Secondary 34M40 34M35 

References

  1. 1.
    Cotti, G., Dubrovin, B.A., Guzzetti, D.: Isomonodromy deformations at an irregular singularity with coalescing eigenvalues (2017). arXiv:1706.04808
  2. 2.
    Dubrovin, B.: Geometry of 2D topological field theories. Lect. Notes Math. 1620, 120–348 (1996)CrossRefGoogle Scholar
  3. 3.
    Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds. In: Proceedings of ICM98, vol. 2, pp. 315–326 (1998)Google Scholar
  4. 4.
    Dubrovin, B.: Painlevé trascendents in two-dimensional topological field theory. In: Conte, R. (ed.) The Painlevé Property, One Century later. Springer, Berlin (1999)CrossRefGoogle Scholar
  5. 5.
    Guzzetti, D.: Inverse problem and monodromy data for three-dimensional Frobenius manifolds. Math. Phys. Anal. Geom. 4, 245–291 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cotti, G.: Coalescence phenomenon of quantum cohomology of grassmannians and the distribution of prime numbers (2016). arXiv:1608.06868
  7. 7.
    Cotti, G., Dubrovin, B.A., Guzzetti, D.: Local moduli of semisimple Frobenius coalescent structures (2017). arXiv:1712.08575
  8. 8.
    Mazzocco, M.: Painlevé sixth equation as isomonodromic deformations equation of an irregular system. CRM Proc. Lect. Notes, Am. Math. Soc. 32, 219–238 (2002)Google Scholar
  9. 9.
    Fokas, A., Its, A., Kapaev, A., Novokshenov, V.: Painlevé Transcendents: The Riemann-Hilbert Approach. AMS (2006)Google Scholar
  10. 10.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (I). Physics D2, 306–352 (1981)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Guzzetti, D.: Stokes matrices and monodromy of the quantum cohomology of projective spaces. Commun. Math. Phys. 207(2), 341–383 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cotti, G., Dubrovin, B.A., Guzzetti, D.: Helix structures in quantum cohomology of Fano manifolds. Ready to appearGoogle Scholar
  13. 13.
    Cotti, G., Guzzetti, D.: Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity. In: Proceedings for the Workshop at CRM. Pisa 13–17 February 2017. Random Matrices: Theory and Applications (2018). https://doi.org/10.1142/S2010326318400038
  14. 14.
    Miwa, T.: Painlevé Property of monodromy preserving deformation equations and the analyticity of \(\tau \) functions. Publ. RIMS, Kyoto Univ. 17, 703–721 (1981)CrossRefGoogle Scholar
  15. 15.
    Hsieh, P.-F., Sibuya, Y.: Note on regular perturbation of linear ordinary differential equations at irregular singular points. Funkcial. Ekvac 8, 99–108 (1966)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Balser, W., Jurkat, W.B., Lutz, D.A.: Birkhoff invariants and stokes’ multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. 71, 48–94 (1979)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Balser, W., Jurkat, W.B., Lutz, D.A.: A general theory of invariants for meromorphic differential equations; Part I. Form. Invariants. Funkcialaj Evacioj 22, 197–221 (1979)zbMATHGoogle Scholar
  18. 18.
    Cotti, G., Guzzetti, D.: Analytic geometry of semisimple coalescent structures. Random Matrices Theory Appl. 6(4), 1740004, 36, 53 (2017)Google Scholar
  19. 19.
    Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé transcendents and reflection groups. Invent. Math. 141, 55–147 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.SISSATriesteItaly

Personalised recommendations