Skip to Main Content


AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution


Differential Galois Theory through Riemann-Hilbert Correspondence: An Elementary Introduction

About this Title

Jacques Sauloy, Institut de Mathématiques de Toulouse, Toulouse, France

Publication: Graduate Studies in Mathematics
Publication Year: 2016; Volume 177
ISBNs: 978-1-4704-3095-5 (print); 978-1-4704-3593-6 (online)
DOI: https://doi.org/10.1090/gsm/177
MathSciNet review: MR3585800
MSC: Primary 34Mxx; Secondary 12H05, 30B10, 30Fxx

PDF View full volume as PDF

Read more about this volume

View other years and volumes:

Table of Contents

PDF Download chapters as PDF

Front/Back Matter

Part 1. A quick introduction to complex analytic functions

Part 2. Complex linear differential equations and their monodromy

Part 3. The Riemann-Hilbert correspondence

Part 4. Differential Galois theory

References [Enhancements On Off] (What's this?)

References
  • Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
  • Yves André, Idées galoisiennes, Histoire de mathématiques, Ed. Éc. Polytech., Palaiseau, 2012, pp. 1–16 (French). MR 2905515
  • D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Mathematics, E22, Friedr. Vieweg & Sohn, Braunschweig, 1994. MR 1276272, DOI 10.1007/978-3-322-92909-9
  • Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 3rd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR 507190
  • George D. Birkhoff, The generalized riemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations, Proc. Amer. Acad. 49 (1913), 521–568.
  • Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
  • Emile Borel, Leçons sur les séries divergentes, 2nd éd. revue et entièrement remaniée avec le concours de G. Bouligand, Collection de monographies sur la théorie des fonctions, Gauthier-Villars, Paris, 1928, 260 pp.
  • Nicolas Bourbaki, Éléments de mathématique. Théories spectrales. Chapitres 1 et 2 (French), reprint of the 1967 original ed., Springer, Berlin, 2007.
  • Jose Cano and Jean-Pierre Ramis, Théorie de Galois différentielle, multisommabilité et phénomènes de Stokes, unpublished, see http://www.math.univ-toulouse.fr/~ramis/ Cano-Ramis-Galois, 2009.
  • Henri Cartan, Elementary theory of analytic functions of one or several complex variables, Éditions Scientifiques Hermann, Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0154968
  • Henri Cartan, Cours de calcul différentiel (French), nouveau tirage ed., Hermann, Paris, 1997.
  • Claude Chevalley, Theory of Lie groups. I, Princeton Mathematical Series, vol. 8, Princeton University Press, Princeton, NJ, 1999. Fifteenth printing; Princeton Landmarks in Mathematics. MR 1736269
  • Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
  • Teresa Crespo and Zbigniew Hajto, Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, vol. 122, American Mathematical Society, Providence, RI, 2011. MR 2778109, DOI 10.1090/gsm/122
  • Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898
  • Pierre Deligne and J. S. Milne, Tannakian categories, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982, pp. 101-228. MR 0654325
  • Régine Douady and Adrien Douady, Algèbre et théories galoisiennes (French), 2ème éd., revue et augmentée ed., Cassini, Paris, 2005.
  • Otto Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York-Berlin, 1981. Translated from the German by Bruce Gilligan. MR 648106
  • William Fulton, Algebraic topology, Graduate Texts in Mathematics, vol. 153, Springer-Verlag, New York, 1995. A first course. MR 1343250, DOI 10.1007/978-1-4612-4180-5
  • Roger Godement, Topologie algébrique et théorie des faisceaux, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Hermann, Paris, 1973 (French). Troisième édition revue et corrigée. MR 0345092
  • Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing, Providence, RI, 2009. Reprint of the 1965 original. MR 2568219, DOI 10.1090/chel/368
  • G. H. Hardy, Divergent series, 2nd (textually unaltered) ed., Chelsea, New York, 1991.
  • Einar Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997. Reprint of the 1976 original. MR 1452105
  • Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008. MR 2363178, DOI 10.1090/gsm/086
  • E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
  • Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig, 1991. A modern theory of special functions. MR 1118604, DOI 10.1007/978-3-322-90163-7
  • Ellis Kolchin, Selected works of Ellis Kolchin with commentary, American Mathematical Society, Providence, RI, 1999. Commentaries by Armand Borel, Michael F. Singer, Bruno Poizat, Alexandru Buium and Phyllis J. Cassidy; Edited and with a preface by Hyman Bass, Buium and Cassidy. MR 1677530
  • Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
  • Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
  • B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 219–226. Dedicated to the memory of Jacques-Louis Lions. MR 1924138, DOI 10.1142/S0252959902000213
  • C. Mitschi and D. Sauzin, Divergent series, summability and resurgence I: Monodromy and resurgence, Lecture Notes in Mathematics, vol. 2153, Springer-Verlag, Berlin, 2016.
  • Juan J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 1999. [2013] reprint of the 1999 edition [MR1713573]. MR 3155796, DOI 10.1007/978-3-0348-8718-2
  • Marius van der Put, Recent work on differential Galois theory, Astérisque 252 (1998), Exp. No. 849, 5, 341–367. Séminaire Bourbaki. Vol. 1997/98. MR 1685624
  • Marius van der Put and Michael F. Singer, Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. MR 1960772, DOI 10.1007/978-3-642-55750-7
  • Jean-Pierre Ramis, Séries divergentes et théories asymptotiques, Bull. Soc. Math. France 121 (1993), no. Panoramas et Synthèses, suppl., 74 (French). MR 1272100
  • Jean-Pierre Ramis and Jacques Sauloy, The $q$-analogue of the wild fundamental group and the inverse problem of the Galois theory of $q$-difference equations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 171–226 (English, with English and French summaries). MR 3335841, DOI 10.24033/asens.2241
  • Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
  • Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
  • Claude Sabbah, Introduction to algebraic theory of linear systems of differential equations, Éléments de la théorie des systèmes différentiels. $\scr D$-modules cohérents et holonomes (Nice, 1990) Travaux en Cours, vol. 45, Hermann, Paris, 1993, pp. 1–80. MR 1603680
  • Claude Sabbah, Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)], EDP Sciences, Les Ulis; CNRS Éditions, Paris, 2002 (French, with French summary). Mathématiques (Les Ulis). [Mathematics (Les Ulis)]. MR 1933784
  • Jacques Sauloy, Galois theory of Fuchsian $q$-difference equations, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, 925–968 (2004) (English, with English and French summaries). MR 2032530, DOI 10.1016/j.ansens.2002.10.001
  • Jacques Sauloy, Analytic study of $q$-difference equations, Galois theories of linear difference equations: An introduction, Mathematical Surveys and Monographs, vol. 211, American Mathematical Society, Providence, RI, 2016, pp. 103–171.
  • J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 0344216
  • I. R. Shafarevich, Basic algebraic geometry, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. MR 0447223
  • Michael F. Singer, Formal solutions of differential equations, J. Symbolic Comput. 10 (1990), no. 1, 59–94. MR 1081261, DOI 10.1016/S0747-7171(08)80037-5
  • Michael F. Singer, An outline of differential Galois theory, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, 1990, pp. 3–57. MR 1038057
  • Michael F. Singer, Introduction to the Galois theory of linear differential equations, Algebraic theory of differential equations, London Math. Soc. Lecture Note Ser., vol. 357, Cambridge Univ. Press, Cambridge, 2009, pp. 1–82. MR 2484905
  • Michael Spivak, Calculus on manifolds. A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR 0209411
  • Hiroshi Umemura, Sur l’équivalence des théories de Galois différentielles générales, C. R. Math. Acad. Sci. Paris 346 (2008), no. 21-22, 1155–1158 (French, with English and French summaries). MR 2464256, DOI 10.1016/j.crma.2008.09.025
  • V. S. Varadarajan, Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 1–42. MR 1339809, DOI 10.1090/S0273-0979-96-00624-6
  • E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge, 1927.