Geodesic
Ambiguity Theory, Old and New
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 259-274.

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This is an introductory survey of some recent developments of "Galois ideas" in Arithmetic, Complex Analysis, Transcendental Number Theory and Quantum Field Theory, and of some of their interrelations. 
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André, Yves. Ambiguity Theory, Old and New. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 1, pp. 259-274. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_1_a13/

[1] André Yves, Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses 17. Société Mathématique de France, Paris, 2004. | MR | Zbl

[2] André Yves, Galois theory, motives, and transcendental number theory, submitted. | DOI | MR | Zbl

[3] Belkale Prakash - Brosnan Patrick, Periods and Igusa local zeta functions, International Mathematics Research Notices. Vol. 2003, no. 49 (2003), 2655-2670. | DOI | MR | Zbl

[4] Bloch Spencer - Esnault Hélène - Kreimer Dirk, On motives associated to graph polynomials, Comm. Math. Phys., 267, no. 1 (2006), 181-225. | DOI | MR | Zbl

[5] Cartier Pierre, A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry [in Les relations entre les mathématiques et la physique théorique, 23-42, Inst. Hautes Âetudes Sci., Bures-sur-Yvette, 1998; Translated from the French by Roger Cooke. Bull. Amer. Math. Soc. (N.S.) 38, no. 4 (2001), 389-408 (electronic). | fulltext EuDML | DOI | MR

[6] Casale Guy, Le groupoïde de Galois de P1 et son irréductibilité, to appear in Commentarii Mathematici Helvetici. | DOI | MR

[7] Connes Alain, Renormalisation et ambiguïté galoisienne. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I. Astérisque No. 296 (2004), 113-143. | MR

[8] Connes Alain - Marcolli Matilde, Noncommutative geometry, quantum fields and motives. American Mathematical Society Colloquium Publications, 55. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008. | MR | Zbl

[9] Connes Alain - Marcolli Matilde, Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory. Frontiers in number theory, physics, and geometry. II (Springer, Berlin, 2007), 617-713. | DOI | MR | Zbl

[10] Connes Alain - Kreimer Dirk, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm. Math. Phys. 210, no. 1 (2000), 249-273. | DOI | MR | Zbl

[11] Deligne Pierre - Goncharov Alexander, Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38, no. 1 (2005), 1-56. | fulltext EuDML | DOI | MR | Zbl

[12] Ehrhardt Caroline, Evariste Galois et la théorie des groupes. Fortune et réélaborations (1811-1910), thèse ENS (2007). | MR

[13] Furusho Hidekazu, Pentagon and hexagon equations, arXiv:math/0702128. | DOI | MR | Zbl

[14] Galois Evariste, Œuvres mathématiques, Gauthiers-Villars, 1951.

[15] Gray Jeremy, Linear differential equations and group theory from Riemann to Poincaré. Second edition. Birkhäuser Boston, Inc., Boston, MA, 2000. | MR | Zbl

[16] Grothendieck Alexandre, Esquisse d'un programme. With an English translation on pp. 243-283. London Math. Soc. Lecture Note Ser., 242, Geometric Galois actions, 1, 5-48, Cambridge Univ. Press, Cambridge, 1997. | MR

[17] Jordan Camille, Mémoire sur les équations différentielles linéaires à intégrale algébrique (1878), Oeuvres II, 13-140.

[18] Katz Nicholas, web page: http://www.monodromy.com/ | MR

[19] Kontsevich Maxim - Zagier Don, Periods. Mathematics unlimited-2001 and beyond (Springer, Berlin, 2001), 771-808. | MR

[20] Lang Serge, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass-London-Don Mills, Ont. (1966). | MR | Zbl

[21] Lautman Albert, Essai sur les notions de structure et d'existence en mathématiques. Reprint, Vrin 2006. | DOI | MR

[22] Lochak Pierre - Schneps Leila, Open problems in Grothendieck-Teichmller theory. Problems on mapping class groups and related topics, 165-186, Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, 2006. | DOI | MR | Zbl

[23] Malgrange Bernard, On nonlinear differential Galois theory. Frontiers in mathematical analysis and numerical methods, 185-196, World Sci. Publ., River Edge, NJ, 2004. | MR

[24] Morales-Ruiz Juan - Ramis Jean Pierre, Galoisian obstructions to integrability of Hamiltonian systems. I, II. Methods Appl. Anal., 8, no. 1 (2001), 33-95, 97-111. | DOI | MR | Zbl

[25] Picard Emile, La vie et l'oeuvre de Joseph Boussinesq, Discours et Notices, Gauthiers-Villars (1936).

[26] Van Der Put Marius - Singer Michael, Galois theory of linear differential equations, Springer Grundlehren der math. Wiss. 328 (2003). | DOI | MR | Zbl

[27] Riemann Bernhard, Beiträge zur Theorie der durch die Gauss'sche Reihe $F(\alpha, \beta, \gamma, x)$ darstellbaren Funktionen, Abh. Kon. Ges. Wiss. Gottingen, VII Math. Classe A-22 (1857). | fulltext EuDML

[28] Schlesinger Ludwig, Handbuch der Theorie der linearen Differentialgleichungen. In zwei Banden. Band II: Theil 1, (Schluss-) Theil 2. (German) Reprint. Bibliotheca Mathematica Teubneriana, Band 31 Johnson Reprint Corp., New York-London 1968. | fulltext EuDML | MR

[29] H. A. Schwarz, Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math., 75 (1872), 292-335. | fulltext EuDML | DOI | MR | Zbl

[30] Umemura Hiroshi, Lie-Drach-Vessiot theory-infinite-dimensional differential Galois theory. CR-geometry and overdetermined systems (Osaka, 1994), 364-385, Adv. Stud. Pure Math., 25, Math. Soc. Japan, Tokyo, 1997. | MR

[31] Weil André, De la métaphysique aux mathématiques (1960), Oeuvres, vol. II, Springer, 408-412.

[32] Singularités irrégulières - Correspondance et documents, Pierre Deligne - Bernard Malgrange - Jean-Pierre Ramis, Documents Mathématiques 5 (2007), SMF. | MR