Ambiguity Theory, Old and New
André, Yves
Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 259-274 / Harvested from Biblioteca Digitale Italiana di Matematica

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.

Publié le : 2009-02-01
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     title = {Ambiguity Theory, Old and New},
     journal = {Bollettino dell'Unione Matematica Italiana},
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André, Yves. Ambiguity Theory, Old and New. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 259-274. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_1_259_0/

[1] André, Yves, Une introduction aux motifs (motifs purs, motifs mixtes, périodes). Panoramas et Synthèses17. Société Mathématique de France, Paris, 2004. | MR 2115000 | Zbl 1060.14001

[2] André, Yves, Galois theory, motives, and transcendental number theory, submitted. | MR 2588609 | Zbl 1219.11109

[3] Belkale, Prakash - Brosnan, Patrick, Periods and Igusa local zeta functions, International Mathematics Research Notices. Vol. 2003, no. 49 (2003), 2655-2670. | MR 2012522 | Zbl 1067.11075

[4] Bloch, Spencer - Esnault, Hélène - Kreimer, Dirk, On motives associated to graph polynomials, Comm. Math. Phys., 267, no. 1 (2006), 181-225. | MR 2238909 | Zbl 1109.81059

[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). | MR 1848254

[6] Casale, Guy, Le groupoïde de Galois de P1 et son irréductibilité, to appear in Commentarii Mathematici Helvetici. | MR 2410777

[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 2135686

[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 2371808 | Zbl 1159.58004

[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. | MR 2290770 | Zbl 1200.81113

[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. | MR 1748177 | Zbl 1032.81026

[11] Deligne, Pierre - Goncharov, Alexander, Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38, no. 1 (2005), 1-56. | MR 2136480 | Zbl 1084.14024

[12] Ehrhardt, Caroline, Evariste Galois et la théorie des groupes. Fortune et réélaborations (1811-1910), thèse ENS (2007). | MR 2537664

[13] Furusho, Hidekazu, Pentagon and hexagon equations, arXiv:math/0702128. | MR 2630048 | Zbl 1257.17019

[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 1751835 | Zbl 0949.01001

[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 1483107

[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 2183396

[19] Kontsevich, Maxim - Zagier, Don, Periods. Mathematics unlimited-2001 and beyond (Springer, Berlin, 2001), 771-808. | MR 1852188

[20] Lang, Serge, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass-London-Don Mills, Ont. (1966). | MR 214547 | Zbl 0144.04101

[21] Lautman, Albert, Essai sur les notions de structure et d'existence en mathématiques. Reprint, Vrin2006. | MR 3775305

[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. | MR 2264540 | Zbl 1222.14046

[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 2172556

[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. | MR 1867495 | Zbl 1140.37352

[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). | MR 1960772 | Zbl 1036.12008

[27] Riemann, Bernhard, Beiträge zur Theorie der durch die Gauss'sche Reihe F(α,β,γ,x) darstellbaren Funktionen, Abh. Kon. Ges. Wiss. Gottingen, VII Math. Classe A-22 (1857).

[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 31Johnson Reprint Corp., New York-London1968. | MR 252187

[29] Schwarz, H. A., Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt, J. Reine Angew. Math., 75 (1872), 292-335. | MR 1579568 | Zbl 05.0146.03

[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 1476252

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

[32] Singularités irrégulières - Correspondance et documents, Deligne, Pierre - Malgrange, Bernard - Ramis, Jean-Pierre, Documents Mathématiques 5 (2007), SMF. | MR 2387754