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The Wadge Hierarchy : Beyond Borel Sets Unicentre CH-1015 Lausanne http://serval.unil.ch Year : 2016 The Wadge Hierarchy : Beyond Borel Sets FOURNIER Kevin FOURNIER Kevin, 2016, The Wadge Hierarchy : Beyond Borel Sets Originally published at : Thesis, University of Lausanne Posted at the University of Lausanne Open Archive http://serval.unil.ch Document URN : urn:nbn:ch:serval-BIB_B205A3947C037 Droits d’auteur L'Université de Lausanne attire expressément l'attention des utilisateurs sur le fait que tous les documents publiés dans l'Archive SERVAL sont protégés par le droit d'auteur, conformément à la loi fédérale sur le droit d'auteur et les droits voisins (LDA). A ce titre, il est indispensable d'obtenir le consentement préalable de l'auteur et/ou de l’éditeur avant toute utilisation d'une oeuvre ou d'une partie d'une oeuvre ne relevant pas d'une utilisation à des fins personnelles au sens de la LDA (art. 19, al. 1 lettre a). A défaut, tout contrevenant s'expose aux sanctions prévues par cette loi. Nous déclinons toute responsabilité en la matière. Copyright The University of Lausanne expressly draws the attention of users to the fact that all documents published in the SERVAL Archive are protected by copyright in accordance with federal law on copyright and similar rights (LDA). Accordingly it is indispensable to obtain prior consent from the author and/or publisher before any use of a work or part of a work for purposes other than personal use within the meaning of LDA (art. 19, para. 1 letter a). Failure to do so will expose offenders to the sanctions laid down by this law. We accept no liability in this respect. FACULTÉ DES HAUTES ÉTUDES COMMERCIALES DÉPARTEMENT DE SYSTÈMES D’INFORMATION The Wadge Hierarchy : Beyond Borel Sets THÈSE DE DOCTORAT présentée à la Faculté des Hautes Etudes Commerciales de l'Université de Lausanne en cotutelle avec l’Université Paris Diderot pour l’obtention du grade de Docteur en Systèmes d’Information et Docteur en Mathématiques par Kevin FOURNIER Co-directeurs de thèse Prof. Jacques DUPARC Prof. Boban VELICKOVIC Jury Prof. Olivier CADOT, Président Prof. Riccardo CAMERLO, Expert interne Prof. Alessandro ANDRETTA, Rapporteur Dr. Hab. Julien MELLERAY, Rapporteur Dr. Hab. Dominique LECOMTE, Expert externe Prof. Alain LOUVEAU, Expert externe Prof. William W. WADGE, Expert externe LAUSANNE 2016 Unicentre CH-1015 Lausanne http://serval.unil.ch Year : 2016 The Wadge Hierarchy: Beyond Borel Sets FOURNIER Kevin FOURNIER Kevin, 2016, The Wadge Hierarchy: Beyond Borel Sets Originally published at : Thesis, University of Lausanne Posted at the University of Lausanne Open Archive http://serval.unil.ch Document URN : Droits d’auteur L'Université de Lausanne attire expressément l'attention des utilisateurs sur le fait que tous les documents publiés dans l'Archive SERVAL sont protégés par le droit d'auteur, conformément à la loi fédérale sur le droit d'auteur et les droits voisins (LDA). A ce titre, il est indispensable d'obtenir le consentement préalable de l'auteur et/ou de l’éditeur avant toute utilisation d'une oeuvre ou d'une partie d'une oeuvre ne relevant pas d'une utilisation à des fins personnelles au sens de la LDA (art. 19, al. 1 lettre a). A défaut, tout contrevenant s'expose aux sanctions prévues par cette loi. Nous déclinons toute responsabilité en la matière. Copyright The University of Lausanne expressly draws the attention of users to the fact that all documents published in the SERVAL Archive are protected by copyright in accordance with federal law on copyright and similar rights (LDA). Accordingly it is indispensable to obtain prior consent from the author and/or publisher before any use of a work or part of a work for purposes other than personal use within the meaning of LDA (art. 19, para. 1 letter a). Failure to do so will expose offenders to the sanctions laid down by this law. We accept no liability in this respect. Jury Alessandro Andretta membre externe Professeur Università degli Studi di Torino Dipartimento di Matematica « Giuseppe Peano » Riccardo Camerlo membre interne Professeur Politecnico di Torino Dipartimento di Scienze Matematiche « Giuseppe Luigi Lagrange » Jacques Duparc co-directeur Professeur Université de Lausanne, Faculté des Hautes Etudes Commerciales Dominique Lecomte membre externe Maître de conférences, Habilité à diriger des recherches Université de Picardie, IUT de l’Oise Alain Louveau membre externe Directeur de recherche émérite CNRS – Université Pierre et Marie Curie – Université Paris Diderot Institut de Mathématiques de Jussieu Julien Melleray membre externe Maître de conférences, Habilité à diriger des recherches Université Lyon 1, Institut Camille Jordan Boban Veličković co-directeur Professeur Université Paris Diderot, Institut de Mathématiques de Jussieu William W. Wadge Professeur University of Victoria, Department of Computer Science Université de Lausanne Faculté des Hautes Études Commerciales Doctorat en Systèmes d’Information Par la présente, je certifie avoir examiné la thèse de doctorat de Kevin FOURNIER Sa thèse remplit les exigences liées à un travail de doctorat. Toutes les révisions que les membres du jury et le-la soussigné-e ont demandées durant le colloque de thèse ont été prises en considération et reçoivent ici mon approbation. Signature : ____________________________ Date : _________________ Prof. Alessandro ANDRETTA Membre externe du jury Université de Lausanne Faculté des Hautes Études Commerciales Doctorat en Systèmes d’Information Par la présente, je certifie avoir examiné la thèse de doctorat de Kevin FOURNIER Sa thèse remplit les exigences liées à un travail de doctorat. Toutes les révisions que les membres du jury et le-la soussigné-e ont demandées durant le colloque de thèse ont été prises en considération et reçoivent ici mon approbation. Signature : Date : 18 janvier 2016 Prof. Jacques DUPARC The Wadge Hierarchy: Beyond Borel Sets Kevin Fournier February 2016 A` la Lune et aux Amis. i “Mathematics is the only avant-garde remaining in the whole province of art. It’s pure art, lad. Art and science. Art, sci- ence and language. Art as much as the art we once called art. It lost its wings after the Babylonians fizzled out. But emerged again with the Greeks. Went down in the Dark Ages. Moslems and Hindus kept it going. But now it’s back, bright as ever.” Don DeLillo, Ratner’s Star. iii Acknowledgment The romantic myth of the mathematician working alone, preferably by can- dlelight late at night before a duel, is still vivid nowadays. Maybe it is one of the numerous human flaws to seek tirelessly for baroque heroic figures – or simply a peculiarity carefully cultivated by the mathematic community itself. I have nonetheless to disappoint the reader right away: I try to keep my work in office hours, and my non-violent principles tend to keep me away from swords and pistols. Most importantly: this thesis is of course a personal and original work, but it should not overshadow the fact that I would not have been able to carry it out all by myself, alone. This thesis is the product of collaborations, interactions, and endless discussions. All the people I have met these last five years, mathematicians or not, have had an impact on my research, and if I take all the blames for any error, mistake or inaccuracy that could occur in this thesis, I should also share with them all the praises its positive outcomes could receive. First and foremost, I would like to thank my two supervisors for their precious guidance and trust. I acknowledge the support of the Swiss National Science Foundation, the University of Lausanne and the CRUS. They provided me with the best possible material conditions to complete this thesis. Iwouldnothavesurvivedtheacademicjunglealltheseyearswithoutthe presence of my dearest colleague Yann Pequignot, who is both an amazing mathematician and a great friend. We formed the perfect team, and I will miss it. Rapha¨elCarroy has always supported me, in countless ways. I thank him for his immense generosity, his energy, and his frankness. IamgreatlyindebtedtothedescriptivesettheoryteamofParis6,andin particular to Dominique Lecomte and Alain Louveau, who provided me with wise and kind advice all along the realization of this thesis. I am much obliged to Alessandro Facchini and Szczepan Hummel who intro- duced me to automata theory, and had the patience to answer all the possible v questions I could have on the subject. I appreciate our collaborations very much. Being a member of the board of the Swiss Graduate Society for Logic and Philosophy of Science has been a very rewarding adventure that I am very happy to have shared with Yann Pequignot, Marion H¨ammerli and Dandolo Flumini. It was a lot of work, but it was worth it, and I am very grateful to Thomas Strahm and Thomas Studer for their support, and for believing in us youngsters. I wish to thank also other students, colleagues and se- nior researchers I met during these years, and who helped me and inspired me, in a way or another: J´er´emie Cabessa, Fabrice Correia, Takako Nemoto, Daisuke Ikegami, Philipp Schlicht, Thilo Weinert, Zolt´an Vidny´anszky, David Chodunsk´y, Laura Fontanella, Sam van Gool, Kentaro Sato, Sylvain Ca- banacq, Rafael Zamora, Jonas Rogger, Jan Walker, Alessandro Andretta, Riccardo Camerlo, Olivier Finkel, Matteo Viale and Silvia Steila. They say teaching is the best way to learn. I do not know if it is a universal truth, but it worked for me and I would like to express my gratitude to all the students I had for their stimulating questions and our interesting discussions. More than the feeling of being helpful, I learned. I had also the chance to count on a solid team of teaching assistants: Chlo´eMorend, Jennifer Lopes, Nathalie Michel, Emilie´ Bruchez, Virginie Michaud and Aline de Montmollin. Ithankthemfortheirhelp,itwasagreatpleasuretoworkwiththem. Ithankmyfamilyfortheirconstantloveandcare:theybelievedinme,andI owe them everything.
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