Discrete versions of the Kadomtsev - Petviashvili hierarchy and the bispectral problem

Acknowledgements First and foremost, I wish to thank my advisor, Professor Luc Haine, for his continual guidance, encouragement and support throughout my graduate studies. I am particularly grateful for introducing me to the subject and the problems and providing me with the knowledge to work on them. I owe him much more than one can say in a few lines. I would like to thank Professor Emil Horozov for numerous illuminating discussions, especially concerning the bispectral problem. I thank Professor Pierre van Moerbeke for his encouragement and several discussions which have opened to me new horizons. I thank Professor Jean-Pierre Françoise who introduced me to the Calogero-Moser systems. I am grateful to Professor Tom Koornwinder for numerous critical remarks which improved some parts of the preliminary version of the thesis, and especially for urging me to add Section 5.6 with the explicit characterization of the bispectral ring. I also thank Professors Juan Carlos Alvarez, Jacques Boël and Philippe Ruelle for accepting to be part of the jury. The support of a FSR grant at the University of Louvain is gratefully acknowledged.