x PREFACE infinite dimensional geometry initiated by the work of Quillen [Q1, Q2] and later Bismut and Freed [BF] on the geometry of families of operators, of Freed on loop groups [Fr], and Maeda, Rosenberg, and Tondeur on the geometry of gauge orbits [MRT1, MRT2], which offer interesting insights into the geometry and topology of infinite dimensional manifolds and bundles (see e.g. [PayR1], [LRST]). These lectures, which are essentially self-contained, are based on joint work (which we refer to with precise references) with various collaborators, among whom Dominique Manchon, Jouko Mickelsson, Steven Rosenberg, Simon Scott, and for- mer Ph.D. students Alexander Cardona, Catherine Ducourtioux, Jean-Pierre Mag- not, Carolina Neira, and Marie-Fran¸ coise Ouedraogo, I would like to thank most warmly. I am also grateful to many students and colleagues in France (Clermont- Ferrand), Burkina Faso (Ouagadougou), Germany (G¨ ottingen, Hannover, Regens- burg and Potsdam12), Colombia (Bogot` a and Villa de Leyva), and Lebanon (Bey- routh), who attended my various courses on regularisation techniques13 which trig- gered this manuscript, for they all contributed in improving this presentation. Let me address my thanks to Ina Kersten in ottingen, Elmar Schrohe in Hannover, and Bernd Ammann in Regensburg for inviting me to deliver a series of lectures on regularisation techniques. I am deeply thankful to Christian Brouder, Nicolas Ginoux, Florian Hanisch, and Carolina Neira for their valuable help in thoroughly reading a previous version of the manuscript. Last but not least, I am very grateful to Rita Paycha who helped me improve the English of this text by her careful reading and to Arthur Greenspoon for his valuable help and immense patience while editing a preliminary version of these notes. The lectures are organised into nine chapters, the first of which reviews ex- tended homogeneous distributions as a preparation for similar techniques intro- duced in the subsequent chapters. Sylvie Paycha 12 I would like to thank Christian Becker, David Hansen, Florian Hanisch, and Tobias J¨urgens in Potsdam for their very constructive comments. 13 For some lecture notes and review articles see [Pa1], [Pa2], [Pa3].
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