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 G¨ 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
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.
would like to thank Christian Becker, David Hansen, Florian Hanisch, and Tobias J¨urgens
in Potsdam for their very constructive comments.
some lecture notes and review articles see [Pa1], [Pa2], [Pa3].