This book presents the text of most of the lectures which were de-
livered at the Meeting Quantum Theories and Geometry which was held
at the Fondation Les Treilles from March 23 to March 27, 1987. The
general aim of this meeting was to bring together mathemati- cians
and physicists who have worked in this growing field of contact
between the two disciplines, namely this region where geometry and
physics interact creatively in both directions. It 1S the strong
belief of the organizers that these written con- tributions will be
a useful document for research people workin~ 1n geometry or
physics. Three lectures were devoted to the deformation approach to
quantum mechanics which involves a modification of both the
associative and the Lie structure of the algebra of functions on
classical phase space. A.Lichnerowicz shows how one can view
classical and quantum statistical mechanics in terms of a
deformation with a parameter inversely propor- tional to
temperature. S.Gutt reviews the physical background of star
products and indicates their applications in Lie groups representa-
tion theory and in harmonic analysis. D.Arnal gives a rigorous
theory Vll viii PREFACI of the star exponential in the case of the
Heisenberg group and shows how this can be extended to arbitrary
nilpotent groups.
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