NSERC grants at Laurentian University Subventions du CRSNG `a l ...

98Patrice SawyerSpherical functions on symmetric spaces and theirapplicationsA grant of $13,000 per year.Une subvention de 13 000 $ par année.Discovery Grant – Subvention à la découverteA symmetric space refers, in plain words, to the kindof “universe” I am studying. The general context of mywork is harmonic analysis of this universe. The basicidea of harmonic analysis, itself a generalization of Fourieranalysis, is to represent general functions in termsof more basic ones, i.e. spherical functions.Historically, such studies concerned spaces such asthe real line (the one-dimensional space); the plane (thetwo-dimensional universe); or space (the three-dimensionaluniverse). A natural extension was then the Euclideann-dimensional space. There are more complicated objects mathematicianswant to work with (the sphere being a natural example). Symmetricspaces are sufficiently complex objects to be of interest as a generalization,while sufficiently well-behaved to make their study relevant and interesting(not to mention accessible!).Not only can one consider questions such as the heat diffusion on a symmetricspace, but it also makes sense to study probability theory using suchobjects. As well, the behaviour of the “basic” functions, the spherical functions,becomes important. Natural assumptions as to how a well-known phenomenonin the Euclidean setting will behave in a different situation areeither confirmed or contradicted.These questions can be addressed by finding explicit formulae for the sphericalfunctions (when possible) or by using general principles.***Un espace symétrique est, en langage simple, le type « d’univers » quej’étudie. Le contexte général de ma recherche est l’analyse harmonique dans