In the series of lecture, I plan to explain some recent topics on quantitative estimate on proper actions with emphasis on their relation to representation theory. I begin with some geometric problems of group actions including properness criterion for reductive homogeneous spaces. In turn, I introduce a “quantification” of proper actions and bring geometric ideas to analytic representation theory such as Harish-Chandra’s temperedness criterion. Basic notion will be illustrated by examples.

I. Is rep theory useful for global analysis on a manifold? — Multiplicity: Approach from PDEs

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II. Tempered homogeneous spaces and tempered subgroups — Dynamical approach

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Ill. Classification theory of non-tempered G / H — Combinatorics of convex polyhedra

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IV. Tempered homogeneous spaces — Interaction with topology and geometry

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© Toshiyuki Kobayashi