Representations of compact Lie groups play a crucial role all over mathematics and physics. Although representations are classified by their highest weight, one often faces the hard problem to relate a priori knowledge on the geometry of a representation to its highest weight. We plan to classify representations whose orbit spaces have boundary. Since these representations occur naturally in different contexts of mathematics, a classification can be the starting point for solving various other problems. In a second project of the proposal we are concerned with non-collapsing phenomena. A non-collapsing phenomenon is present if a certain class of Riemannian manifolds has a uniform lower bound on the volume. Once one has established such a result, the understanding of this class of Riemannian manifolds as a whole improves significantly. Many recent progress has been made in the field. We propose that the recent progress should allow to attack the Klingenberg Sakai conjecture, which asserts that for each manifold a non-collapsing phenomenon is present in the moduli space of positively pinched metrics.
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