Representations whose orbit spaces have boundary

紧李群的表示在数学和物理中起着至关重要的作用。虽然表示按其最高权重进行分类，但人们经常面临将表示的几何形状的先验知识与其最高权重相关联的难题。我们计划分类表示的轨道空间有边界。由于这些表示在不同的数学环境中自然发生，因此分类可以成为解决各种其他问题的起点。在该提案的第二个项目中，我们关注的是非坍缩现象。如果某类黎曼流形具有一致的体积下界，则存在非坍缩现象。一旦建立了这样一个结果，对这类黎曼流形的理解就有了显著的提高。最近在这一领域取得了许多进展。我们建议，最近的进展应该允许攻击Klingenberg Sakai猜想，它断言，对于每个流形的非崩溃现象是目前在模空间的正捏度量。

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.