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，但人们经常面临一个难题，即如何将关于表示几何的先验知识与其最高权重联系起来。我们计划对轨道空间有边界的表示进行分类。由于这些表示法自然出现在不同的数学环境中，因此分类可以作为解决各种其他问题的起点。在提案的第二个项目中，我们关注的是非坍塌现象。如果某类黎曼流形在体积上有一致的下界，则存在非折叠现象。一旦建立了这样的结果，人们对这类黎曼流形的理解作为一个整体就会显著提高。最近在这一领域取得了许多进展。我们认为，最近的进展应该允许攻击Klingenberg Sakai猜想，该猜想断言对于每个流形，在正收缩度量的模空间中存在非折叠现象。

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.