Alexandrov Geometry and Its Relatives

亚历山德罗夫几何从公理的角度研究度量空间。 这些公理与欧几里得的公理非常相似，主要区别是将某个等式换成了所谓的比较不等式。 它使我们有可能考虑各种各样的度量空间，这在现代几何中是重要的，并使用自欧几里得以来已知的同样古老的直觉来研究它们。 这个理论在群论中有着丰富的应用，特别是在群作用和黎曼几何中。 奇异空间自然出现在许多问题中，亚历山德罗夫几何的优势在于能够直接处理许多这样的空间。 PI还旨在建议博士生完成两本教科书的编写（一个高级研究生和另一个本科生水平），继续编辑服务的高度重视的数学杂志，几何和功能分析，并参加在宾夕法尼亚州立大学的MASS夏季本科生研究计划.拟议的研究分为以下主要部分：PI与合作者引入的一种新型度量比较的研究。 满足这些比较的空间还远未被理解。通过等距群和具有最优输运连续性的空间，已经发现了一些强联系。 具有曲率界的光滑黎曼流形对多面体空间的逼近。这一主题可能提供一个工具，可以在分段线性世界以及在光滑的世界是有用的。 研究一种特定的构造，它提供了具有非正曲率的亚历山德罗夫空间和欧几里得空间上的离散等距群作用之间的联系。 研究黎曼流形坍缩序列的曲率积分的极限行为。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Alexandrov geometry studies metric spaces from an axiomatic point of view. These axioms are very similar to the axioms of Euclid, the main difference is that a certain equality is exchanged to the so called comparison inequality. It makes it possible to consider a wide variety of metric spaces, which are important in modern geometry, and to study them using the same old intuition known since Euclid. This theory has fruitful applications in group theory, in particular group actions and also in Riemannian geometry. Singular spaces appear naturally in many problems and the advantage of Alexandrov geometry lies in the ability to work with many such spaces directly. The PI also aims to advise PhD students, complete writing two textbooks (one advanced graduate and another undergraduate level), continue editorial service for the highly regarded mathematical journal, Geometric and Functional Analysis, and participate in the MASS summer undergraduate research program at Penn State.The proposed research is divided into the following main parts: The study of a new type of metric comparisons introduced by PI with collaborators. The spaces satisfying these comparisons are far from being understood. Some strong connections to the quotients by groups of isometries and to the spaces with continuity property of optimal transport have been found already. The approximation of polyhedral spaces by smooth Riemannian manifold with a curvature bound. This subject might provide a tool that could be useful in the piecewise linear world as well as in the smooth world. The study of a certain construction that provides a link between Alexandrov spaces with non-positive curvature and discrete isometric group actions on the Euclidean space. The study of the limit behavior of curvature integrals for collapsing sequences of Riemannian manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.