Alexandrov Geometry and Its Relatives

亚历山大几何学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，主要的不同之处在于，一定的等式被交换成了所谓的比较不等式。它使我们有可能考虑在现代几何中很重要的各种各样的度量空间，并用自欧几里得以来已知的相同的旧直觉来研究它们。这一理论在群论，特别是群作用和黎曼几何中有着广泛的应用。奇异空间自然而然地出现在许多问题中，而亚历山大几何的优势在于能够直接处理许多这样的空间。该研究还旨在为博士生提供建议，完成两本教科书的编写(一本是高级研究生，另一本是本科生)，继续为备受推崇的数学期刊《几何与泛函分析》提供编辑服务，并参与宾夕法尼亚州立大学的大规模暑期本科生研究项目。拟议的研究分为以下主要部分：由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。已经发现了一些与等距群的商和具有最优运输连续性质的空间的强联系。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会提供一个工具，在分段线性世界和平稳世界中都是有用的。研究具有非正曲率的Alexandrov空间与欧氏空间上离散等距群作用之间的联系的某种结构的研究。对黎曼流形折叠序列的曲率积分极限行为的研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.