Alexandrov Geometry and Its Relatives

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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.