This three-year, $364,931 National Science Foundation Division of Mathematical Sciences Project Grant supports research in comparison geometry and the mentoring of students at the University of California, Riverside. The principal investigator and collaborators will investigate three fundamental problems in Riemannian geometry: the diffeomorphism stability question, the pinching problem in positive curvature, and the construction of manifolds with almost non-negative curvature. Specifically, the team aims to establish piecewise linear stability in all dimensions and develop tools to show that some limit spaces are not diffeomorphically stable. They will also use these tools to construct exotic spheres with the prescribed properties. Additionally, the award will improve the Abresch-Meyer injectivity radius estimate using the Jacobi field comparison lemma. Broader impacts include training and mentoring students, disseminating mathematical findings, and advancing diversity, equity and inclusion initiatives. The work directly supports the goals of the NSF Mathematical and Physical Sciences program to strengthen the nation's scientific enterprise through increased understanding of major problems in these fields.