Speaker: Professor  Xiaolong Li (Auburn University) 

Title: Sphere Theorems in Geometry

Abstract: Sphere theorems are central results in Riemannian geometry that reveal how curvature shapes the topology of a manifold. A classical example is the quarter-pinched sphere theorem of Berger and Klingenberg, later strengthened by Brendle and Schoen using the Ricci flow, which shows that any manifold whose sectional curvature is quarter-pinched must be diffeomorphic a spherical space form. In this talk, I will begin with an overview of classical sphere theorems and the geometric ideas behind them. I will then describe several recent developments that extend these results by introducing new curvature conditions guaranteeing spherical topology. These include a resolution of Nishikawa’s conjecture on the curvature operator of the second kind and a partial solution to Yau’s 1990 pinching problem by linking sectional and isotropic curvature.