Xialong Li, Wichita State University
In 1986, Nishikawa conjectured that a closed Riemannian manifold with positive (or nonnegative) curvature operator of the second kind is diffeomorphic to a spherical space form (or a Riemannian locally symmetric space, respectively). In this talk, I will begin with an overview of sphere theorems and their rigidity phenomena in geometry to set the stage for Nishikawa’s conjecture. Then I will outline the recent proof of Nishikawa’s conjecture due to Cao-Gursky-Tran, myself, and Nienhaus-Petersen-Wink. Beyond this result, I will explore various generalizations, including analogous results for Kähler manifolds and my latest work on sphere theorems under negative lower bounds for the curvature operator of the second kind. Some problems and potential directions for future research will be mentioned along the way.