In 1986, Nishakawa conjectured that a closed Riemannian manifold with positive (resp. nonnegative) curvature operator of the second is diffeomorphic to a spherical space form (resp. a Riemannian locally symmetric space). In this talk, I will present the recent resolution of this conjecture by Cao-Gursky-Tran and myself, and its further improvements by Nienhaus-Petersen-Wink and myself, as well as some analogue results for Kahler manifolds obtained by Nienhaus-Petersen-Wink-Wylie and myself. Along the way, I will mention some questions and conjectures.

Xiaolong Li is currently an Assistant Professor at Wichita State University. He obtained his Ph.D degree under the supervision of Prof. Lei Ni and Prof. Ben Chow at University of California, San Diego. After that, he spent three years as a Visiting Assistant Professor working under the supervision of Prof. Richard Schoen at University of California, Irvine and one year as a Britton Postdoctoral Fellow at McMaster University. His research interests include Differential Geometry, Geometric Analysis and Partial Differential Equations.