Abstract: In 2012, O'Grady introduced a filtration on the Chow group of a K3 surface. Using this filtration, he further conjectured a correspondence between sheaves and cycles on a K3 surface, which generalizes earlier work of Huybrechts. 
In this talk, we will first review recent joint work with Qizheng Yin and Xiaolei Zhao on the solution of generalized O'Grady conjecture concerning derived categories of K3 surfaces. Then we will discuss how to extend O'Grady's filtration to the K3 category constructed by Kuznetsov associated a cubic fourfold. We propose a conjecture relating Kuznetsov's K3 category to one cycles on the cubic fourfold. Evidence will be provided and discussed. The second part is joint work in progress with Qizheng Yin.