Generic K3 categories and Hodge theory

In this talk I will focus on two examples of K3 categories: bounded derived categories of (twisted) coherent sheaves and K3 categories associated with smooth cubic fourfolds. The group of autoequivalences of the former has been intensively studied over the years (work by Mukai, Orlov, Bridgeland and others), whereas the investigation of the latter has only just began. As a motivation, I shall recall Mukai's classification of finite groups of automorphisms of K3 surfaces and its more recent derived version which involves the Leech lattice. In the second half I will discuss work in progress describing the group of autoequivalences of the very general cubic K3 category in terms of Hodge theory.