Seminario Nacional de Geometría Algebraica 
Lisa Marquand, Courant Institute of Mathematical Sciences, NYU.
Two cubic fourfolds are Fourier Mukai partners if there is an equivalence between their Kuznetsov components. It has been suggested by Huybrechts that a pair of Fourier Mukai cubic fourfold partners are birational, however examples of such phenomena are rare.
In an attempt to investigate this phenomena, we develop an algorithm for counting the number of Fourier Mukai partners of a given cubic fourfold, and run it on cubics with automorphisms. We prove that a cubic fourfold admitting a symplectic involution has 1120 non-trivial Fourier-Mukai partners, all birational but non-isomorphic. We'll explain the 1120 in terms of root systems and non-syzygetic pairs of cubic scrolls. 
https://www.matem.unam.mx/~lozano/eseminar.html