Student AG Seminar: A study of derived equivalences via Matsui spectra and the Fourier-Mukai locus

A celebrated result of Bondal-Orlov shows that a smooth projective scheme $X$ with (anti-)ample canonical bundle can be recovered from the triangulated structure of $\operatorname{Perf} X$. Recent work of Ito-Matsui gives a new proof of this theorem and introduces the Fourier-Mukai locus associated to $X$, which is roughly a gluing of its derived equivalences, with a view toward providing a categorical approach to questions in birational geometry. I will explain these constructions and, time permitting, discuss examples in the context of abelian varieties.