Explore recent advancements in constructing derived equivalences and semi-orthogonal decompositions from birational geometry in this hour-long lecture by David Favero from the University of Alberta. Delve into the relationship between partial compactifications of group actions and Fourier-Mukai functors, examining their role in comparing geometric invariant theory quotients. Investigate the potential of deriving these Fourier-Mukai functors to provide a general comparison in singular settings, as viewed through the lens of derived algebraic geometry.