Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool people "don't care" so much about plain old derived categories anymore, preferring instead to look at souped up things like dg-categories/A-infinity categories/... (for example, triangulated vs. dg/A-infinity). At this level, FM partners are no longer equivalent.

From the point of view of dg-categories/infinity categories, what does it mean for two varieties to be Fourier-Mukai partners? Why keep studying these things?

Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool people "don't care" so much about plain old derived categories anymore, preferring instead to look at souped up things like dg-categories/A-infinity categories/... (for example, triangulated vs. dg/A-infinity). At this level, FM partners are no longer equivalent.

From the point of view of dg-categories/infinity categories, what does it mean for two varieties to be Fourier-Mukai partners? Why keep studying these things?