  This note aims to clarify the deep relationship between birational
modifications of a variety and semiorthogonal decompositions of its derived
category of coherent sheaves. The result is a conjecture on the existence and
properties of canonical semiorthogonal decompositions, which is a
noncommutative analog of the minimal model program. We identify a mechanism for
constructing semiorthogonal decompositions using Bridgeland stability
conditions, and we propose that through this mechanism the quantum differential
equation of the variety controls the conjectured semiorthogonal decompositions.
We establish several implications of the conjectures: one direction of
Dubrovin's conjecture on the existence of full exceptional collections; the
$D$-equivalence conjecture; the existence of new categorical birational
invariants for varieties of positive genus; and the existence of minimal
noncommutative resolutions of singular varieties. Finally, we verify the
conjectures for smooth projective curves by establishing a previously
conjectured description of the stability manifold of $\mathbb{P}^1$.
