Speaker: Daniel Halpern-Leistner (Cornell University)

Abstract: The manifold of Bridgeland stability conditions parameterizes a homological structure on a triangulated category that is analogous to a Kaehler structure on a projective variety. Recently, I have proposed a "noncommutative minimal model program" in which the quantum differential equation of a projective variety determines paths toward infinity in the stability manifold of that variety, and that these paths can be used to define canonical (semiorthogonal)decompositions of its derived category.
In fact, these paths converge in a certain partial compactification of the stability manifold, the space of "augmented stability conditions." In order to define this partial compactification, I will introduce a structure on a triangulated category that we call a multi-scale decomposition, which generalizes a semiorthogonal decomposition, and a new moduli space of multi-scale lines that is closely related to the moduli spaces of multi-scale differentials which are of recent interest in dynamics. The main conjecture about the space of augmented stability conditions is that it is a manifold with corners (in a specific way that I will explain). One consequence: If this conjecture holds for any smooth and proper dg-category, then any stability condition on a smooth and proper dg-category admits proper moduli spaces of semistable objects.
The plan for the lectures is, loosely:
1) The noncommutative MMP
2) The space of n-pointed multi-scale lines (lecture on Wednesday will be given by Alekos Robotis)
3) The space of augmented stability conditions
4) Structure of the boundary: the manifold-with-corners conjecture and consequences