Ed Segal

Title: Semi-orthogonal decompositions and discriminants

Abstract:
The derived category of a toric variety can usually be decomposed into smaller pieces, 
by passing through different birational models and applying the `windows' theory relating VGIT 
and derived categories. There are many choices involved and the decompositions are not unique. 
We prove a Jordan-Holder result, that the multiplicities of the pieces are independent of choices.

If the toric variety is Calabi-Yau then there are no decompositions, instead the theory produces 
symmetries of the derived category. Physics predicts that these all these symmetries form an action 
of the fundamental group of the `FI parameter space'. I'll explain why our Jordan-Holder result is 
necessary for this prediction to work, and state a conjecture (based on earlier work of Aspinwall-Plesser-Wang) 
relating our multiplicities to the geometry of the FI parameter space. This is joint work with Alex Kite.