L. Katzarkov: “Gaps, Spectra, and Applications”
Here are my notes from Katzarkov’s talk in Bonn:
Summary:
- Clemens-Griffiths showed that the 3-dimensional cubic is not rational by showing that its intermediate Jacobian is not the Jacobian of a curve; we suggest analogs of this.
- Homological Mirror Symmetry and the perverse sheaf of vanishing cycles
- detecting rationality via monodromy properties on the LG mirror (Gross-Katzarkov, Pryzalkowski, Golyshev)
- spectra of triangulated categories; examples
- Theorem:
is rational of dimension 
implies that the spectrum of 
has no gaps of size greaeter than 
- the outlook for 4-dimensional cubics